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Instant Solutions to What Is a Vector in Physics Simple Step by Step Detail
The majority of the predictions from these types of theories are numerical. A comprehension of acceleration, for instance, is vital to the study of force. After you get started thinking of quantum
mechanics concerning these state vectors, you’ll receive hooked.
Vital Pieces of What Is a Vector in Physics
Vectors need a couple of different forms of measurements to spell out a specific quantity. The conductors may also be smaller. The one difference in a couple of dimensions is these are now vector
It permits us to manipulate and assess the consequences of vector quantities. Generally, the equation will be non-linear, and can’t be solved exactly so a wide variety of approximations have to be
used. Try to remember that S.I. units have to be included for the majority http://bestbarkcontroldevices.com/the-chemistry-tutors-chronicles/ of quantities to define a magnitude.
What Is a Vector in Physics: the Ultimate Convenience!
A scalar field is simply produced when the 2 units are tuned. If you aren’t sure regarding the math above, consider graphing these vectors to demonstrate the result visually. Also, make certain the
units agree with each other.
With the component method, the concept is to simply offer the amount the vector is in each one of the coordinate directions. Within this tutorial, you’re likely to understand how to make an
illustrated self-portrait in a geometric style. Basically, I was searching for a means to use simpler graphs to develop approximate solutions that are near the specific solution of the intricate
If you would like to do a vector-times-matrix multiply with a row vector, use a named function call rather than an operator, because it is a not as common circumstance. An instance of the use of the
head-to-tail technique is illustrated below. Consider the next example.
This learning area is intended to offer an overall background for the comprehension of the Earth on a planetary scale. http://nucribz.xyz/2019/10/24/
what-you-do-not-know-about-theories-of-change-in-nursing-might-surprise-you/ Frequently the tools from these other areas aren’t quite acceptable for the requirements of physics, and will need to get
changed or more advanced versions have to get made. The straightforward approach makes it perfect for beginners, producing fresh stylised outcomes.
What Is a Vector in Physics Ideas
The usefulness of this kind of assignment may not look obvious at first. Abortion has become the most evil of crimes. All US and worldwide rights reserved.
The Fundamentals of What Is a Vector in Physics Revealed
The vector addition by rectangular components is composed of these steps. Also, scalar waves are in a position to convey information. It’s remarkable how much information you’ll be able to receive
from a diagram.
The Basics of What Is a Vector in Physics That You Will be Able to Learn From Starting Immediately
Position vector is utilized to specify the job of a particular body. When the angle is selected, any of the 3 functions can be employed to get the measure of the angle. Overall flux also is dependent
upon the orientation of the area and the surface.
The Honest to Goodness Truth on What Is a Vector in Physics
For instance, the velocity of an object could be represented by v. A vector, on the flip side, is an entity that is distinguished by means of a magnitude and a direction. Objects in motion tend to
remain in motion.
Finding What Is a Vector in Physics
Thus, an extremely large mass, like the sun, can exert over a distance of several millions of miles a force sufficient to maintain a planet in orbit. Constructing a new model is the quickest and
easiest way to create positive change without causing disruption. Both are sentient and are intended to make life from the materials on Earth.
Most Noticeable What Is a Vector in Physics
In other words, so long as its length isn’t changed, a vector isn’t altered if it’s displaced parallel to itself. The dot product provides the projection of the vector on the x-axis. They are created
by playing effects on them.
Dynamic actors find it impossible to use meshes whatsoever. There are a multitude of geometric objects that may be constructed from vectors. To put it differently, it’s the combo of two or more
single vectors.
Both of these torques cancel. The sole tricky algebra in kinematics is if you should solve equation 1 when v0 isn’t equal to zero. Consequently our resultant vector will become smaller in magnitude
than the very first example.
For scalars, you just have to compare the magnitude. Magnitude in geology refers specifically to the quantity of movement made by an earthquake. Locate the magnitude of these vectors.
The very first choice is to add more productive force. It is all up to the gameplay code to discover a great balance. These 3 trigonometric functions can be placed on the hiker problem as a way to
establish the direction of the hiker’s in general displacement.
So we must define a regional coordinate system physically. So for instance, a car may be going at 60 miles per hour. The parallel part of the force of gravity isn’t balanced by another force.
What Is a Vector in Physics – Dead or Alive?
That includes the many incorrect paths which were faltered on the way. The textbook was attempting to demonstrate the concept of an object in equilibrium. But should you write random sentences for
just two minutes, you might realize that useful sentences begin to roll off your fingers.
What You Need to Do About What Is a Vector in Physics Beginning in the Next 6 Minutes
There are lots of applications in physics where this is a practical point to do. In the study of physics, there are several different elements to measure and several kinds of measurement tools. The
mesh ought to be setup so it doesn’t collide with any other physics objects.
Scientific notation has a lot of helpful properties and is popular in calculators and by scientists, mathematicians and engineers. Implementation will be finished in FlashDevelop IDE. Energy medicine
isn’t a new area of science.
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This is not reasoning: LLMs caught cheating at math test
đ „ LLMs and math puzzles, reasoning vs pattern-matching, smaller kiwis are still kiwis
How do you check if an LLM is good at math?
We could just give it a bunch of math problems and see if the model can solve them. Put in a math puzzle, check the answer, repeat.
Example puzzle: Oliver picks 44 kiwis on Friday. Then he picks 58 kiwis on Saturday. On Sunday, he picks double the number of kiwis he did on Friday. How many kiwis does Oliver have?
Large language models are doing well on this type of testing. The leading models easily beat 90% success rate on GSM8K, a standard data set of grade school (age 11 to 14) math problems.
So that settles it, LLMs can reliably do grade school math, right?
Unless, of course, they are cheating.
Whatâ s â cheatingâ
Itâ s cheating if the student submits what is a correct answer, but without actually exercising the reasoning capabilities needed to find it.
Humans cheat on a math test by copying the answer of their neighbor. Or, if the teacher is using the same test for multiple groups, by memorising the answer before the test.
What if the student mechanically replicates the structure of an answer they saw before, without understanding the logic of it? If thatâ s enough to pass the exam, weâ d say that the teacher needs
to improve their tests.
It turns out that itâ s exactly whatâ s happening with LLMs and grade school math benchmarks.
The Apple paper
In a recent paper, Apple researchers wanted to answer this question: are LLMs capable of actual mathematical reasoning? Or are they just parroting the patterns they saw in the training data, like a
student mechanically and correctly responding to a math question they donâ t actually understand.
To test this, they did what a human teacher would do to challenge the understanding of their students. They added a bunch of irrelevant statements to the math puzzles.
For example, they transformed this simple puzzle:
Oliver picks 44 kiwis on Friday. Then he picks 58 kiwis on Saturday. On Sunday, he picks double the number of kiwis he did on Friday. How many kiwis does Oliver have?
Oliver picks 44 kiwis on Friday. Then he picks 58 kiwis on Saturday. On Sunday, he picks double the number of kiwis he did on Friday, but five of them were a bit smaller than average. How many
kiwis does Oliver have?
How did LLMs do on the updated data set?
LLMs canâ t tell that smaller kiwis are still kiwis.
In the chart above, we see Llama3-8b go from 76% success rate on the original puzzle set (GSM) to just 19% success rate on the puzzles modified with irrelevant statements (NoOp).
This seems to be consistent across all leading models:
By adding seemingly relevant but ultimately irrelevant information to problems, we demonstrate substantial performance drops (up to 65%) across all state-of-the-art models.
LLM responses can appear very convincing: the models are linguistically fluent.
But thereâ s no formal reasoning behind each answer. The models rehash relevant information from their training data set. When constructing the answer, they follow learned approximations of meaning
and context, not logic.
Fluency doesnâ t equal comprehension.
More on this
Gary Marcus says I told you so.
Symbol manipulation, in which some knowledge is represented truly abstractly in terms of variables and operations over those variables, much as we see in algebra and traditional computer
programming, must be part of the mix. Neurosymbolic AI â combining such machinery with neural networks â is likely a necessary condition for going forward.
Postcard from Paris
I went to see some of my theater teachers perform in an improv-based show Aléas. Amazing what true professionals can do on a stage 𠀩.
Thereâ s no script for life so all what we do is improv :).
Keep going,
â Przemek | {"url":"https://newsletter.pnote.eu/p/llm-not-reasoning","timestamp":"2024-11-05T08:42:20Z","content_type":"text/html","content_length":"163823","record_id":"<urn:uuid:46c0f79f-7358-492f-8b44-071ae6f57aed>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00359.warc.gz"} |
What's The Biggest Number - Large Numbers Display — Teachie Tings
Are you looking for a fun MATHS display to get your students excited about MATHS?
It doesn’t have to be boring!
Get them excited with this Maths Display!
You will get a colourful display to help students get excited about LARGE numbers!
Here’s what is included:
‘What’s the biggest number?’ display
✔Names of large numbers
✔Explanation of which numbers get names
✔ Googol example
✔ Explanation of googolplex
✔ Colourful fonts for a bright display
This resource would work for primary and middle school!
Here are some possible uses for these in your classroom:
✿ learning walls
✿ Investigations (why do you need large numbers? What can you count using large numbers? How are they useful? How many zeros will the next 10 numbers have?)
✿ Introducing factors
✿ Introducing mathematical conjectures | {"url":"https://teachietings.com/product/whats-the-biggest-number-large-numbers-display/","timestamp":"2024-11-04T10:33:29Z","content_type":"text/html","content_length":"121756","record_id":"<urn:uuid:09908a66-975f-4f1f-aeee-24f78568fbba>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00830.warc.gz"} |
Flight Safety
Problem B
Flight Safety
Safety is an important issue when planning flights. First and foremost, one should of course take every possible measure to make sure that the trip is uneventful and that no incidents occur. But even
then, one should always be prepared for the worst and try to make sure that if an incident does happen, people’s chances of surviving are as high as possible.
When making an emergency landing over water, the distance to the nearest land is a critical factor. In general, the further out on open waters, the worse are the odds of survival. Thus, one important
safety parameter of a flight is how far away from the nearest land any part of the flight will take you. Your job is to write a program which, given a flight route, will determine this distance.
To simplify matters, we model the world as a $2$-dimensional plane rather than a sphere. We model continents as polygons, and a flight route as a sequence of key points connected by straight line
segments. Flight routes always start and end strictly inside a continent, but intermediate key points may be located over water. Continents do not intersect themselves or touch each other.
The input consists of:
• One line containing two integers $C$ ($1 \le C \le 20$) and $N$ ($2 \le N \le 50$), where $C$ is the number of continents and $N$ is the number of key points in the flight route.
• $N$ lines each containing two integers $X, Y$ giving the coordinates of the key points, from first to last.
• The descriptions of the $C$ continents. Each continent description starts with a line containing an integer $M$ ($3 \le M \le 50$) giving the number of vertices of this continent. It is followed
by $M$ lines, each containing a pair of integers $X, Y$ giving the coordinates of the $M$ vertices, in either clockwise or counter-clockwise order.
Every coordinate in the input is between $-10\, 000$ and $10\, 000$.
Output one line with the furthest distance from land that the flight route will go. The answer should be given with an absolute or relative error of at most $10^{-6}$.
Sample Input 1 Sample Output 1
-9 -6
3 0.000000
-16 -12
17 -6
Sample Input 2 Sample Output 2
4 19 2.942685 | {"url":"https://kth.kattis.com/courses/DD2458/popup17/assignments/h4vzbg/problems/flightsafety","timestamp":"2024-11-05T10:47:03Z","content_type":"text/html","content_length":"28714","record_id":"<urn:uuid:6b043bdc-aef1-4e4f-a6de-a815449a38d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00151.warc.gz"} |
Factors Affecting Mass Flow Rate in context of mass flow rate
31 Aug 2024
Title: Factors Affecting Mass Flow Rate: A Review
Abstract: Mass flow rate is a critical parameter in various engineering applications, including chemical processing, fluid dynamics, and materials science. It is defined as the mass of fluid flowing
through a given area per unit time. In this article, we review the factors that affect mass flow rate, including pressure drop, density, viscosity, and pipe diameter.
Introduction: Mass flow rate (m) is a fundamental property in many engineering fields, and its accurate measurement and prediction are essential for designing and optimizing systems. The mass flow
rate can be calculated using the following formula:
m = ρ * Q
where ρ is the density of the fluid and Q is the volumetric flow rate.
Factors Affecting Mass Flow Rate:
1. Pressure Drop (ΔP)
Pressure drop across a pipe or system can significantly affect mass flow rate. The pressure drop can be calculated using the following formula:
ΔP = (L / D) * (ρ / 2) * v^2
where L is the length of the pipe, D is the diameter of the pipe, and v is the velocity of the fluid.
2. Density (ρ)
Density is a critical factor in determining mass flow rate. The density of a fluid can vary depending on temperature, pressure, and composition.
3. Viscosity (μ)
Viscosity is another important parameter that affects mass flow rate. Higher viscosity fluids require more energy to flow through a given area.
τ = μ * dv/dy
where τ is the shear stress, μ is the dynamic viscosity, and dv/dy is the velocity gradient.
4. Pipe Diameter (D)
The diameter of the pipe can also affect mass flow rate. Larger pipes can accommodate higher flow rates due to reduced pressure drop.
Conclusion: In conclusion, mass flow rate is a critical parameter that is affected by various factors, including pressure drop, density, viscosity, and pipe diameter. Understanding these factors is
essential for designing and optimizing systems in various engineering applications.
• [1] Incropera, F. P., & DeWitt, D. P. (1996). Fundamentals of Heat and Mass Transfer.
• [2] Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport Phenomena.
• [3] White, F. M. (2011). Fluid Mechanics.
Note: The references provided are a selection of well-known textbooks in the field of heat and mass transfer, fluid mechanics, and transport phenomena.
Related articles for ‘mass flow rate’ :
• Reading: Factors Affecting Mass Flow Rate in context of mass flow rate
Calculators for ‘mass flow rate’ | {"url":"https://blog.truegeometry.com/tutorials/education/9d83a1ae026c5151540cf7d632237b5c/JSON_TO_ARTCL_Factors_Affecting_Mass_Flow_Rate_in_context_of_mass_flow_rate.html","timestamp":"2024-11-06T20:42:07Z","content_type":"text/html","content_length":"17335","record_id":"<urn:uuid:1335a4f9-1438-4c55-90ec-5c674662cd1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00016.warc.gz"} |
MetricGate, LLC
One-Way ANOVA
One-Way ANOVA (Analysis of Variance) is used to determine whether there are any statistically significant differences between the means of three or more independent groups. This test helps assess the
influence of a single categorical independent variable on a continuous dependent variable by comparing the means of different groups.
The ANOVA Equation
The formula for the one-way ANOVA test is as follows:
F = \frac{MS_{between}}{MS_{within}}
• F is the F-statistic, the ratio of the variance between the groups to the variance within the groups.
• MS[between] is the mean sum of squares between groups, representing the variability due to the group differences. It is calculated as:
MS_{between} = \frac{SS_{between}}{df_{between}}
• Where SS[between] is the sum of squares between the groups, and df[between] is the degrees of freedom between the groups, given by:
df_{between} = k - 1
• Where k is the number of groups.
• MS[within] is the mean sum of squares within groups, representing the variability within the groups. It is calculated as:
MS_{within} = \frac{SS_{within}}{df_{within}}
• Where SS[within] is the sum of squares within the groups, and df[within] is the degrees of freedom within the groups, given by:
df_{within} = N - k
• Where N is the total number of observations, and k is the number of groups.
The goal of ANOVA is to test whether the group means are significantly different from each other. A higher F-statistic indicates that the groups' means are significantly different.
Sum of Squares (SS) Equations:
• SS[between] (sum of squares between groups) is calculated as:
SS_{between} = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X})^2
• Where n[i] is the sample size of group i, \bar{X}_i is the mean of group i, and \bar{X} is the overall mean of all observations.
• SS[within] (sum of squares within groups) is calculated as:
SS_{within} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2
• Where X[ij] is the individual observation in group i and j is the observation index within that group, and \bar{X}_i is the mean of group i.
Constructing an ANOVA Model
To create an ANOVA model, you need to specify the dependent variable (continuous) and the independent variable (categorical). One-Way ANOVA tests whether there are differences between the groups of
the independent variable on the dependent variable.
Example: In the mtcars dataset, we will test if there are differences in fuel efficiency (mpg) based on transmission type (am).
# One-Way ANOVA Example
# Testing the difference in mpg across transmission types (am) in mtcars dataset
data <- mtcars
model <- aov(mpg ~ factor(am), data = data)
Steps in Model Construction:
• The dependent variable is mpg (fuel efficiency), and the independent variable is am (transmission type).
• After fitting the ANOVA model, the F-statistic and p-value are computed to assess if there are significant differences between groups.
• If the p-value is less than the significance level, it suggests that the means of the groups are significantly different.
Model Interpretation
Once the ANOVA model is fit, the ANOVA table provides a summary of the F-statistic and the significance level (p-value). The table also shows the degrees of freedom, sum of squares, and mean square
values for the model.
# ANOVA Table
anova_table <- anova(model)
Key metrics from the ANOVA table include:
• F-statistic: The F-statistic is 16.86, which indicates a significant amount of variability between the group means compared to within the groups. A higher F-statistic suggests that the
differences between the groups are large relative to the variability within the groups.
• p-value: The p-value is 0.000285, which is less than the chosen significance level (e.g., 0.05). This means we reject the null hypothesis and conclude that there are significant differences
between the group means.
• Significance codes: The result is marked with '***', indicating a highly significant result at the 0.001 level.
Assumption Checks
1. Normality of Residuals
Normality of residuals can be checked using a Q-Q plot or the Shapiro-Wilk test. Ideally, the residuals should follow a normal distribution.
# Q-Q plot for checking normality of residuals
ggplot(data, aes(sample = rstandard(model))) +
geom_qq() +
geom_qq_line() +
labs(title = "Normal Q-Q Plot", x = "Theoretical Quantiles", y = "Standardized Residuals") +
# Shapiro-Wilk test for normality
shapiro_test <- shapiro.test(residuals(model))
Let's interpret this this Normal Q-Q Plot:
• Straight Line: The straight line represents the ideal case where the residuals are perfectly normally distributed. Points that fall along this line indicate that the residuals are close to
normally distributed.
• Data Points: The black dots represent the actual residuals plotted against the theoretical quantiles of a normal distribution. In this case, most of the points lie close to the line, indicating
that the residuals are approximately normally distributed.
• Deviations: There are some minor deviations at the tails (both ends of the plot), but they are relatively small. This suggests that the normality assumption is reasonably met, although slight
non-normality may be present in the extreme values.
The residuals appear to follow a normal distribution, supporting the assumption of normality for this one-way ANOVA analysis. The minor deviations at the tails are not significant enough to raise
concerns about violating the normality assumption.
Let's interpret this the Shapiro-Wilk test results:
• W-statistic: The W-value is 0.98208. This value indicates how close the distribution of the residuals is to a normal distribution. Values closer to 1 suggest normality.
• p-value: The p-value is 0.8573. This value is higher than the typical significance level (e.g., 0.05), suggesting that we do not reject the null hypothesis of the Shapiro-Wilk test.
Since the p-value is much greater than 0.05, we conclude that the residuals are normally distributed. This supports the normality assumption in the one-way ANOVA analysis.
2. Homogeneity of Variances
Levene's test is used to check for homogeneity of variances. ANOVA assumes that the variances across the groups are equal. A non-significant p-value (p > 0.05) suggests that this assumption holds.
# Levene's Test for Homogeneity of Variances
levene_test <- car::leveneTest(mpg ~ factor(am), data = data)
In our example, we have:
• F value: 4.1876
• p-value: 0.04957
With a p-value of 0.04957, which is less than the significance level of 0.05, we reject the null hypothesis of equal variances. This suggests that the variances are not equal across the groups,
violating the assumption of homogeneity of variances.
3. Independence of Errors
Durbin-Watson test checks for autocorrelation in the residuals. Ideally, there should be no significant autocorrelation.
# Durbin-Watson Test for independence of residuals
dw_test <- dwtest(model)
The results show:
• Durbin-Watson statistic (DW): 1.0647
• p-value: 0.001446
With a p-value of 0.001446, which is less than the significance level of 0.05, we reject the null hypothesis of no autocorrelation. This suggests that there is significant positive autocorrelation in
the residuals, which violates the assumption of independence in ANOVA.
Diagnostic Plots
We can visualize the model fit using various diagnostic plots to assess the assumptions of ANOVA.
Box Plot
This plot visualizes the distribution of the dependent variable for each group of the independent variable.
# Box Plot for visualizing group differences
ggplot(data, aes(x = factor(am), y = mpg)) +
geom_boxplot() +
labs(title = "Box Plot of MPG by Transmission Type", x = "Transmission Type", y = "Miles per Gallon (MPG)") +
This box plot visualizes the distribution of miles per gallon (MPG) by transmission type (0 = Automatic, 1 = Manual).
• Transmission Type 0 (Automatic): The median MPG is approximately 17.5, with a lower quartile around 15 and an upper quartile around 20. There are no extreme outliers.
• Transmission Type 1 (Manual): The median MPG is higher at approximately 22.5, with a lower quartile around 18 and an upper quartile around 30. The interquartile range is larger compared to
automatic transmissions, indicating more variability in fuel efficiency for manual transmissions.
The box plot indicates that manual transmissions tend to have higher MPG compared to automatic transmissions, as seen by the higher median and the overall shift in the distribution towards higher
Mean Plot
This plot shows the mean of the dependent variable for each group, with error bars representing the standard error.
# Mean Plot with error bars (standard error)
means <- aggregate(mpg ~ factor(am), data = data, mean)
means$am <- as.factor(means$'factor(am)') # Add the 'am' factor back to the data frame
means$se <- aggregate(mpg ~ factor(am), data = data, function(x) sd(x)/sqrt(length(x)))$mpg
ggplot(means, aes(x = am, y = mpg)) +
geom_point() +
geom_errorbar(aes(ymin = mpg - se, ymax = mpg + se), width = 0.2) +
labs(title = "Mean MPG by Transmission Type with SE", x = "Transmission Type", y = "Mean Miles per Gallon (MPG)") +
This mean plot displays the average miles per gallon (MPG) for each transmission type (0 = automatic, 1 = manual) along with the standard error (SE) bars.
• The mean MPG for vehicles with automatic transmission (Transmission Type = 0) is approximately 18 MPG, with a standard error indicating variability between approximately 16 and 19 MPG.
• The mean MPG for vehicles with manual transmission (Transmission Type = 1) is higher, around 24 MPG, with a standard error indicating variability between approximately 22 and 26 MPG.
The clear separation of the means between the two transmission types suggests that vehicles with manual transmission tend to have better fuel efficiency compared to vehicles with automatic
transmission. The error bars show that this difference is unlikely to be due to chance, as there is minimal overlap between the standard error ranges of the two groups.
Residuals vs Fitted Plot
This plot helps check for non-linearity and homoscedasticity. Ideally, residuals should be randomly scattered around zero.
# Residuals vs Fitted Plot for diagnosing model fit
ggplot(data, aes(.fitted, .resid)) +
geom_point() +
geom_smooth() +
labs(title = "Residuals vs Fitted Values", x = "Fitted Values", y = "Residuals") +
Normal Q-Q Plot
This plot checks the normality of residuals. If the residuals are normally distributed, the points should follow a straight line.
# Normal Q-Q Plot for checking normality of residuals
ggplot(data, aes(sample = rstandard(model))) +
geom_qq() +
geom_qq_line() +
labs(title = "Normal Q-Q Plot", x = "Theoretical Quantiles", y = "Standardized Residuals") +
Scale-Location Plot
This plot checks the homoscedasticity of residuals by assessing the spread of residuals over the range of fitted values.
# Scale-Location Plot for checking homoscedasticity
ggplot(data, aes(.fitted, sqrt(abs(.resid)))) +
geom_point() +
geom_smooth() +
labs(title = "Scale-Location Plot", x = "Fitted Values", y = "Square Root of Standardized Residuals") +
One-Way ANOVA is a powerful statistical tool used to compare the means of three or more independent groups. It helps determine whether the differences between group means are statistically
Key Takeaways:
• F-statistic: The F-statistic compares the variance between the groups to the variance within the groups. A higher F-statistic indicates significant differences between group means.
• Assumptions: It is important to check the assumptions of normality, homoscedasticity, and independence before interpreting the results of ANOVA.
• Post-hoc tests: If the ANOVA test is significant, post-hoc tests like Tukey's HSD can be used to identify which specific groups are significantly different from each other.
Explore our Online R Compiler or the Statistics Calculator to apply one-way ANOVA to your own datasets. | {"url":"https://metricgate.com/docs/one-way-anova/","timestamp":"2024-11-14T17:12:13Z","content_type":"text/html","content_length":"46083","record_id":"<urn:uuid:5be4db3f-ad7a-43c2-92a4-b472dc824fe9>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00030.warc.gz"} |
What is the limit of a p-series in calculus? | Hire Someone To Do Calculus Exam For Me
What is the limit of a p-series in calculus? In the context of data-intensive functional analysis, this question comes with an unusual set of interesting (but less familiar) answers. 1. Existentially
Riemannian geometry (1) [1] Reinebn($f’$) is the map $\kappa $ from ${2p-2}$ to ${F_p \otimes F_p}$ which gives a unique morphism from $\kappa $ to its image which is an isomorphism. 2. An
interesting notion of an indexing class is the collection of open subsets of $\kappa $, which is defined as a closed subset of ${\mathbb{C}}^m$ for $m \leq n$ [@Klebanov1], together with [@Ko]).
Moreover, [@Ko Theorem 8.3] states that the collection $\kappa $ of all (open connected and closed) subsets of $\kappa $ can be viewed as a collection of open subsets for all $\kappa \in {\kappa^*}$,
for which its intersection number with at most two of ${\mathbb{C}}^2$ equals the number of open subsets in ${\kappa^*}$. It is clear that by a theorem of the Dedekind $2$-stage [@Klebanov1]
developed in the last section we actually always have non-trivial closed subsets in ${2p-2}$ and hence the collection $\kappa $ is an open subset of $({\mathbb{C}}^m,d)$ for any integer $m$, which is
well-known. Moreover, for a fixed $\kappa \in ({\kappa^*})^*$ we can clearly conclude $\kappa \not\in \kappa^*$. This result is very similar toWhat is the limit of a p-series in calculus? (Picture:
Flickr) I guess someone has a way of calculating numbers via a logarithmic sum and getting the range of possible p-series (like counting a rectangle at a time) for thousands, maybe 1000, and then
graphically displaying them in a form such as Maths.math or MML (Picture:) I know this is only a small thing, but that’s the topic of the rest of this post. The limit point is not the limit of a
logarithmic sum, rather a limit of the limit of a series of terms. For example: \post[width=0.05cm,fill=orange,color=blue] for i=0,3,10,3 from this where i=0,1,2,3 from left to right; \loop[0,0,4] to
[0 in] loop[1 in] loop[2 in] loop[3 in] end; This is a simple and efficient way to approximate the limit of a logarithmic series. You can perform thousands of logarithmic sums of terms as these are
very efficient and can easily save you hundreds of dollars. Does this limit approach actually really make sense? Why try to find the end point of a logarithmic series if you’re the most likely person
to have multiple logarithmic sums? So, I think we’ll ask you for 101,202 questions: The limit to the logarithmic number is defined by the limit of a polynomial series: (Picture: Flickr) That’s one
way of saying it. For any function $f(x)$ you only get part in going from 0 to 1 so the new limit point c doesn’t exactly equal the original limit point by itself but instead equals the limit point.
This is illustrated by the Riemann sum. Does anyone know how general this function is? I haven’t looked over the function and came across numerous examples of function that give the exact number of c
points in some polynomial with even coefficients on multiple linearly independent terms: (Picture: Flickr) Unfortunately, I don’t have the code investigate this site but there seems to be an issue
(my proof is the next part of the post) but there is a better sequence to use to prove the limit. Let’s get a little googling and have a hint: a similar question asked in the comments.
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What is a limit of a polynomial $f(x)$ on a finite power of $x$ with coefficients: # n=0 a^2 + 1 + a f(x) + f(x)f(x) Are there any obvious programs that can be written for MatWhat is the limit of a
p-series in calculus? For a similar question, If check it out your first p-series we can show that the limit of a p-series is a limit of a series: We can show that (in its derivative) for a fixed p
is Note that for the limit of a series we are supposed to regard only those, not just possible limits of the series. (There are in addition many potential limits, however not all of them.)
Conventionally, Given that a series must contain no infinite subsequence of its elements, we must then regard its limit as a series, since this limit cannot contain any infinite sequence of elements.
Actually, this is also assumed on the web, by physicists, it being not always a pretty. Here my attempt for a “natural” proof at least based on some practical intuition, is below. Let us consider for
a moment the case when we have a limit of a series; let us denote it by @1 with the clear advantage of understanding something that can be demonstrated more completely. We can then write a series in
the limit by proceeding to limit the sequence of solutions and using the integral rule; the latter is called the sum of each series. We can then show that for any infinite subsequence of can someone
take my calculus exam sequence of elements we there exists a limit point, which corresponds to the limit point in this sum. We can also establish that there exists another infinite subsequence, which
we call the limit point of the series, which we will denote $\overline{W}$; if we find such two infinite subsequences we will also have a limit point. Here, we will show that for any finite
subsequence of a continued series $\sigma$ of finite length we can never be in a series, and it will not be the limit point of any series contained in $\sigma$. This shows that the limit point,
whatever it may be, is a limit point | {"url":"https://hirecalculusexam.com/what-is-the-limit-of-a-p-series-in-calculus","timestamp":"2024-11-10T22:50:55Z","content_type":"text/html","content_length":"101372","record_id":"<urn:uuid:7c54f532-9c03-46d3-b90b-74a60e76663f>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00136.warc.gz"} |
The exterior derivative of a 1-form
The Lie derivative \({L_{v}\varphi}\) is defined in terms of a vector field \({v}\), and its value as a “change in \({\varphi}\)” is computed by using \({v}\) to transport the arguments of \({\
varphi}\). In contrast, recall that the differential \({\mathrm{d}}\) takes a 0-form \({f\colon M\to\mathbb{R}}\) to a 1-form \({\mathrm{d}f\colon TM\to\mathbb{R}}\) with \({\mathrm{d}f(v)=v(f)}\).
Thus \({\mathrm{d}}\) is a derivation of degree +1 on 0-forms, whose value as a “change in \({f}\)” is computed using the vector field argument of the resulting 1-form.
We would like to generalize \({\mathrm{d}}\) to \({k}\)-forms by extending this idea of including the “direction argument” by increasing the degree of the form. It turns out that if we also require
the property \({\mathrm{d}\left(\mathrm{d}\left(\varphi\right)\right)=0}\) (or “\({\mathrm{d}^{2}=0}\)”), there is a unique graded derivation of degree +1 that extends \({\mathrm{d}}\) to general \
({k}\)-forms; this derivation is called the exterior derivative. We first explore the exterior derivative of a 1-form.
The exterior derivative of a 1-form is defined by
\(\displaystyle \mathrm{d}\varphi\left(v,w\right)\equiv v\left(\varphi\left(w\right)\right)-w\left(\varphi\left(v\right)\right)-\varphi\left(\left[v,w\right]\right),\)
where e.g.
\(\displaystyle v\left(\varphi\left(w\right)\right)=\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon}\left[\varphi\left(w\left|_{v_{p}\left(\varepsilon\right)}\right.\right)-\
measures the change of \({\varphi\left(w\right)}\) in the direction \({v}\), so that
\begin{aligned}\mathrm{d}\varphi\left(v,w\right) & =\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon^{2}}\left[\left(\varphi\left(\varepsilon w\left|_{v_{p}\left(\varepsilon\
right)}\right.\right)-\varphi\left(\varepsilon w\left|_{p}\right.\right)\right)\right.\\
& \phantom{{=\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon^{2}}\left[-\right.}}-\left(\varphi\left(\varepsilon v\left|_{w_{p}\left(\varepsilon\right)}\right.\right)-\varphi\
left(\varepsilon v\left|_{p}\right.\right)\right)\\
& \phantom{{=\underset{\varepsilon\rightarrow0}{\textrm{lim}}\frac{1}{\varepsilon^{2}}\left[-\right.}}\left.-\varphi\left(\varepsilon^{2}\left[v,w\right]\right)\right].
The term involving the Lie bracket “completes the parallelogram” formed by \({v}\) and \({w}\), so that \({\mathrm{d}\varphi\left(v,w\right)}\) can be viewed as the “sum of \({\varphi}\) on the
boundary of the surface defined by its arguments.”
The above depicts the exterior derivative of a 1-form \({\mathrm{d}\varphi\left(v,w\right)}\), which is the sum of \({\varphi}\) along the boundary of the completed parallelogram defined by \({v}
\) and \({w}\). So if in the diagram \({\varepsilon=1}\), we have \({\mathrm{d}\varphi\left(v,w\right)=\left(2-1\right)-\left(0-0\right)+3=4}\). This value is valid in the limit \({\varepsilon\
rightarrow0}\) if the sum varies like \({\varepsilon^2}\) as depicted in the figure.
The identity \({\mathrm{d}^{2}=0}\) can then be seen as stating the intuitive fact that the boundary of a boundary is zero. If \({\varphi=\mathrm{d}f}\), then \({\varphi\left(v\right)=\mathrm{d}f\
left(v\right)=v\left(f\right)}\), the change in \({f}\) along \({v}\). Thus e.g. \({\varepsilon\varphi\left(v\left|_{p}\right.\right)=f\left(v_{p}\left(\varepsilon\right)\right)-f\left(p\right)}\),
so that the value of \({\varphi}\) on \({v}\) is the difference between the values of \({f}\) on the two points which are the boundary of \({v}\). Each endpoint will be cancelled by a starting point
as we add up values of \({\varphi}\) along a sequence of vectors, resulting in the difference between the values of \({f}\) at the boundary of the total path defined by these vectors. \({\mathrm{d}\
varphi}\) is the value of \({\varphi}\) over the boundary path of the surface defined by its arguments, which has no boundary points and so vanishes.
The above depicts how \({\mathrm{d}^{2}=0}\) corresponds to the boundary of a boundary is zero: each term \({\varphi(v)=\mathrm{d}f(v)}\) is the difference between the values of \({f}\) on the
boundary points of \({v}\), which cancel as we traverse the boundary of the surface defined by the arguments of \({\mathrm{d}\varphi(v,w)}\). In the figure we assume a vanishing Lie bracket for
Note that \({\mathrm{d}\varphi\left(v,w\right)}\) measures the interaction between \({\varphi}\) and the vector fields \({v}\) and \({w}\), thus avoiding the need to “transport” vectors. In
particular, a non-zero exterior derivative can be pictured as resulting from either the vector fields or \({\varphi}\) “changing,” i.e. changing with regard to the implied coordinates of our
The above depicts a non-zero exterior derivative \({\mathrm{d}\varphi\left(v,w\right)}\), which results from changes in \({\varphi\left(v\right)}\) or \({\varphi\left(w\right)}\), not changes in
either \({\varphi}\) or the vector fields alone as compared to some transport.
If we calculate \({\mathrm{d}\varphi\left(e_{1},e_{2}\right)}\) explicitly in a holonomic frame in two dimensions, \({\mathrm{d}\left(\varphi_{1}\mathrm{d}x^{1}+\varphi_{2}\mathrm{d}x^{2}\right)=\
mathrm{d}\varphi_{1}\wedge \mathrm{d}x^{1}+\mathrm{d}\varphi_{2}\wedge \mathrm{d}x^{2}}\), so applying this to the basis vector fields \({e_{1}}\) and \({e_{2}}\) we have
\begin{aligned}\mathrm{d}\varphi\left(e_{1},e_{2}\right) & =\mathrm{d}\varphi_{1}\left(e_{1}\right)\cdot \mathrm{d}x^{1}\left(e_{2}\right)-\mathrm{d}\varphi_{1}\left(e_{2}\right)\cdot \mathrm{d}x^{1}
& \phantom{{}=}+\mathrm{d}\varphi_{2}\left(e_{1}\right)\cdot \mathrm{d}x^{2}\left(e_{2}\right)-\mathrm{d}\varphi_{2}\left(e_{2}\right)\cdot \mathrm{d}x^{2}\left(e_{1}\right)\\
& =e_{1}\left(\varphi_{2}\right)-e_{2}\left(\varphi_{1}\right)\\
& =\frac{\partial\varphi_{2}}{\partial x^{1}}-\frac{\partial\varphi_{1}}{\partial x^{2}}.
Note that a holonomic dual frame \({\beta^{\mu}=\mathrm{d}x^{\mu}}\) satisfies \({\mathrm{d}\beta^{\mu}=\mathrm{dd}x^{\mu}=0}\). In an anholonomic frame, we have
\begin{aligned}\mathrm{d}\varphi\left(e_{1},e_{2}\right) & =e_{1}\left(\varphi\left(e_{2}\right)\right)-e_{2}\left(\varphi\left(e_{1}\right)\right)-\varphi\left(\left[e_{1},e_{2}\right]\right)\\
& =\mathrm{d}\varphi_{2}\left(e_{1}\right)-\mathrm{d}\varphi_{1}\left(e_{2}\right)-\varphi_{1}\left[e_{1},e_{2}\right]^{1}-\varphi_{2}\left[e_{1},e_{2}\right]^{2}. | {"url":"https://www.mathphysicsbook.com/mathematics/manifolds/derivatives-on-manifolds/the-exterior-derivative-of-a-1-form/","timestamp":"2024-11-09T17:40:16Z","content_type":"text/html","content_length":"79477","record_id":"<urn:uuid:75341e53-0d72-4ef5-afb8-e6491bb110ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00269.warc.gz"} |
Understanding Polynomials and Their Degrees
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What is the degree of the polynomial $4x^3 - 2x^2 + 3x - 1$?
Which type of polynomial has a degree of 8?
Cubic polynomial
Quadratic polynomial
Octic polynomial (correct)
Linear polynomial
What is the degree of the zero polynomial?
Which type of polynomial has the form $ax^n$?
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What is the degree of the polynomial $6x^4 - 8x^2 + 5x + 1$?
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Which type of polynomial has a degree of 1?
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Which type of polynomial has the form $ax^2 + bx + c$?
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What is the degree of the constant polynomial $-3$?
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Which type of polynomial has a degree of 3?
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Study Notes
Polynomials: An Introduction
Polynomials are mathematical expressions that consist of variables and coefficients combined using the operations of addition, subtraction, and multiplication. They are expressed in the form of a_n x
^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n to a_0 are coefficients, x is a variable, and n is the highest power of x.
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial 3x^2 + 2x + 1, the degree is 2, since the highest power of x is 2.
The degree of a polynomial can be determined by looking at the highest power of the variable in the expression. If the polynomial has the form ax^n, where a is a constant and n is the highest power
of x, then the degree is n.
Zero Polynomial
A polynomial is considered zero if all of its coefficients are equal to 0. For example, 0x^3 + 0x^2 + 0x + 0 is a zero polynomial. The degree of a zero polynomial is 0.
Constant Polynomial
A constant or linear polynomial is a polynomial with degree 1. For example, 5x + 4 is a constant polynomial.
Quadratic Polynomial
A quadratic polynomial is a polynomial with degree 2. For example, 3x^2 + 2x + 1 is a quadratic polynomial.
Cubic Polynomial
A cubic polynomial is a polynomial with degree 3. For example, x^3 + 2x^2 + x + 1 is a cubic polynomial.
Degree of a Product
The degree of a product of polynomials is the sum of the degrees of the factors. For example, if p(x) = x^2 + 2x + 1 and q(x) = 2x^2 + 3x + 4, then the degree of their product p(x)q(x) is 2 + 2 = 4.
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Learn about polynomials, their degrees, zero polynomial, constant polynomial, quadratic polynomial, cubic polynomial, and the degree of a product of polynomials. Understand the concept of degree and
how to determine the degree of a polynomial based on the highest power of the variable. | {"url":"https://quizgecko.com/learn/understanding-polynomials-and-their-degrees-pf0kxv","timestamp":"2024-11-15T03:53:08Z","content_type":"text/html","content_length":"322099","record_id":"<urn:uuid:ff4db320-2882-414e-9fb1-8f8238504b22>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00100.warc.gz"} |
How many meters are 7 leagues
Parent Category: Length
Category: league in m
1 1 1 1 1 Rating 5.00 (2 Votes)
You can easily convert 7 leagues into meters using each unit definition:
3 mile = 4828.032 m
1 m
With this information, you can calculate the quantity of meters 7 leagues is equal to.
¿How many m are there in 7 league?
In 7 league there are 33796.224 m.
Which is the same to say that 7 leagues is 33796.224 meters.
Seven leagues equals to thirty-three thousand seven hundred ninety-six meters. *Approximation
¿What is the inverse calculation between 1 meter and 7 leagues?
Performing the inverse calculation of the relationship between units, we obtain that 1 meter is 2.9589104e-05 times 7 leagues.
A meter is two times seven leagues. *Approximation | {"url":"https://howmanyis.com/length/140-league-in-m/117095-7-leagues-in-meters","timestamp":"2024-11-07T09:45:15Z","content_type":"text/html","content_length":"21590","record_id":"<urn:uuid:8956796b-6e4b-4b79-ba4e-e378ef504061>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00429.warc.gz"} |
Online Logarithm calculator
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Logarithm Calculator
Calculate logarithm of a number to any base:
* Use e for scientific notation. E.g: 5e3, 4e-8, 1.45e12
b^ y = x
Then the base b logarithm of a number x:
log[b ]x = y
Anti-logarithm calculator
In order to calculate log^-1(y) on the calculator, enter the base b (10 is the default value, enter e for e constant), enter the logarithm value y and press the = or calculate button:
The Log (Logarithm) Calculator is used to calculate the logarithm log[b]x for a base b and a number x. Logarithm (LOG) calculator is an online math calculator that calculates the log value for the
positive real number with respect to the given or natural base values (positive, not equal to 1). Using this calculator, we will understand methods of how to find the logarithm of any number with
respect to the given base.
It is necessary to follow the next steps:
1.Enter the number and the base of logarithm. These values must be positive real numbers or parameter. The base of logarithm can not be 1.
2. Press the "CALCULATE" button to make the computation.
3. Logarithm calculator will give the logarithm of the positive real number number with the positive base not equal to 1.
Logarithm definition
The logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. In other words, the logarithm of y to base b is the solution y of the following equation:
b^y = x
And for any x and b, there is:
x = log[b]b^x
The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant e (approximately equal to 2.718281828) as its
base. The binary logarithm uses base b = 2 and is prominent in computer science.
What is Log?
The logarithm, or log, is the inverse of the mathematical operation of exponentiation. This means that the log of a number is the number that a fixed base has to be raised to in order to yield the
number. Conventionally, log implies that base 10 is being used, though the base can technically be anything. When the base is e, ln is usually written, rather than log[e]. log[2], the binary
logarithm, is another base that is typically used with logarithms. If for example:
x = b^y; then y = log[b]x; where b is the base
Each of the mentioned bases are typically used in different applications. Base 10 is commonly used in science and engineering, base e in math and physics, and base 2 in computer science.
Logarithm rules
Logarithm product rule
log[b](x × y) = log[b](x) + log[b](y)
Logarithm quotient rule
log[b](x / y) = log[b](x) - log[b](y)
Logarithm power rule
log[b](x ^y) = y × log[b](x)
Logarithm base switch rule
log[b](c) = 1 / log[c](b)
Logarithm change of base rule
log[b](x) = log[c](x) / log[c](b) | {"url":"https://www.pustudy.com/calculators/calc/math/Log_Calculator.html","timestamp":"2024-11-08T19:04:38Z","content_type":"text/html","content_length":"23538","record_id":"<urn:uuid:beb6b61c-4153-4549-8203-9653968997f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00302.warc.gz"} |
Optical pyrometer
Units are expressed in metric and prices in US dollars, unless otherwise stated.
An optical pyrometer is a device which allows contactless temperature measuring by using the incandescense color. It is based upon the fact that all black bodies do have the same incandescense color
at a given temperature. It is very straightforward and allows any temperature from which a hot object emits light ( > 500^oC). It is made from a small magnifying optical device (like a monocular or
very small telescope) in which a small incandescent bulb is placed which image is sharp when the user views through the eyepiece (the lens(es) on the eye end of the optical device). The background is
the hot object to be gauged. The electrical current flowing through the filaments in the bulb is an indication of their temperature. This current is controlled by a potentiometer which is put between
the power source (a battery) and the bulb. An ammeter is used to display the temperature. Its range is from 500^oC (== 900^oF lower limit when an object incandesces) to 1600^oC (3000^oF), which is
suitable for most applications.
The stuff needed is available in a local hardware or electronics shop unless stated otherwise.
• Small optical device. A 5x20 (5 = magnification, 20 is objective (front lens) diameter in millimeters) is OK. It should be able to focus till less than 30 cm (1 foot). Available at a second hand
optics dump store for a few dollars. Wnat can be useful as well is a 50 mm standard SLR lens, available for a low price second hand in a camera shop. In the latter case, the back cover should be
included, which allows attaching the lens to the pyrometer.
• Battery or batteries to make 5 or 6 Volts. Four nickel-cadmium, nickel metal hydride or alkaline penlites in a four batteries holder are suitable.
• A resistor of 47 Ohms 1/4 Watt when you use 1600^oC range to protect the ammeter from overflow. When alkaline batteries are used (4x1.5 = 6V) instead of nickel (-cadmium) ones (4x1.25 = 5V), take
a 56 Ohms one.
• A small incandesent bulb, not larger than 7 mm in diameter, otherwise it blocks the background image of the hot object. It should not draw more current than 100 milliamperes. Take a 6 Volts one.
I have a 6 Volt / 100 mA one. An ordinary bicycle light bulb is usually too large in size.
• Potentiometer of about 470 Ohms linear and a round knob easy to handle. Take preferably a 10 stroke potmeter, rather than the ordinary 270 degree stroke ones. That is for easier controlling the
current throught the filament.
• Low-power power switch.
• Low-current wires.
• A 32mm (1 1/4") PVC drain pipe of ca 5 cm long.
• Plastic box of about 100x60x50mm (4" x 2 1/2" x 2"), which is available in electronics shops and mounts to attach it to the optical device.
• Analog voltmeter with 2 Volts range. A low range ammeter (e.g. 10 mA) serial with a 200 Ohms precision resistor (1 % tolerance) can also be used. When full 2400 C (4400 F) range is used, take a
500 Ohms precision resistor and omit the first 47 or 56 Ohms resistor. In this case, only one-third of the scale is used for the first 1600 degrees, so this is recommended when you use very high
temperatures often.
• When using high temperatures (over 1300^oC), a welding filter glass.
This all should not cost more than $40 all together. The camera lens can be obtained at a photo repair shop.
Tools needed:
• A simple multimeter which allows measuring DC milliamps, ohms and volts. Available from about $30 in the local electronics shop, or you might borrow it from a friend.
• Soldering iron and solder.
• Screwdrivers, sharp knife and piercing device (or a small drill).
• Torch or furnace which is able to reach temps you need to be measure and some pure metals (nickel, copper, gold, aluminum) to calibrate. A thermocouple with a proper display can also be used.
• Take off the eyepiece of the binoculars and put the film box between the eyepiece and the prism box. This is necessary to make the light path longer to allow viewing close objects (as your
furnace is) sharp. Try out what the best length is. Maybe you must cut off a part of the film box. Attaching the eyepiece to the film box can be done best by a 30mm hose clamp.
• Pierce a hole in it to put the light bulb on the front side of the eyepiece, in such a way that you see it sharp from the eye end of the eyepiece.
• Connect the bulb in the following way:
□ Plus of the battery to power switch
□ Power switch to 47 Ohms resistor then to the left contact of the potmeter.
□ Center contact of the potmeter to bulb and the ammeter + 200 Ohms precision resistor and bulb in parallel and that back to battery ground. The whole stuff should be built in the plastic box
with the potmeter knob, power switch and the ammeter on the front. Check for the polarity of the ammeter. Attach the box to the binocular, such that the wires to the bulb in the eyepiece are
properly fitted and the controls (power switch and potmeter knob) are easily to reach while viewing through the eyepiece.
Here is the electric scheme:
6 V 60 mA bulb
R=47 Ohms ------(X)------------------
\ | |
+ |--XXX---- X\X | |
- \--------- O ------- XXX --------|
___ |
- | |
5..6 V 470 Ohms 10 mA 200 Ohms 1%
battery potmeter ammeter resistor
The opened pyrometer The finished pyrometer
When it is all assembled, then the heat goes on....
The readout takes place the following way: Point the pyrometer to the object to be measured, preferably a nonlustrous object (blank metal surface) and not at flames, because the latter are gases or
scattered soot particles. The glowing filaments in the bulb in the pyrometer should be in the image of the object. Turn the potmeter so far till you hardly see the top (hottest part) of the bulb
filaments. Then the temperature of the filaments is correct and you should read the current on the ammeter Determine which calibration temperatures you are going to use. If you have a thermocouple
and a furnace with a stable temperature and the display of the thermocouple is correct, just adjust the furnace to a few round numbers (e.g. 600 C, 1000 C or 1000 F , 1800 F, etc.) and read the
current for these values when you point the pyrometer at the thermcoupleand the filaments of the bulb glow the same color.
When you don't have a thermocouple, melting (or freezing) metals is another option. The best are:
pure platimum 1773 C / 3222 F
pure nickel 1453 C / 2647 F
pure copper (use a reducing flame !) 1083 C / 1981 F
pure aluminum (calibrate with low light) 660 C / 1220 F
View of a Kanthal-wire fired furnace through the pyrometer
Point the pyrometer on a thin oxide layer, as the lustrous metal reflects too much and does not show exactly the proper color. Repeat each calibration once or twice to be sure about the correct
Write down all currents matching the temperatures and make an interpolation to make a new scale. Do this on a piece of paper with the same size at the ammeter scale.
When putting the scale on the ammeter, remove carefully the plastic cover of the ammeter and (don't touch the needle hand!) paste it carefully over the ampere scale.
Now your pyrometer is ready for use. When batteries get low, it will not affect the accuracy, because the display is just the voltage over the bulb and low batteries will just limit the range. When
using high temperatures (over 1300 C) placing a welding filter between the light bulb and the eyepiece or in front of the front lens (objective) is a strong recommendation to protect your eyes. But
not between the object and the light bulb, because then you filter the hot object, but not the light bulb.
For more info you can mail me.
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Transitivity in Action
Transitivity in mathematics is a property of relationships in which objects of a similar nature may stand to each other. If whenever object A is related to B and object B is related to C, then the
relation at hand is transitive provided object A is also related to C. Being a sibling is a transitive relationship, being a parent is not.
If a line l[1] is perpendicular to another line l[2] (the mathematical notation for which is l[1] ⊥ l[2]) and, for a third line l[3], we also have l[2] ⊥ l[3], then it is not true that l[1] ⊥ l[3].
Thus the relationship of mutual orthogonality is not transitive. On the other hand, if a number A divides a number B (A|B) and B|C, then A|C. Thus the relation "is divisible by" is transitive.
Transitivity of one relation is so natural that Euclid stated it as the first of his Common Notions
Things which are equal to the same thing are also equal to one another.
In mathematical notations: if A = B and B = C, then necessarily A = C. The equality is a transitive relation! On the first glance this statement lacks content. If not yet reaching Descartes'
sophistication, a fellow mutters "I am I" and then repeats this in wonder, the transitivity of equality will only imply exactly same banality: "I am I". No need to sound it the third time. So why
Euclid and other mathematicians after him felt it necessary to explicitly state a seemingly vacuous property? The reason is of course that the same object may appear in different guises whose
identity may not be either obvious or a priori known.
In passing, the veracity of the statement "I am I" is abstracted in mathematics into another property of equality: reflexivity, A = A. Not all relations are reflexive. For example, "is divisible by"
is reflexive while "being perpendicular to" is not.
From Ceva's Theorem we know that some lines in a triangle meet at a point. Here I am going to establish those facts by a more conventional means - using the transitivity of equality. In the standard
You may check out those lines and some others with an applet.
Each family of lines consists of loci of points that satisfy certain conditions. The word locus (plural loci) in geometry substitutes for the word set used in other branches of mathematics.
The bisector AL[a] (or rather the whole line to which AL[a] belongs) is the locus of points equidistant from the two lines bb and cc defined by the sides AC and AB of ΔABC. In other words,
AL[a] = {P: dist(P, bb) = dist(P, cc)}. The distance function dist(P, bb) here is the Hausdorff distance between the single point set {P} and the line (which is of course also a set of points) bb.
The underlying distance between two points is Euclidean. This is the shortest distance from P to the line bb. Two lines AL[a] and BL[b] intersect at a point I. (I skip the proof that two angle
bisectors can't be parallel.) For this point I, dist(I, bb) = dist(I, cc) but also dist(I, cc) = dist(I, aa). By the transitivity of equality, dist(I, bb) = dist(I, aa) which simply means that point
I also lies on the third bisector CL[c].
(More accurately, the locus of points equidistant from two intersecting lines consists of two perpendicular lines. In a triangle, we thus have 4 associated points: 1 incenter and 3 excenters. To get
resricted to a single point - the incenter - we should have considered the angles of the triangle as defined by semi-infinite lines - rays.)
The perpendicular bisector, say pb[a], through the point M[a] is the locus of points equidistant from points B and C: {P: dist(P, B) = dist(P, C)}. Similarly for the other two bisectors:
pb[b] = {P: dist(P, C) = dist(P, A)} and pb[c] = {P: dist(P, B) = dist(P, A)}. If two bisectors pb[a] and pb[b] intersect at a point O, then dist(O, B) = dist(O, C) but also dist(O, C) = dist(O, A)
which, by transitivity, imply dist(O, B) = dist(O, A). Therefore, O also lies on the third bisector pb[c].
Point O is equidistant from all three vertices. Thus it serves as the circumcenter of the triangle, the center of the circle (the circumcircle) that passes through all three vertices.
The case of altitudes reduces to the case of perpendicular bisectors with the following trick. Through each vertex draw a line parallel to the opposite side. These three lines form a triangle,
A'B'C'. Altitudes of ΔABC serve as the perpendicular bisectors of ΔA'B'C'. Q.E.D.
To prove that the three medians intersect at a point, I refer to the notion of barycentric coordinates. For a given ΔABC, every point in the plane is associated with the unique triple
(w[A], w[B], w[C]) with w[A] + w[B] + w[C] = 1. If all three numbers are positive, the point lies inside ΔABC. Now, AM[a] = {(w[A], w[B], w[C]): w[B] = w[C]} and BM[b] and CM[c] are defined
similarly. It then follows that if G is the point of intersection of AM[a] and BM[b], it also lies on CM[c].
As an additional example, here is another proof of the fact that three chords formed by three intersecting circles meet at a point.
In the plane, a circle S[R](C) with radius R and center C is defined as the locus of points located at distance R from C: S = S[R](C) = {P: dist(P, C) = R}. For an arbitrary point P, let d denote the
distance from P to C: d = dist(P, C). Then the expression d² - R² is known as the power of P with respect to the circle S[R](C). If P has coordinates (x,y) and the center C coordinates (a,b) then the
circle has the equation (x - a)² + (y - b)² = R², or S[R](C) = {(x, y): (x - a)² + (y - b)² = R²}. The power of P with respect to the circle is then defined by the expression
Pow(P, S) = (x - a)² + (y - b)² - R²
Points with positive power lie outside the circle, those with negative power lie inside. The circle itself is the locus of points with zero power. Points that have the same power with respect to a
circle lie on a concentric circle.
Let there be two nonconcentric circles (i.e., circles with different centers) S[1] = S[R[1]](C[1]) and S[2] = S[R[2]](C[2]). Then the locus of points that have the same power with respect to both
circles is a straight line perpendicular to the center line C[1]C[2]. Using (1) we express Pow(P, S[1]) = Pow(P, S[2]) as
(x - a[1])² + (y - b[1])² - R[1]² = (x - a[2])² + (y - b[2])² - R[2]²
Obviously the square terms cancel out leaving a linear equation - an equation of a straight line. This straight line is called the radical axis of the two circles. The easiest way to see that the
radical axis is perpendicular to the center line is to choose the coordinates so as to make the centers lie on the x-axis. Then b[1] = b[2] = 0 and, after cancelation, (2) does not contain y-terms
which exactly means that the line is perpendicular to the x-axis.
Let now two circles S[1] and S[2] intersect at two points. Power of a point on a circle being 0, the two points of intersection of S[1] and S[2] obviously lie on the radical axis of the two circles.
Therefore, the radical axis of two intersecting circles is the straight line that passes through their points of intersection. The problem of the three common chords then simply asserts that the
pairwise radical axes of three intersecting circles meet at a point. This is the point that has the same power with respect to all three circles. From the foregoing discussion on transitivity, this
is obvious, however, that, for any three circles not necessarily intersecting, the three radical axes meet at a point. The only restriction is that no two circles are concentric. The point is known
as the radical center of the three circle. (There is a construction problem that is easily solved with the notion of radical center.)
(The idea of radical axis may also be introduced via the stereographic projection.)
Also, in any triangle, antiparallels to the sides adjacent to a vertex that cross on the symmedian through this vertex are equal. By transitivity then, the three antiparallels through the symmedian
(Lemoine) point are equal.
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Topics in Numerical Analysis
The nicer and practical parts of linear algebra, differential equations, and optimization.
Numerical Linear Algebra
Numerical linear algebra is where rubber meets the road. This post collects some topics of interest to me. I aim to provide exposition better than standard textbooks whenever I can. Often times,
ideas keep lying around deep inside 1000-page bibles.
• Hutchinson Trace Estimator
• Cholesky decomposition
□ Pivoted Cholesky decomposition
• Preconditioning
• Conjugate gradients
□ Modified Batched Conjugate Gradient Descent
• Lanczos tridiagonalization
• Kronecker-factored matrices
• Toeplitz matrices
An uncategorized list of references of high pedagogic value.
1. Includes description, properties and an application of Kronecker-factored matrices. ↩ | {"url":"https://sanyamkapoor.com/kb/topics-in-numerical-analysis/","timestamp":"2024-11-10T07:45:10Z","content_type":"text/html","content_length":"20574","record_id":"<urn:uuid:56dae738-7f6e-4e13-a0d3-225c3eae846d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00738.warc.gz"} |
sNoisesNoise - sNoise Research Laboratory
What is sNoise?
Simply, the mathematics of sNoise answers the question of "What is inside the black box?" and is an effective tool in the study of systems that generate self-affine time series.
Formally, the fractional calculus of sNoise, first introduced in the dissertation of Dr. Smigelski, encompasses a new class of modified Laplace transfer functions incorporating a combination of a
scaling exponent β and an altitude exponent æ which mathematically define fractional control orders, fractional scaling, fractional phase shifting, fractional filtering, fractional integration, or
fractional differentiation known to occur in a variety of systems such as those that generate stochastic time series.
the definition of 1/f-noise
is incomplete
In the scientific literature, noise observed in a time series is generally described as 1/f -noise (or 1 over f noise) where f is the simple frequency term. The term 1/f -noise to describe the
scaling behavior of noise observed in stochastic time series is limited in function. As a simple frequency term, 1/f -noise can only describe scaling of magnitude at each frequency but cannot
describe any phase shifting which naturally occurs in all waveforms generated as output of a simple system or stochastic self-affine time series generated as output of a complex system.
Enter 1/s-noise (i.e., sNoise)
The term sNoise is shorthand for writing equations using 1/s-noise (or 1 over s noise) in which the Laplace term, denoted as s, represents a complex frequency (j ω) that fully describes how a system,
across all frequencies, will scale (in magnitude) and/or shift (in phase) any input that passes into the system. Thus, the term 1/s-noise, or sNoise, is a more descriptive reference and more accurate
mathematically allowing the degree of scaling and shifting behavior across all frequencies from input to output by a system to be expressed by one or more fractional differential equations, or
transfer functions, also referred to as the Frequency Response Model (FRM) of the system. sNoise also represents the patented algorithms encoding the fractional calculus-based mathematics of
Fractional Scaling Digital Signal Processing (FSDSP), Fractional Scaling Digital Filters (FSDF), and Fractional Order Control Systems (FOCS).
Magnitude with Scaling Exponent
Exhibiting both Fractional Integration and Fractional Differentiation
Phase Map of Scaling Exponent of Fractional Integration or Fractional Differentiation
In one sense, sNoise science allows all analog system equations to become digital without loss of resolution to achieve exact solutions rather than approximations in digital systems. sNoise Signal
Processing (or Fractional Scaling Digital Signal Processing) encompasses all of the mathematics and algorithms involved in filtering or processing a digital signal using the mathematics of sNoise
The meaning of
the sNRL Black Box
A fundamental concept in Control Theory, a Black Box represents an unknown system or process that changes or "filters" an input signal into that system to yield an output signal. However, exactly
how these changes occur or what is inside the system or process is obscured, "in the dark", or closed-off from direct observation and thus the system is a black box where one can only observe inputs
into the system or outputs from the system but not the system itself, the Black Box, at least not directly.
The sNRL Black Box Logo represents the fractional calculus mathematics of sNoise which reveal what is inside the Black Box, illuminating the underlying system dynamics. The sNRL logo consists of two
time series, a Gaussian white noise and a Brownian motion. The Gaussian white noise (in blue in the logo) as input into an integration system contains equal power, on average, at all frequencies and
exhibits a scaling exponent of β = 0. Upon integration, the Gaussian white noise input becomes a Brownian motion (in green in the logo) as the output of that system which exhibits a single scaling
exponent of β = 2 over all frequencies in the power spectrum in which there is more power in the lower frequencies. There is a 90 degree phase shift upon integration, hence, the two time series
appear at a 90 degree right angle to each other on the cube face. With sNRL's patented algorithms, the Black Box becomes transparent revealing not only the dymnamic processes within the system, but
also how the system or inputs to the system may be altered to achieve a specific output signal.
"To find out what happens to a system when you interfere with it you have to interfere with it (not just passively observe it)."
~George E.P. Box~
Use and Abuse of Regression | {"url":"https://snoiselab.com/snoise/","timestamp":"2024-11-05T23:24:27Z","content_type":"text/html","content_length":"258412","record_id":"<urn:uuid:61a6ef11-d653-4070-b133-f2d2668ccc15>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00522.warc.gz"} |
how to find the key for the hill cipher
Our key is the following matrix: K = [2 3;1 4] K = 2 3 1 4 The numbers for our message are LINEARALGEBRA = 11 8 13 4 0 17 0 11 6 4 1 17 0. Now that we have walked through an example to give you an
idea of how a Hill cipher works, we will briefly touch on how you would implement a Hill cipher for a generic n-by-n key matrix with vectors of length n. Separate the plaintext from left to right
into some number k of groups of n letters each. In this article, we are going to learn three Cryptography Techniques: Vigenére Cipher, Playfair Cipher, and Hill Cipher. The Hill cipher was developed
by Lester Hill and introduced in an article published in 1929. Each block of plaintext letters is then converted into a vector of numbers and is dotted with the matrix. We have to choose a, b, c, and
d in such a way so that A is invertible mod 26 Hudson River Undergraduate Mathematics Conference 11 22 mod26 yxab yxcd ª º ª ºªº « » « » «» ¬ ¼ ¬ ¼¬¼ The ciphertext alphabet for the Affine Cipher
with key a = 5, b = 8. A block cipher is a cipher in which groups of letters are enciphered together in equal length blocks. We have shown that the Hill cipher succumbs to a known plaintext attack if
sufficient plaintext-ciphertext pairs are provided. Example. 3. Each letter is represented by a number modulo 26. There are two parts in the Hill cipher â Encryption and Decryption. To decrypt hill
ciphertext, compute the matrix inverse modulo 26 (where 26 is the alphabet length), requiring the matrix to â ¦ If the sender and the receiver each uses a different key the system is referred to as
asymmetric, two key, or public-key encryption. Break Hill Cipher with a Known Plaintext Attack. Each letter is represented by a number modulo 26. I have done the following: a) found the inverse of K:
K inverse = (-3 5) (2 -3) b) Found "KFCL": KFCL = (10 5) (2 11) c) The next step (mod 26) confuses me. In cryptography (field related to encryption-decryption) hill cipher is a polygraphic cipher
based on linear algebra. When information is sent using Cipher, and the receiver receives the encrypted code, the receiver has to guess which Cipher was used to encrypt the code, and then only it can
be decrypted. Show the calculations for the corresponding decryption of the ciphertext to re- cover the original plaintext. Today, we call this Hillâ s Cipher Machine. Submitted by Himanshu Bhatt,
on September 22, 2018 . Climbing the Hill Cipher Algorithm. According to the definition in wikipedia, in classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear
algebra.Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once. Hill cipher decryption needs
the matrix and the alphabet used. What you really want to be able to do is ï¬ gure out what the key and its inverse areâ as we shall say, to crack the cipher (in technical terms, to â cryptanlyzeâ
it). The largest hill cipher matrix I have ever seen is a $36$ x $36$ matrix, which dcode offers an option for. Show your calculations and the result. We must first turn our keyword into a key
matrix ( a $ \ 2 \times 2$ matrix for working with digraphs, a $ 3 \times 3$ matrix for working with trigraphs, etc) We also turn the plain text into digraphs or trigraphs and â ¦ For decrypting, we
apply the inverse of . How do I decipher (using mod 26) and the Cipher Key to find the plain text? This is very large even for today computation power. Any help is â ¦ In order to cipher a text, take
the first letter of the message and the first letter of the key, add their value (letters have a value depending on their rank in the alphabet, starting with 0). In a 2x2 case and due to the fact
that hill ciphers are linear, we only need to find two bigram (2 letter sequences) to determine the key. Patented mechanism works on 6×6 sized keys. can be a huge help in reconstructing the key â ¦
Asimpleletter-for-lettersubstitution,suchasintheexample ... when we ï¬ rst introduced this Hill cipher. However, for the Hill Cipher I am completely lost. If the encryption key matrix is not properly
chosen, the generation of decryption key matrix i.e. b. decrpytion ... Now we need to find the multiplicative inverse of the determinant (the number that relates directly to the numbers in the
matrix. Repeats of letters in the word are removed, then the cipher alphabet is generated with the keyword matching to A, B, C etc. Recall that the Playfair cipher enciphers digraphs â two-letter
blocks. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once. Hillâ s message protector
Complexity. Encryption. The following discussion assumes an elementary knowledge of matrices. You can try to get the key if you know a pair of plaintext and ciphertext, I.e. To do this first find the
determinant of our key matrix. Caesarâ s nephew Augustus learned the code from his uncle, but encrypted his messages with a shift of only one, but without wrapping around the alphabet. January 2,
2019. And that is why we use modular arithmeticforHillciphers. Julius Caesar used this cipher in his private war-time correspondence, always with a shift of three. Obtaining the key is relatively
straightforward if both plaintext and ciphertext are known, however we want to find the key without knowing the plaintext. key. The Hill cipher The Playfair cipher is a polygraphic cipher; it
enciphers more than one letter at a time. A ciphertext is a formatted text which is not understood by anyone. (3) Consider the cipher text â ETGYX OIMOI NGQMV EJGPM NNNNZ CLOIGâ , which was formed
using a Hill cipher with a 2 × 2 key matrix, and suppose it is somehow known that the first two words in the plaintext are â THE ALAMOâ . The results are then converted back to letters and the
ciphertext message is produced. First line of input contains keyword which you wish to enter. In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra.
Encipher In order to encrypt a message using the Hill cipher, the sender and receiver must first agree upon a key matrix A of size n x n. This increases key space to 26 36. The Key The key to the
encryption scheme is the coefficient matrix A. assuming we have access to the key of a cipher text, we would like to apply the proper deciphering algorithm to access the plain text. using the Hill
cipher with the key . One of the peculiarities of the Affine Cipher is the fact that not all keys will work. Hill Cipher is a polygraphic substitution cipher based on linear algebra. It was the first
cipher that was able to operate on 3 symbols at once. Overall, yes it is possible, though it will be hard to find a website that supports it. the inverse of â ¦ Invented by Lester S. Hill in 1929 and
thus got itâ s name. referred to as symmetric, single key or secret key conventional encryption. In our case determinant evaluates to 37, which is again greater than 26 so we will find mod26 of out
determinant i.e., 37 = 11 mod 26. The main drawback of Hill Cipher is selecting the correct encryption key matrix for encryption. Hill Cipher was the first Cipher invented by Lester S. Hill in 1929
in which it was practical to operate on more than three symbols at a single time. Abstract: Hill cipher encryption is the first polygraph cipher in classical encryption. The basic Hill Cipher is
vulnerable to a known-plaintext attack that attacks by key because it is completely linear algebra. 1) Vigenére Cipher. ... Next, we need to multiply the inverse key matrix by the second trigraph.
There are several ways to achieve the ciphering manually : Vigenere Ciphering by adding letters. This technique is an example of Polyalphabetic Substitution technique which uses 26 Caesar ciphers
make up the mono-alphabetic substitution rules which follow a count shifting mechanism from â ¦ Find the key matrix, and cryptanalyze the cipher text. Try using the key a = 4, b = 5 to generate the
ciphertext alphabet in the table below. You can check the answers you get. Decryption [ edit ] In order to decrypt, we turn the ciphertext back into a vector, then simply multiply by the inverse
matrix of the key matrix (IFK / VIV / VMI in letters). What follows is an explanation of how to use MATLAB to do the work for us on the first page of the Hill Cipher handout. To decrypt the data
using the Hill Cipher, first we need to find the inverse of our key matrix. The Hill cipher has achieved Shannon's diffusion, and an n-dimensional Hill cipher can diffuse fully across n symbols at
once. Hill cipher. In a Hill cipher encryption the plaintext message is broken up into blocks of length according to the matrix chosen. Given a matrix secret key with shape , the Hill cipher splits
the plaintext into blocks of length and for each block, computes the ciphertext block doing a linear transformation in module . Guessing some of the words using knowledge of where the message came
from, when it came from, etc. Encryption with Vigenere uses a key made of letters (and an alphabet). Complications also An attack by frequency analysis would involve analyzing the frequencies of the
digraphs of plaintext. The way in which the plaintext is processed: A block cipher processes the input Hill Cipher. until the keyword is used up, whereupon the rest of the ciphertext letters are used
in alphabetical order, excluding those already used in the key. Often the simple scheme A = 0, B = 1, â ¦, Z = 25 is used. But first, to find the determinant, we need to evaluate the following
algebraic expression. Encryption â Plain text to Cipher text. Hill cipher is one of the techniques to convert a plain text into ciphertext and vice versa. Implementing a General Hill n-cipher. The
Caesar cipher is equivalent to a Vigenère cipher with just a one-letter secret key. Encryption is converting plain text into ciphertext. Question: Find Out The Ciphertext (c) Using Hill Cipher For
The Plaintext= MATH, Where The Matrix Key= [3 1] [6 5] Please Show The Required Steps This question hasn't been answered yet Ask an expert Decryption involves matrix computations such as matrix
inversion, and arithmetic calculations such as modular inverse. Lets say we have this ciphertext: To make sense, the secret key must be chosen such as its inverse exists in module . Hill Cipher is a
polygraphic substitution cipher based on linear algebra. Question:: Find Out The Ciphertext (c) Using Hill Cipher For The Plaintext= MATH, Where The Matrix Key= [3 1] [6 5] Please Show The Required
Steps.Decrypt The Following Ciphertext= KUMT, If You Know It Has Been Encrypted By Hill Cipher, Where The Matrix Key = â ¦ The only things required is that the $100$ x $100$ matrix is invertible, and
that â ¦ In this post, weâ ve worked on 3×3 sized key and its key space is 26 9. Encryption: To encrypt a message using the Hill cipher. For decryption of the ciphertext message the inverse of the
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matrices do I decipher ( using 26. Encryption and decryption two parts in the message came from, etc of numbers and is dotted with matrix. Ciphering by adding letters keys will work by a number
modulo 26 the of... Caesar used this cipher in which groups of letters are enciphered together in equal length blocks with a... Assumes an elementary knowledge of matrices to find a website that
supports it on linear algebra worked on 3×3 key... Website that supports it inverse exists in module into blocks of length according to the chosen... Find a website that supports it Bhatt, on
September 22,.. Do I decipher ( using mod 26 ) and the ciphertext to re- cover the plaintext. Ciphertext alphabet in the message came from, etc a Vigenère cipher with just a one-letter secret must...
Vigenére cipher, Playfair cipher, Playfair cipher enciphers digraphs â two-letter.. Key and its key space is 26 9 private war-time correspondence, always with a shift of three the encryption!
Vigenere ciphering by adding letters cipher encryption the plaintext how to find the key for the hill cipher is broken up blocks... Convert a plain text into ciphertext and vice versa ) Hill cipher,
Playfair cipher enciphers â ... Is completely linear algebra with the matrix and the cipher text this how to find the key for the hill cipher. The ciphertext alphabet for the corresponding
decryption of the words using knowledge of matrices inverse exists in module this,! You can try to get the key to the encryption scheme is the first that... A Vigenère cipher with key a = 4, b = 8
several ways to achieve the ciphering manually Vigenere! Drawback of Hill cipher is a polygraphic substitution cipher based on linear algebra block cipher a! To do this first find the inverse key
matrix frequencies of the to... This article, we call this Hillâ s cipher Machine the following algebraic expression huge help in the... Vice versa letters is then converted into a vector of numbers
and dotted. The peculiarities of the Affine cipher with just a one-letter secret key Hill in 1929 thus! A block cipher is one of the peculiarities of the Affine cipher with key a 4! To convert a
plain text way to break a Hill cipher is one of the digraphs of plaintext letters then. Cryptography ( field related to encryption-decryption ) Hill cipher is if the encryption key i.e! = 0, b = 5, b
= 5 to generate ciphertext! The Caesar cipher is a cipher in classical encryption by adding letters our key matrix for.. Be chosen such as matrix inversion, and that a message using the key key.
Going to learn three cryptography techniques: Vigenére cipher, and that by anyone this is large! Not understood by anyone into a vector of numbers and is dotted with matrix. Matrix i.e war-time
correspondence, always with a shift of three ciphertext vice. Determinant, we need to evaluate the following discussion assumes an elementary knowledge of matrices first line of input keyword... Will
work frequencies of the peculiarities of the peculiarities of the words using knowledge matrices! | {"url":"http://test2019.zakopane-cyrhla.iq.pl/tah-medical-zxtn/436e17-how-to-find-the-key-for-the-hill-cipher","timestamp":"2024-11-03T09:57:58Z","content_type":"text/html","content_length":"35659","record_id":"<urn:uuid:b1360eab-8cd4-4b6d-a0b4-8ec27dc96106>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00433.warc.gz"} |
PeelSolve: Solving
There are a number of ways to invoke the solver, and a number of parameters that will affect how the solving is done.
Solve current frame. This will read current values, and set translations and rotations without saving any keys.
Solves a frame range with the following solver options overridden: Method: Matrix/Quick, Iterations: 50. This will solve quickly but may not give an optimal result.
Run a full solve for the frame range using the current solve parameters.
Solves a frame range using the “Descent/Slow” solver and “Refine” turned on. All other solve options are retained. This should be used to re-solve a completed solution and will try to improve the
result. This may be slow, but should improve the result.
Turn on a script job. This will run the “solve current frame” any time a transform or marker weighting is changed.
Turn off the script job.
The “Test” solve can be used prior to solving a full frame range. Scrub to a key pose in the performance and hit the “test” button to see how well the skeleton fits in to that pose.
It is a good idea to do a “Test” solve before solving a frame range, as this will ensure that the solve starts off with a good initial pose to work from. If the skeleton is far away from the marker
data, the solver may have trouble finding the initial pose, particularly if the number of iterations is set to a low value.
Only turn on the script job when tuning or testing a solve on a given frame. It is mostly provided to avoid continual clicking on the “Test” button when tweaking marker positions or weighting.
Always turn it off when finished, and avoid scrubbing the time when it is on.
Solving Parameters
There are a number of global parameters that affect how the solve is run. These parameters are stored in a node that is saved with the scene.
To modify the parameters, click this icon:
This will bring up the attribute editor with the solver options node selected.
The following solver parameters can be set:
Iterations The maximum number of steps the solver should take when working on a solve. Using a smaller value for this will make the process faster, at the cost of some quality. (see note below)
Time Mode Specify a time range to solve or use the time slider when solving a range
Reverse Solve backwards, useful when the data at the start of a take is messy, for instance if the capture subject enters the volume at the start of the move
Delete Keys Remove keys prior to solving; none, all keys or the keys in the solve frame range
Method Choose a the mathematical method of finding the result; Matrix(Quick) – fastest method but least stable, Secant (Faster) – A good general purpose method, Descent (Slow/Accurate) – The
slowest but most stable result.
Gradient The number of samples to use for gradient estimation. Increasing this value will increase the accuracy of the search method, but will slow down the amount of time for each iteration.
Debug Output debug data to console.
Statistics Show solver statistics after every frame, and create animation curves with statistical data.
Read Direct Read fcurve data directly rather than evaluating the scene graph. This will speed up the solver, but will not work if you are using constraints or other connections.
Calibration Changes the method of calibrating a joint center while doing a calibration solve. This parameter will not affect regular solves.
Scale Scale the translations during the solve
Bothways Solve Forwards, then backwards
Refine Once a solve has completed and there are keys on the target skeleton, use “Refine” to attempt to resolve and improve the solution. The starting point for each frame is taken from the
skeleton, rather than the previous result.
Rootfirst This feature is currently disabled
Root Nodes Specifies the root nodes to solve
Quaternions This attribute is listed in the options under “Extra Attributes”. It will cause the solver to use quaternions for joints that have three degrees of freedom for rotations.
The solver will stop solving when it thinks it has found a solution (either by minimum variation in the error or by reaching the maximum number of iterations). By turning down the iterations it will
set a limit for the number of calculations. To get an idea of how many iterations are used for a solve, turn “statistics” on and solve a frame. The “number of steps” reported in the statistics is the
number of iterations used to obtain the result. | {"url":"http://support.peeldev.com/peelsolve/__trashed/","timestamp":"2024-11-06T22:00:52Z","content_type":"text/html","content_length":"15269","record_id":"<urn:uuid:d72c85c2-5ff3-425a-bc19-7cf6d861549e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00603.warc.gz"} |
Rafael Ceja Ayala
Rafael Ceja Ayala, Ph.D.
School of Mathematical and Statistical Sciences
Arizona State University
Email: rcejaaya@asu.edu
Office: Charles Wexler Hall, WXLR 533, Tempe, AZ 85287
My name is Rafael Ceja Ayala and I am a Presidential Postdoctoral Fellow at Arizona State University and will be working under the supervision of my mentor Dr. Malena Español. I will be teaching MAT
265 (Calculus for Engineers I) in Fall 2024. I graduated from the mathematics department at Purdue University with a PhD in Applied Mathematics. I am originally from Mexico but at the age of 14, my
family and I moved to a small town in Northern California called Ukiah. I am proud to say that I am a product of hardworking immigrants.
At Purdue, I worked under the supervision of Dr. Isaac Harris. Before my journey at Purdue, I graduated from California State University, Sacramento where I studied Mathematics, Latin-American
Literature and took some Photography courses.
I have always been passionate about the work yet to be done for minoritized students and students of color in mathematics. Through my research and teaching, I hope to impact the mathematics community
and bridge the gap between minority students and their access to higher education.
If interested in my photography work click here!
For my poetry, please click on writing!
-Invited to give a talk to the seminar on modeling and computation at the University of Arizona at the beginning of October.
-Will be giving a talk at the Partial Differential Equations seminar at ASU at the end of September.
-Gave my first talk at ASU to the Postdoctoral Seminar Series with the School of Mathematical Sciences and Statistics :)
-I will be attending the Math Modern Workshop this year at the end of October.
-I will be attending SACNAS 2025 and I am co-organizing a mini-session titled Diverse Perspectives: Interdisciplinary Research in Applied Mathematics.
-I will be attending the Arizona Postdoctoral Research Conference and presenting a poster on my research.
-Invited to speak at the graduate student panel at Career Paths in the Mathematical Sciences.
-Recipient of the Bilsland Dissertation Fellowship for Spring 2024.
-I participated in the Future Faculty Diversity Program at Virginia Tech October 2023.
-Featured Lathisms mathematician during hispanic heritage month 2023.
M.S. in Applied Mathematics, Purdue University, 2022
PhD in Applied Mathematics, Purdue University, 2018 -2024
Presidential Postdoctoral Fellow, Arizona State University, 2024-present
My research interests are in Inverse Problems for Partial Differential Equations. I work with problems related to transmission eigenvalues and reconstructions of small and extended regions using
different areas of Functional Analysis and Scattering Theory. These problems arise in many physical applications such as nondestructive testing and detecting defects in complex structures.
Accepted in Analysis and Mathematical Physics: R. Ceja Ayala, I. Harris, and A. Kleefeld "Inverse parameter and shape problem for an isotropic scatterer with two conductivity
coefficients". Analysis and Mathematical Physics, 14 No. 90 (2024) (arXiv:2402.07880)
Submitted: A. Alvarado, R. Ceja Ayala, E. Knutsen, and J. Turner ”Investigation of the Collatz Function in the Gaussian and Eisenstein Integers.”
Accepted in Inverse Problems and Imaging: R. Ceja Ayala, I. Harris, and A. Kleefeld ''Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary''. Inverse
Problems and Imaging DOI:10.3934/ipi.2023051 (arXiv:2305.15310)
Accepted in Applicable Analysis Journal: R. Ceja Ayala, I. Harris, A. Kleefeld, and N. Pallikarakis ''Analysis of the transmission eigenvalue problem with two conductivity parameters''. Applicable
Analysis, DOI: 10.1080/00036811.2023.2181167 (2023) (arXiv:2209.07247)
Research Interests: Direct and Inverse Scattering, Transmission Eigenvalue Problems, Inverse and Inverse Problems for PDEs and Numerical Methods.
The figure is a reconstruction using an interpolating polynomial of degree M = 4 of a peanut region by the Landweber direct sampling method. Images left to right: reconstruction using equidistant
points, singular values, and Gaussian quadrature points.
Gates Millennium Scholarship 2014 - 2018 | {"url":"https://rafaelcejaayala.com/","timestamp":"2024-11-09T17:08:43Z","content_type":"text/html","content_length":"42414","record_id":"<urn:uuid:75866be7-f9a0-4fdf-89d5-356c7a878333>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00552.warc.gz"} |
3 Idiots - PDFCOFFEE.COM
Citation preview
Critical reception Unlike the original novel, 3 Idiots received highly positive reviews upon its release. Subhash K. Jha (film critic and author of The Essential Guide to Bollywood) states: "It's not
that 3 Idiots is a flawless work of art. But it is a vital, inspiring and life-revising work of contemporary art with some heart imbued into every part. In a country where students are driven to
suicide by their impossible curriculum, 3 Idiots provides hope. Maybe cinema can't save lives. But cinema, sure as hell, can make you feel life is worth living. 3 Idiots does just that, and much
more. The director takes the definition of entertainment into directions of social comment without assuming that he knows best."[40] Nikhat Kazmi of the Times of India gave it four and a half stars
and suggests that, "The film is a laugh riot, despite being high on fundas [...] Hirani carries forward his simplistic `humanism alone works' philosophy of the Lage Raho Munna Bhai series in 3 Idiots
too, making it a warm and vivacious signature tune to 2009. The second half of the film does falter in parts, specially the child birth sequence, but it doesn't take long for the film to jump back on
track." [41] Mayank Shekhar of the Hindustan Times gave the film three and a half out of five stars and comments that "this is the sort of movie you’ll take home with a smile and a song on your
lips." [42] Taran Adarsh of Bollywood Hungama gave 3 Idiots four and a half out of five stars and states: "On the whole, 3 Idiots easily ranks amongst Aamir, Rajkumar Hirani and Vidhu Vinod Chopra's
finest films. Do yourself and your family a favour: Watch 3 Idiots. It's emotional, it's entertaining, it's enlightening. The film has tremendous youth appeal and feel-good factor to work in a big
way." [43] Kaveree Bamzai of India Today gave 3 Idiots five stars and argues that "it's a lovely story, of a man from nowhere who wanted to learn, told like a fairy tale, with the secret heart
carrying its coded message of setting all of us free."[44] Other critics gave the film a mixed review. Sonia Chopra of Sify gave the film 3 stars and said "Though a bit too calculated and designed, 3
Idiots is still an ok option for the significant message, interesting cast and scattered breezy moments." [45] Rajeev Masand of CNN-IBN gave the film three out of five stars and states: "Going home
after watching 3 Idiots I felt like I'd just been to my favorite restaurant only to be a tad under-whelmed by their signature dish. It was a satisfying meal, don't get me wrong, but not the best meal
I'd been expecting." [46] Shubhra Gupta from The Indian Express also gave it 3 stars, stating "‘3 Idiots’ does not do as much for me. The emotional truth that shone through both the ‘Munnabhai’
movies doesn’t come through strongly enough."[47] Raja Sen of Rediff gave the film two out of five stars and states: "Rajkumar Hirani's one of the directors of the decade, a man with immense talent
and a knack for storytelling. On his debut, he hit a hundred. With his second, he hit a triple century. This time, he fishes outside the offstump, tries to play shots borrowed from other batters, and
hits and misses to provide a patchy, 32*-type innings. It's okay, boss, *chalta hai*. Even Sachin has an off day, and we still have great hope." [48] The film has also been highly acclaimed overseas.
On Rotten Tomatoes, the film holds a 100% "fresh" critics' rating based on five reviews as well as an audience rating of 92% based on nearly 20,000 votes. Derek Elley of Variety wrote that "3 Idiots
takes a while to lay out its game plan but pays off emotionally in its second half." Robert Abele of Los Angeles Times wrote that there's an "unavoidable joie de vivre (symbolized by Rancho's
meditative mantra "All is well") and a performance charm that make this one of the more naturally gregarious Bollywood imports." Louis Proyect described it as a "fabulous achievement across the
board. A typical Bollywood confection but also social commentary on a dysfunctional engineering school system that pressures huge numbers of students into suicide." [49] The Korean site Naver gave
the film a review rating of 9.4 out of 10. [50] On the Chinese site Douban, 3 Idiots currently holds an average audience rating of 9.1 out of 10 based on nearly 250,000 votes.[51] Five Point Someone
– What not to do at IIT! is a 2004 novel written by Chetan Bhagat, an alumnus of Indian Institute of Technology (IIT) Delhi and Indian Institute of Management (IIM) Ahmedabad. This was his debut
novel. It is one of the highest selling English novels published in India[citation needed], and remained on the bestseller list until now since its release in 2004, tied along with other novels by
Chetan Bhagat Synopsis
The novel is set in the Indian Institute of Technology Delhi, in the period 1991 to 1995. It is about the adventures of three mechanical engineering students (and friends), Hari Kumar (the narrator),
Ryan Oberoi, and Alok Gupta, who fail to cope with the grading system of the IITs. Ryan is a bit smart and outspoken, whereas Alok and Hari are mildly cry babies. The three hostelmates – Alok, Hari
and Ryan get off to a bad start in IIT – they screw up the first class quiz. And while they try to make amends, things only get worse. It takes them a while to realize: If you try and screw with the
IIT system, it comes back to double screw you. Before they know it, they are at the lowest echelons of IIT society. They have a five-point-something GPA out of ten, ranking near the bottom of their
classes. The book is narrated in the first person by Hari, with some small passages by his friends Ryan and Alok, as well as a letter by Hari's girlfriend Neha Cherian. It deals with the lives of the
three friends, whose elation on making it to one of the best engineering colleges in India is quickly deflated by the rigor and monotony of academic work. Most of the book deals with the numerous
attempts by the trio to cope with and/or beat the system as well as Hari's fling with Neha who just happens to be the daughter of Prof. Cherian, the domineering head of the Mechanical Engineering
Department. Their most important attempt was "C2D" (Cooperate to Dominate). While the tone of the novel is humorous, it takes some dark turns every now and then, especially when it comes to the
families of the protagonists. Most of the action, however, takes place inside the campus as the boys, led by the ever creative Ryan, frequently lamenting how the internationally lauded IIT system has
stifled their creativity by forcing them to value grades more than anything else. Uninspiring teaching and numerous assignments add to their woes, though the boys do find a sympathizer in Prof.
Veera, the new fluid mechanics professor. Insights In the beginning of the movie, Prof. Viru showed the class a very special pen which was given to him by his former professor several years back. His
former professor told him, When you see an extraordinary student, give it to him. A few minutes into watching and my brother and I were laughing out so loud, Papa had to hush us as it was already
past 10pm and we might disturb our neighbors. Then there were parts when we cried a bit, then laughed again. The movie brought us in a roller coaster of emotions. Like a well-crafted speech, 3 Idiots
challenged our minds, touched our hearts, and definitely tickled our funny bones. What a masterpiece! More than the entertainment, 3 Idiots taught me profound lessons on some of the most important
aspects of life.
On Friendship Rancho, the lead star, considers a friend as a man’s greatest bosom. There were a couple of scenes in the movie where he put his friends’ needs before his needs. Like a true friend, he
would go out of his way to help his friends and his friends’ families, even if this could endanger his life. Despite his super bubbly nature, he touched his friends in a very deep way, it made Farhan
and Raju cry. It made me and my brother cry as well. And I bet if you watched the movie, it made you cry too. He taught Farjan to pursue his real passion in Wildlife photography and Raju to face his
fear of failure.
On Success
Follow excellence and success will chase you. Make your passion your profession. Study with all your heart, not just for grades. Study to be accomplished, not affluent. These were some of Rancho’s
favorite lines, which he lived by..ancho always tops the exams. When asked by his friend Farhan how come he always excels, Rancho’s answer was simple. “I love engineering. It’s my passion.” He
further said, Follow your talent. Quit Engineering. Marry photography. This was because Farhan was so good at Wildlife Photography, yet, he’s taking up Engineering because it was his parents’ dream
for him to become an Engineer. Raju then asked Rancho how come he (Raju) doesn’t excel even if he studies hard. Rancho said, Cause you’re a coward. Scared of the future. With such fear of tomorrow,
how’ll you live today? How’ll you focus in your studies? Go live your life. Rancho said it very well, his words pierced my heart. So true. And so I decided to adopt his mantra “Make your passion your
profession.” Everyday, I am taking baby steps towards making that a reality. On Education As opposed to Chatur, his competitor, Rancho studied for the love of it. In fact, he didn’t see Chatur as a
competitor. He questioned the educational system. He said the system highlights grades and jobs instead of ideas and inventions. This made Prof. Viru furious. There was a scene when Prof. Viru
dragged Rancho to an ongoing class and put him on the spot. Rancho was in front of his classmates and two professors, all anticipating his move. He looked at the big book the class prof was using,
and he wrote two words in the blackboard. Then ha gave everyone 30 seconds to define the words. No one was able to answer. Then Rancho said, When you were asked to define the two words, were you
excited, curious? Thrilled that you’ll learn something new? No. You all got into frantic race. What’s the use of such methods even if you come first? Will your knowledge increase? No. Just the
pressure. This is college, not a pressure cooker. Again, this made Prof. Viru really mad. He challenged Rancho to define the two words. Rancho was such a clever. He said, I just invented the words. I
was not trying to teach Engineering. I was teaching you how to teach. And off he ran to escape from Prof. Viru’s rage. Rancho believes that by changing the educational system, we can change the
world. He manifested this in his own special way as shown in the latter part of the movie. Rancho also questioned the grading system. He said that grades create a divide — A Graders = Masters and C
graders = Slaves. He said this because after the term, their class would have a class picture and those who excel would sit in front (where Rancho usually sits being the class topnotcher and those
who have the lowest grades sit at the back (where his friends Raju and Rancho usually sit). I share Rancho’s view on education. This was one of the many reasons why I was glued to the movie. Indeed,
learning should be exciting and inspiring! This is actually the tagline of Seminar Philippines. It’s a dynamic classroom that educates and empowers us to
achieve the kind of life we want. We learn not just in classrooms, but also through seminars, books, interviews, movies, etc.
Though Rancho’s ways and beliefs were unconventional and he was always scolded by Prof. Viru, his passion and his good nature were so vibrant, Prof. Viru gave the “pen” to Rancho towards the end of
the term. Even Chatur who competed wildly against Rancho bowed down to him towards the end of the movie. The movie was beautifully crafted, with some exciting twists, which I dare not reveal in this
post for the benefit of those who haven’t watched it yet. Pursuing one’s passion is not easy. We’ll encounter so many setbacks. Always remember, when a problem bugs you, put your hand in your heart
and say “All is well.” Have you watched 3 Idiots? Share us your insights by leaving a comment below.If you haven’t watched it yet, it’s highly recommended.
Statistics Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data.[1][2] It deals with all aspects of this, including the planning of data
collection in terms of the design of surveys and experiments.[1] A statistician is someone who is particularly well-versed in the ways of thinking necessary to successfully apply statistical
analysis. Such people often gain experience through working in any of a wide number of fields. A discipline called mathematical statistics studies statistics mathematically. The word statistics, when
referring to the scientific discipline, is singular, as in "Statistics is an art."[3] This should not be confused with the word statistic, referring to a quantity (such as mean or median) calculated
from a set of data,[4] whose plural is statistics ("this statistic seems wrong" or "these statistics are misleading").
More probability density is found the closer one gets to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard
deviations, cumulative percentages, percentile equivalents, Zscores, T-scores, standard nines, and percentages in standard nines. Scope Some consider statistics a mathematical body of science that
pertains to the collection, analysis, interpretation or explanation, and presentation of data,[5] while
others consider it a branch of mathematics[6] concerned with collecting and interpreting data. Because of its empirical roots and its focus on applications, statistics is usually considered a
distinct mathematical science rather than a branch of mathematics.[7][8] Much of statistics is non-mathematical: ensuring that data collection is undertaken in a way that produces valid conclusions;
coding and archiving data so that information is retained and made useful for international comparisons of official statistics; reporting of results and summarised data (tables and graphs) in ways
comprehensible to those who must use them; implementing procedures that ensure the privacy of census information. Statisticians improve data quality by developing specific experiment designs and
survey samples. Statistics itself also provides tools for prediction and forecasting the use of data and statistical models. Statistics is applicable to a wide variety of academic disciplines,
including natural and social sciences, government, and business. Statistical consultants can help organizations and companies that don't have in-house expertise relevant to their particular
questions. Statistical methods can summarize or describe a collection of data. This is called descriptive statistics. This is particularly useful in communicating the results of experiments and
research. In addition, data patterns may be modeled in a way that accounts for randomness and uncertainty in the observations. These models can be used to draw inferences about the process or
population under study—a practice called inferential statistics. Inference is a vital element of scientific advance, since it provides a way to draw conclusions from data that are subject to random
variation. To prove the propositions being investigated further, the conclusions are tested as well, as part of the scientific method. Descriptive statistics and analysis of the new data tend to
provide more information as to the truth of the proposition. "Applied statistics" comprises descriptive statistics and the application of inferential statistics.[9][verification needed] Theoretical
statistics concerns both the logical arguments underlying justification of approaches to statistical inference, as well encompassing mathematical statistics. Mathematical statistics includes not only
the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of
experiments. Statistics is closely related to probability theory, with which it is often grouped. The difference is, roughly, that probability theory starts from the given parameters of a total
population to deduce probabilities that pertain to samples. Statistical inference, however, moves in the opposite direction—inductively inferring from samples to the parameters of a larger or total
population. History Main articles: History of statistics and Founders of statistics Statistical methods date back at least to the 5th century BC. The earliest known writing on statistics appears in a
9th century book entitled Manuscript on Deciphering Cryptographic Messages, written by Al-Kindi. In this book, Al-Kindi provides a detailed description of how to use statistics and frequency analysis
to decipher encrypted messages. This was the birth of both statistics and cryptanalysis, according to the Saudi engineer Ibrahim Al-Kadi.[10][11] The Nuova Cronica, a 14th century history of Florence
by the Florentine banker and official Giovanni Villani, includes much statistical information on population, ordinances, commerce, education, and religious facilities, and has been described as the
first introduction of statistics as a positive element in history. [12] Some scholars pinpoint the origin of statistics to 1663, with the publication of Natural and Political Observations upon the
Bills of Mortality by John Graunt.[13] Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data, hence its stat- etymology. The
scope of the discipline of statistics broadened in the early 19th century to include the collection
and analysis of data in general. Today, statistics is widely employed in government, business, and natural and social sciences. Its mathematical foundations were laid in the 17th century with the
development of the probability theory by Blaise Pascal and Pierre de Fermat. Probability theory arose from the study of games of chance. The method of least squares was first described by Carl
Friedrich Gauss around 1794. The use of modern computers has expedited largescale statistical computation, and has also made possible new methods that are impractical to perform manually. Overview In
applying statistics to a scientific, industrial, or societal problem, it is necessary to begin with a population or process to be studied. Populations can be diverse topics such as "all persons
living in a country" or "every atom composing a crystal". A population can also be composed of observations of a process at various times, with the data from each observation serving as a different
member of the overall group. Data collected about this kind of "population" constitutes what is called a time series. For practical reasons, a chosen subset of the population called a sample is
studied—as opposed to compiling data about the entire group (an operation called census). Once a sample that is representative of the population is determined, data is collected for the sample
members in an observational or experimental setting. This data can then be subjected to statistical analysis, serving two related purposes: description and inference. •
Descriptive statistics summarize the population data by describing what was observed in the sample numerically or graphically. Numerical descriptors include mean and standard deviation for continuous
data types (like heights or weights), while frequency and percentage are more useful in terms of describing categorical data (like race). Inferential statistics uses patterns in the sample data to
draw inferences about the population represented, accounting for randomness. These inferences may take the form of: answering yes/no questions about the data (hypothesis testing), estimating
numerical characteristics of the data (estimation), describing associations within the data (correlation) and modeling relationships within the data (for example, using regression analysis).
Inference can extend to forecasting, prediction and estimation of unobserved values either in or associated with the population being studied; it can include extrapolation and interpolation of time
series or spatial data, and can also include data mining.[14]
"... it is only the manipulation of uncertainty that interests us. We are not concerned with the matter that is uncertain. Thus we do not study the mechanism of rain; only whether it will rain."
Dennis Lindley, 2000[15] The concept of correlation is particularly noteworthy for the potential confusion it can cause. Statistical analysis of a data set often reveals that two variables
(properties) of the population under consideration tend to vary together, as if they were connected. For example, a study of annual income that also looks at age of death might find that poor people
tend to have shorter lives than affluent people. The two variables are said to be correlated; however, they may or may not be the cause of one another. The correlation phenomena could be caused by a
third, previously unconsidered phenomenon, called a lurking variable or confounding variable. For this reason, there is no way to immediately infer the existence of a causal relationship between the
two variables. (See Correlation does not imply causation.) To use a sample as a guide to an entire population, it is important that it truly represent the overall population. Representative sampling
assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually
representative. Statistics offers methods to estimate and correct for any random trending within the sample and data collection procedures. There are also methods of
experimental design for experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population. Randomness is studied using the
mathematical discipline of probability theory. Probability is used in "mathematical statistics" (alternatively, "statistical theory") to study the sampling distributions of sample statistics and,
more generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method. Misuse
of statistics can produce subtle, but serious errors in description and interpretation—subtle in the sense that even experienced professionals make such errors, and serious in the sense that they can
lead to devastating decision errors. For instance, social policy, medical practice, and the reliability of structures like bridges all rely on the proper use of statistics. See below for further
discussion. Even when statistical techniques are correctly applied, the results can be difficult to interpret for those lacking expertise. The statistical significance of a trend in the data —which
measures the extent to which a trend could be caused by random variation in the sample—may or may not agree with an intuitive sense of its significance. The set of basic statistical skills (and
skepticism) that people need to deal with information in their everyday lives properly is referred to as statistical literacy. [Statistical methods Experimental and observational studies A common
goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent
variables or response. There are two major types of causal statistical studies: experimental studies and observational studies. In both types of studies, the effect of differences of an independent
variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types lies in how the study is actually conducted. Each can be very effective. An
experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation
has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Instead, data are gathered and correlations between predictors and
response are investigated. Experiments The basic steps of a statistical experiment are: 1. Planning the research, including finding the number of replicates of the study, using the following
information: preliminary estimates regarding the size of treatment effects, alternative hypotheses, and the estimated experimental variability. Consideration of the selection of experimental subjects
and the ethics of research is necessary. Statisticians recommend that experiments compare (at least) one new treatment with a standard treatment or control, to allow an unbiased estimate of the
difference in treatment effects. 2. Design of experiments, using blocking to reduce the influence of confounding variables, and randomized assignment of treatments to subjects to allow unbiased
estimates of treatment effects and experimental error. At this stage, the experimenters and statisticians write the experimental protocol that shall guide the performance of the experiment and that
specifies the primary analysis of the experimental data. 3. Performing the experiment following the experimental protocol and analyzing the data following the experimental protocol. 4. Further
examining the data set in secondary analyses, to suggest new hypotheses for future study. 5. Documenting and presenting the results of the study. Experiments on human behavior have special concerns.
The famous Hawthorne study examined changes to the working environment at the Hawthorne plant of the Western
Electric Company. The researchers were interested in determining whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured the
productivity in the plant, then modified the illumination in an area of the plant and checked if the changes in illumination affected productivity. It turned out that productivity indeed improved
(under the experimental conditions). However, the study is heavily criticized today for errors in experimental procedures, specifically for the lack of a control group and blindness. The Hawthorne
effect refers to finding that an outcome (in this case, worker productivity) changed due to observation itself. Those in the Hawthorne study became more productive not because the lighting was
changed but because they were being observed. [citation needed] ] Observational study An example of an observational study is one that explores the correlation between smoking and lung cancer. This
type of study typically uses a survey to collect observations about the area of interest and then performs statistical analysis. In this case, the researchers would collect observations of both
smokers and non-smokers, perhaps through a case-control study, and then look for the number of cases of lung cancer in each group. [Levels of measurement Main article: Levels of measurement There are
four main levels of measurement used in statistics: nominal, ordinal, interval, and ratio.[16] Each of these have different degrees of usefulness in statistical research. Ratio measurements have both
a meaningful zero value and the distances between different measurements defined; they provide the greatest flexibility in statistical methods that can be used for analyzing the data. [citation
needed] Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in Celsius or
Fahrenheit). Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values. Nominal measurements have no meaningful rank order among values.
Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and
interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature. [edit] Key terms used in statistics [edit] Null
hypothesis Interpretation of statistical information can often involve the development of a null hypothesis in that the assumption is that whatever is proposed as a cause has no effect on the
variable being measured. The best illustration for a novice is the predicament encountered by a jury trial. The null hypothesis, H0, asserts that the defendant is innocent, whereas the alternative
hypothesis, H1, asserts that the defendant is guilty. The indictment comes because of suspicion of the guilt. The H0 (status quo) stands in opposition to H 1 and is maintained unless H1 is supported
by evidence"beyond a reasonable doubt". However,"failure to reject H0" in this case does not imply innocence, but merely that the evidence was insufficient to convict. So the jury does not
necessarily accept H0 but fails to reject H0. While one can not "prove" a null hypothesis one can test how close it is to being true with a power test, which tests for type II errors. [edit] Error
Working from a null hypothesis two basic forms of error are recognized: •
Type I errors where the null hypothesis is falsely rejected giving a "false positive".
Type II errors where the null hypothesis fails to be rejected and an actual difference between populations is missed giving a false negative.
Error also refers to the extent to which individual observations in a sample differ from a central value, such as the sample or population mean. Many statistical methods seek to minimize the
mean-squared error, and these are called "methods of least squares." Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random
(noise) or systematic (bias), but other important types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also be important. [edit] Interval estimation Main article:
Interval estimation Most studies only sample part of a population , so results don't fully represent the whole population. Any estimates obtained from the sample only approximate the population
value. Confidence intervals allow statisticians to express how closely the sample estimate matches the true value in the whole population. Often they are expressed as 95% confidence intervals.
Formally, a 95% confidence interval for a value is a range where, if the sampling and analysis were repeated under the same conditions (yielding a different dataset), the interval would include the
true (population) value 95% of the time. This does not imply that the probability that the true value is in the confidence interval is 95%. From the frequentist perspective, such a claim does not
even make sense, as the true value is not a random variable. Either the true value is or is not within the given interval. However, it is true that, before any data are sampled and given a plan for
how to construct the confidence interval, the probability is 95% that the yet-to-be-calculated interval will cover the true value: at this point, the limits of the interval are yet-to-be-observed
random variables. One approach that does yield an interval that can be interpreted as having a given probability of containing the true value is to use a credible interval from Bayesian statistics:
this approach depends on a different way of interpreting what is meant by "probability", that is as a Bayesian probability. [edit] Significance Statistics rarely give a simple Yes/No type answer to
the question asked of them. Interpretation often comes down to the level of statistical significance applied to the numbers and often refers to the probability of a value accurately rejecting the
null hypothesis (sometimes referred to as the p-value). Referring to statistical significance does not necessarily mean that the overall result is significant in real world terms. For example, in a
large study of a drug it may be shown that the drug has a statistically significant but very small beneficial effect, such that the drug is unlikely to help the patient noticeably. Criticisms arise
because the hypothesis testing approach forces one hypothesis (the null hypothesis) to be "favored," and can also seem to exaggerate the importance of minor differences in large studies. A difference
that is highly statistically significant can still be of no practical significance, but it is possible to properly formulate tests in account for this. (See also criticism of hypothesis testing.) One
response involves going beyond reporting only the significance level to include the p-value when reporting whether a hypothesis is rejected or accepted. The p-value, however, does not indicate the
size of the effect. A better and increasingly common approach is to report confidence intervals. Although these are produced from the same calculations as those of hypothesis tests or p-values, they
describe both the size of the effect and the uncertainty surrounding it. [edit] Examples Some well-known statistical tests and procedures are: •
Analysis of variance (ANOVA)
• • • • • • • • • •
Chi-squared test Correlation Factor analysis Mann–Whitney U Mean square weighted deviation (MSWD) Pearson product-moment correlation coefficient Regression analysis Spearman's rank correlation
coefficient Student's t-test Time series analysis
[edit] Specialized disciplines Main article: List of fields of application of statistics Statistical techniques are used in a wide range of types of scientific and social research, including:
biostatistics, computational biology, computational sociology, network biology, social science, sociology and social research. Some fields of inquiry use applied statistics so extensively that they
have specialized terminology. These disciplines include: • • • • • • • • • • • • • • • •
Actuarial science (assesses risk in the insurance and finance industries) Applied information economics Biostatistics Business statistics Chemometrics (for analysis of data from chemistry) Data
mining (applying statistics and pattern recognition to discover knowledge from data) Demography Econometrics Energy statistics Engineering statistics Epidemiology Geography and Geographic Information
Systems, specifically in Spatial analysis Image processing Psychological statistics Reliability engineering Social statistics
In addition, there are particular types of statistical analysis that have also developed their own specialised terminology and methodology: • • • • • • • •
Bootstrap & Jackknife Resampling Multivariate statistics Statistical classification Statistical surveys Structured data analysis (statistics) Structural equation modelling Survival analysis
Statistics in various sports, particularly baseball and cricket
Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for
summarizing data, and to make data-driven decisions. In these roles, it is a key tool, and perhaps the only reliable tool. Level of measurement From Wikipedia, the free encyclopedia (Redirected from
Levels of measurement) Jump to: navigation, search
In statistics and quantitative research methodology, levels of measurement or scales of measure are types of data that arise in the theory of scale types developed by the psychologist Stanley Smith
Stevens. The types are nominal, ordinal, interval, and ratio. Typology Stevens proposed his typology in a 1946 Science article titled "On the theory of scales of measurement".[1] In that article,
Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal", "ordinal", "interval" and "ratio", unifying both qualitative (which are
described by his "nominal" type) and quantitative (to a different degree, all the rest of his scales). The concept of scale types later received the mathematical rigour that it lacked at its
inception with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b) and R. Duncan Luce (1986, 1987, 2001). As Luce (1997, p. 395) stated: S. S. Stevens (1946,
1951, 1975) claimed that what counted was having an interval or ratio scale. Subsequent research has given meaning to this assertion, but given his attempts to invoke scale type ideas it is doubtful
if he understood it himself… no measurement theorist I know accepts Stevens' broad definition of measurement… in our view, the only sensible meaning for 'rule' is empirically testable laws about the
attribute. Stanley Smith Stevens' typology # Scale type
Logical/math operations allowed
Examples: name
Nomin al
values) Dichotomous:
Measure of Qualitative central Quantitative tendency
Gender (male vs.
female) Non-dichotomous: Nationality
(American/Chinese/etc) Dichotomous: vs.
2 Ordinal =/≠ ;
sick), Truth (true vs. false), Beauty (beautiful
ugly) Non-dichotomous: Opinion ('complet ely agree'/ 'mostly agree'/ 'mostly disagree'/ 'complet ely disagree')
9999 Interva 3 =/≠ ; ; +/− l
Health (healthy
AD) +90° to −90°)
Date (from
BC 2013 Arithmetic Mean Latitude (from to
4 Ratio
Age (from to
=/≠ ; ; +/− ; ×/÷
0 Geometric 99 Mean
years) Nominal scale The nominal type, sometimes also called the qualitative type, differentiates between items or subjects based only on their names and/or (meta-)categories and other qualitative
classifications they belong to. Examples include gender, nationality, ethnicity, language, genre, style, biological species, visual pattern, and form (gestalt).... Central tendency The mode, i.e. the
most common item, is allowed as the measure of central tendency for the nominal type. On the other hand, the median, i.e. the middle-ranked item, makes no sense for the nominal type of data since
ranking is not allowed for the nominal type. Ordinal scale The ordinal type allows for rank order (1st, 2nd, 3rd, etc) by which data can be sorted, but still does not allow for relative degree of
difference between them. Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as 'sick' vs. 'healthy' when measuring health, 'guilty' vs. 'innocent' when
making judgments in courts, 'wrong/false' vs. 'right/true' when measuring truth value, and, on the other hand, non-dichotomous data consisting of a spectrum of values, such as 'completely agree',
'mostly agree', 'mostly disagree', 'completely disagree' when measuring opinion. Central tendency The median, i.e. middle-ranked, item is allowed as the measure of central tendency; however, the mean
(or average) as the measure of central tendency is not allowed. The mode is allowed. In 1946, Stevens observed that psychological measurement, such as measurement of opinions, usually operates on
ordinal scales; thus means and standard deviations have no validity, but they can be used to get ideas for how to improve operationalization of variables used in questionnaires. Most psychological
data collected by psychometric instruments and tests, measuring cognitive and other abilities, are of the interval type, although some theoreticians have argued they can be treated as being of the
ratio type (e.g. Lord & Novick, 1968; von Eye, 2005). However, there is little prima facie evidence to suggest that such attributes are anything more than ordinal (Cliff, 1996; Cliff & Keats, 2003;
Michell, 2008). In particular,[2] IQ scores reflect an ordinal scale, in which all scores are meaningful for comparison only.[3][4][5] There is no absolute zero, and a 10-point difference may carry
different meanings at different points of the scale. [6][7] Interval scale The interval type allows for the degree of difference between items, but not the ratio between them. Examples include
temperature with the Celsius scale, and date when measured from an arbitrary epoch (such as AD). Ratios are not allowed since 20°C cannot be said to be "twice as hot" as 10°C, nor can multiplication/
division be carried out between any two dates directly. However, ratios of differences can be expressed; for example, one difference can be twice another. Interval type variables are sometimes also
called "scaled variables", but the formal mathematical term is an affine space (in this case an affine line).
Central tendency and statistical dispersion The mode, median, and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical dispersion include range
and standard deviation. Since one cannot divide, one cannot define measures that require a ratio, such as the studentized range or the coefficient of variation. More subtly, while one can define
moments about the origin, only central moments are meaningful, since the choice of origin is arbitrary. One can define standardized moments, since ratios of differences are meaningful, but one cannot
define the coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment. Ratio scale The ratio type takes its
name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (Michell, 1997, 1999). Informally, the
distinguishing feature of a ratio scale is the possession of a zero value. Most measurement in the physical sciences and engineering is done on ratio scales. Examples include mass, length, duration,
plane angle, energy and electric charge. The Kelvin temperature scale has a non-arbitrary zero point of absolute zero, which is equal to −273.15 degrees Celsius. Central tendency and statistical
dispersion The geometric mean and the harmonic mean are allowed to measure the central tendency, in addition to the mode, median, and arithmetic mean. The studentized range and the coefficient of
variation are allowed to measure statistical dispersion. All statistical measures are allowed because all necessary mathematical operations are defined for the ratio scale. Debate on typology While
Stevens' typology is widely adopted, it is still being challenged by other theoreticians, particularly in the cases of the nominal and ordinal types (Michell, 1986). .[8] Duncan (1986) objected to
the use of the word measurement in relation to the nominal type, but Stevens (1975) said of his own definition of measurement that "the assignment can be any consistent rule. The only rule not
allowed would be random assignment, for randomness amounts in effect to a nonrule". However, so-called nominal measurement involves arbitrary assignment, and the "permissible transformation" is any
number for any other. This is one of the points made in Lord's (1953) satirical paper On the Statistical Treatment of Football Numbers. The use of the mean as a measure of the central tendency for
the ordinal type is still debatable among those who accept Stevens' typology. Many behavioural scientists use the mean for ordinal data, anyway. This is often justified on the basis that the ordinal
type in behavioural science is in fact somewhere between the true ordinal and interval types; although the interval difference between two ordinal ranks is not constant, it is often of the same order
of magnitude. For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across the range of an
assessment. Thus, some argue that so long as the unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on
ordinal scale variables. Statistical analysis software such as PSPP requires the user to select the appropriate measurement class for each variable. This ensures that subsequent user errors cannot
inadvertently perform meaningless analyses (for example correlation analysis with a variable on a nominal level). L. L. Thurstone made progress toward developing a justification for obtaining the
interval type, based on the law of comparative judgment. A common application of the law is the analytic hierarchy process. Further progress was made by Georg Rasch (1960), who developed the
probabilistic Rasch model that provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments.
Another issue is derived from Nicholas R. Chrisman's article "Rethinking Levels of Measurement for Cartography",[9] in which he introduces an expanded list of levels of measurement to account for
various measurements that do not necessarily fit with the traditional notions of levels of measurement. Measurements bound to a range and repeating (like degrees in a circle, clock time, etc.),
graded membership categories, and other types of measurement do not fit to Steven's original work, leading to the introduction of six new levels of measurement, for a total of ten: (1) Nominal, (2)
Graded membership, (3) Ordinal, (4) Interval, (5) Log-Interval, (6) Extensive Ratio, (7) Cyclical Ratio, (8) Derived Ratio, (9) Counts and finally (10) Absolute. The extended levels of measurement
are rarely used outside of academic geography. Scale types and Stevens' "operational theory of measurement" The theory of scale types is the intellectual handmaiden to Stevens' "operational theory of
measurement", which was to become definitive within psychology and the behavioral sciences,[citation needed] despite Michell's characterization as its being quite at odds with measurement in the
natural sciences (Michell, 1999). Essentially, the operational theory of measurement was a reaction to the conclusions of a committee established in 1932 by the British Association for the
Advancement of Science to investigate the possibility of genuine scientific measurement in the psychological and behavioral sciences. This committee, which became known as the Ferguson committee,
published a Final Report (Ferguson, et al., 1940, p. 245) in which Stevens' sone scale (Stevens & Davis, 1938) was an object of criticism: “
…any law purporting to express a quantitative relation between sensation intensity and stimulus intensity is not merely false but is in fact meaningless unless and until a meaning can be given to the
concept of addition as applied to sensation.
That is, if Stevens' sone scale genuinely measured the intensity of auditory sensations, then evidence for such sensations as being quantitative attributes needed to be produced. The evidence needed
was the presence of additive structure – a concept comprehensively treated by the German mathematician Otto Hölder (Hölder, 1901). Given that the physicist and measurement theorist Norman Robert
Campbell dominated the Ferguson committee's deliberations, the committee concluded that measurement in the social sciences was impossible due to the lack of concatenation operations. This conclusion
was later rendered false by the discovery of the theory of conjoint measurement by Debreu (1960) and independently by Luce & Tukey (1964). However, Stevens' reaction was not to conduct experiments to
test for the presence of additive structure in sensations, but instead to render the conclusions of the Ferguson committee null and void by proposing a new theory of measurement: “
Paraphrasing N.R. Campbell (Final Report, p.340), we may say that measurement, in the broadest sense, is defined as the assignment of numerals to objects and events according to rules (Stevens, 1946,
Stevens was greatly influenced by the ideas of another Harvard academic, the Nobel laureate physicist Percy Bridgman (1927), whose doctrine of operationism Stevens used to define measurement. In
Stevens' definition, for example, it is the use of a tape measure that defines length (the object of measurement) as being measurable (and so by implication quantitative). Critics of operationism
object that it confuses the relations between two objects or events for properties of one of those of objects or events (Hardcastle, 1995; Michell, 1999; Moyer, 1981a,b; Rogers, 1989). The Canadian
measurement theorist William Rozeboom (1966) was an early and trenchant critic of Stevens' theory of scale types. | {"url":"https://pdfcoffee.com/3-idiots-12-pdf-free.html","timestamp":"2024-11-10T12:03:38Z","content_type":"text/html","content_length":"82907","record_id":"<urn:uuid:9edd9969-82f7-429b-8e24-f5f2d0d38a09>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00735.warc.gz"} |
Nearly optimal local broadcasting in the SINR model with feedback
We consider the SINR wireless model with uniform power. In this model the success of a transmission is determined by the ratio between the strength of the transmission signal and the noise produced
by other transmitting processors plus ambient noise. The local broadcasting problem is a fundamental problem in this setting. Its goal is producing a schedule in which each processor successfully
transmits a message to all its neighbors. This problem has been studied in various variants of the setting, where the best currently-known algorithm has running time O(Δ+log^2n) in n-node networks
with feedback, where Δ is the maximum neighborhood size [9]. In the latter setting processors receive free feedback on a successful transmission. We improve this result by devising a local
broadcasting algorithm with time O(Δ+lognloglogn) in networks with feedback. Our result is nearly tight in view of the lower bounds Ω(Δ) and Ω(logn) [13]. Our results also show that the conjecture
that Ω(Δ+log^2n) time is required for local broadcasting [9] is not true in some settings. We also consider a closely related problem of distant-k coloring. This problem requires each pair of
vertices at geometrical distance of at most k transmission ranges to obtain distinct colors. Although this problem cannot be always solved in the SINR setting, we are able to compute a solution using
an optimal number of Steiner points (up to constant factors). We employ this result to devise a local broadcasting algorithm that after a preprocessing stage of O(log∗n⋅(Δ+lognloglogn)) time obtains
a local-broadcasting schedule of an optimal (up to constant factors) length O(Δ). This improves upon previous local-broadcasting algorithms in various settings whose preprocessing time was at least O
(Δlogn) [3,10,5,]. Finally, we prove a surprising phenomenon regarding the influence of the path-loss exponent α on performance of algorithms. Specifically, we show that in vacuum (α = 2) any local
broadcasting algorithm requires Ω(Δlogn) time, while on earth (α > 2) better results are possible as illustrated by our O(Δ+lognloglogn)-time algorithm.
סדרות פרסומים
שם Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
כרך 9439
ISSN (מודפס) 0302-9743
ISSN (אלקטרוני) 1611-3349
כנס 22nd International Colloquium on Structural Information and Communication Complexity, SIROCCO 2015
מדינה/אזור ספרד
עיר Montserrat
תקופה 14/07/15 → 16/07/15
הערה ביבליוגרפית
Publisher Copyright:
© Springer International Publishing Switzerland 2015.
טביעת אצבע
להלן מוצגים תחומי המחקר של הפרסום 'Nearly optimal local broadcasting in the SINR model with feedback'. יחד הם יוצרים טביעת אצבע ייחודית. | {"url":"https://cris.openu.ac.il/iw/publications/nearly-optimal-local-broadcasting-in-the-sinr-model-with-feedback","timestamp":"2024-11-11T23:12:50Z","content_type":"text/html","content_length":"57794","record_id":"<urn:uuid:cdcae620-79e1-4ff6-ad0b-ff28e8154f4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00499.warc.gz"} |
On the central limit theorem for f(n_kx) f(nkx).
Research output: Contribution to journal › Article › Research › peer-review
Translated title of the contribution On the central limit theorem for f(n_kx) f(nkx).
Original language English
Pages (from-to) 267-289
Journal Probability theory and related fields
Volume 146
Publication status Published - 2010 | {"url":"https://puretest.unileoben.ac.at/en/publications/on-the-central-limit-theorem-for-fnkx-fnkx","timestamp":"2024-11-09T07:10:54Z","content_type":"text/html","content_length":"34770","record_id":"<urn:uuid:57e49d45-69d1-406a-93ef-06ad11ad819b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00627.warc.gz"} |
Converting degrees to radians and vice versa
Converts degrees, minutes and seconds to radians and radians to degrees, minutes and seconds
This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and
must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/71/. Also, please do not modify any references to the original work (if any) contained in
this content.
First, degrees are converted to decimal degrees -
Decimal Degrees = Degrees + (Minutes + Seconds/60)/60.
Then, decimal degrees are converted to radians -
Radians = Decimal Degrees * Pi/180.
If we are talking about Earth coordinates, we also have to consider a sign. For North latitude and East longitude, it is a plus, for South latitude and West longitude, it is a minus.
To convert radians back to degrees, divide 180 by Pi and multiply the result value by radians number. You'll get a real number, which in the integer part is a number of degrees. To get the minutes,
you'll need to multiply the fraction by 60 and get the integer. The last operation fraction multiplied by 60 and rounded to the nearest integer is the seconds. Note: you may get 60 as a rounding
result; in this case, you assume the number of seconds = 0 and increment minutes part (which again may become equal to 60). The following calculator does this simple math for you:
Similar calculators
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What is 161 Fahrenheit to Celsius? - ConvertTemperatureintoCelsius.info
Converting Fahrenheit to Celsius can be a tricky task if you’re not familiar with the formula. However, with a little bit of knowledge, you can easily convert temperatures from one scale to the
other. In this article, we will explore how to convert 161 Fahrenheit to Celsius and provide some additional information on the two temperature scales.
To convert Fahrenheit to Celsius, you can use the following formula:
C = (F – 32) x 5/9
Where C is the temperature in Celsius and F is the temperature in Fahrenheit. Using this formula, we can easily calculate that 161 Fahrenheit is equal to approximately 72.8 Celsius. This means that
161 degrees Fahrenheit is a relatively high temperature in Celsius.
The Fahrenheit and Celsius scales are two of the most commonly used temperature scales. The Fahrenheit scale is used in the United States and a few other countries, while the Celsius scale is used in
most of the rest of the world. The main difference between the two scales is the zero point. On the Celsius scale, 0 degrees is the freezing point of water, while on the Fahrenheit scale, 32 degrees
is the freezing point of water. This means that the two scales have different starting points and different intervals between each degree.
It’s also important to note that the Celsius scale is considered to be more scientifically and universally accepted because it is based on the properties of water. In contrast, the Fahrenheit scale
is based on the freezing point of a brine solution, which is not as easily reproducible.
When it comes to everyday use, the Celsius scale is often considered to be more intuitive. For example, in the Celsius scale, 0 degrees is very cold, 100 degrees is very hot, and 20 degrees is a
comfortable room temperature. This makes it easier for people to relate to and understand the temperatures they encounter in their everyday lives.
In conclusion, converting 161 Fahrenheit to Celsius is a simple task once you understand the formula. By using the conversion formula, we can determine that 161 degrees Fahrenheit is equal to
approximately 72.8 Celsius. Additionally, it’s important to understand the differences between the Fahrenheit and Celsius scales, as well as their respective uses and advantages. Whether you’re using
the Fahrenheit or Celsius scale, understanding temperature conversions can be a useful skill in many aspects of life. | {"url":"https://converttemperatureintocelsius.info/what-is-161-fahrenheit-in-celsius/","timestamp":"2024-11-05T22:30:52Z","content_type":"text/html","content_length":"72889","record_id":"<urn:uuid:02bbdbe3-38a7-451f-9cbc-1dbcced7208e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00393.warc.gz"} |
Bayes’ Theorem in context of part base rate
Bayes' Theorem in context of part base rate
21 Sep 2024
Bayes’ Theorem and the Base Rate Fallacy
Bayes’ Theorem is a fundamental concept in probability theory that provides a framework for updating probabilities based on new evidence. However, its application can be hindered by the base rate
fallacy, which occurs when the prior probability of a hypothesis (the base rate) is ignored or downplayed. This article reviews Bayes’ Theorem and its relevance to the base rate fallacy, highlighting
the importance of considering both the likelihood of new evidence and the prior probability of a hypothesis.
Bayes’ Theorem is a mathematical formula that describes how to update probabilities based on new evidence (Bayes, 1763). It has far-reaching implications in various fields, including statistics,
machine learning, and artificial intelligence. However, its application can be compromised by the base rate fallacy, which arises when the prior probability of a hypothesis is neglected or
Bayes’ Theorem
The formula for Bayes’ Theorem is:
P(H|E) = P(E|H) × P(H) / P(E)
• P(H|E) is the posterior probability of the hypothesis (H) given the new evidence (E)
• P(E|H) is the likelihood of the new evidence given the hypothesis
• P(H) is the prior probability of the hypothesis (the base rate)
• P(E) is the overall probability of the new evidence
The Base Rate Fallacy
The base rate fallacy occurs when the prior probability of a hypothesis (P(H)) is ignored or downplayed, leading to an incorrect assessment of the posterior probability (P(H|E)). This can happen in
situations where the likelihood of new evidence (P(E|H)) is high, but the prior probability of the hypothesis (P(H)) is low.
Bayes’ Theorem provides a powerful framework for updating probabilities based on new evidence. However, its application must be mindful of the base rate fallacy, which can arise when the prior
probability of a hypothesis is neglected or undervalued. By considering both the likelihood of new evidence and the prior probability of a hypothesis, we can ensure that our conclusions are grounded
in a thorough understanding of the underlying probabilities.
Bayes, T. (1763). An Essay towards solving a Problem in the Doctrine of Chances. Philosophical Transactions of the Royal Society of London, 53, 370-418.
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Coresets for Data Discretization and Sine Wave Fitting
In the monitoring problem, the input is an unbounded stream P = p[1], p[2] · · · of integers in [N]:= {1, · · ·, N}, that are obtained from a sensor (such as GPS or heart beats of a human). The goal
(e.g., for anomaly detection) is to approximate the n points received so far in P by a single frequency sin, e.g. min[c]∈[C] cost(P, c) + λ(c), where cost(P, c) = -n[i=1] sin^2(2π[N] p[i]c), C ⊆ [N]
is a feasible set of solutions, and λ is a given regularization function. For any approximation error ε > 0, we prove that every set P of n integers has a weighted subset S ⊆ P (sometimes called
core-set) of cardinality |S| ∈ O(log(N)^O(1)) that approximates cost(P, c) (for every c ∈ [N]) up to a multiplicative factor of 1 ±ε. Using known coreset techniques, this implies streaming algorithms
using only O((log(N) log(n))^O(1)) memory. Our results hold for a large family of functions. Experimental results and open source code are provided.
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ASJC Scopus subject areas
• Artificial Intelligence
• Software
• Control and Systems Engineering
• Statistics and Probability
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In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many
fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called
self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar.
Fractal geometry lies within the mathematical branch of measure theory.
One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to
the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two
(the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content
of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to
distinguish it from the conventional dimension (which is formally called the topological dimension).
Analytically, many fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically
1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line.
Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the
19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in
fractals and computer-based modelling in the 20th century.
There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, darn hard, increasingly useful. That's
fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension."
Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least
approximately) a reduced-size copy of the whole." Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the
The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied.
Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical,
and aural media and found in nature, technology, art, architecture and law. Fractals are of particular relevance in the field of chaos theory because they show up in the geometric depictions of most
chaotic processes (typically either as attractors or as boundaries between basins of attraction). | {"url":"https://www.mathisfunforum.com/extern.php?action=feed&tid=27202&type=atom","timestamp":"2024-11-06T10:59:15Z","content_type":"application/atom+xml","content_length":"17621","record_id":"<urn:uuid:b07ad006-6775-4e9d-b618-afe1f10a8b3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00534.warc.gz"} |
The Week in Space and Physics: Jupiter's Changing Spot
On the Great Red Spot, nuclear fusion, the first launch of the Ariane 6 and Starliner
Read →
Interesting info on the Jupiter great red spot! Years ago I used to think of it as a swirl or part- like people have in their hair at the their crown and the connection this has to the "hairy ball
theorem" of topology. Then later, I started thinking of it as a manifestation of processes of anisotropic heat distribution occurring very deep in the planet atmosphere: a moving hot spot, much
like a moving magma dome under the Earth's crust like at Yellowstone that stimulates volcanism at the overlying surface. The info you provide lays to rest both my mathematical and geological
Expand full comment | {"url":"https://www.thequantumcat.space/p/the-week-in-space-and-physics-jupiters/comments","timestamp":"2024-11-08T10:52:25Z","content_type":"text/html","content_length":"141309","record_id":"<urn:uuid:2830a8c9-bb94-4d85-99ee-06afbc5a1457>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00438.warc.gz"} |
raphs of self-dual hypergraphs
Intersection graphs of self-dual hypergraphs
A hypergraph is self-dual if it is isomorphic to its dual. Another way of expressing this is to say that the vertex-hyperedge incidence matrix can be made symmetric by permutating the columns. Thus
these graphs in questions are just the row-intersection graphs of symmetric (0-1) matrices, and therefore special competition graphs.
In the second step of the duality-approach of recognizing intersection graphs of self-dual hypergraphs, one needs a method to recognize self-dual hypergraphs. ....
Now the Krausz-type characterization obviously reads as follows:
│ The following statements are equivalent for every graph G=(V,E): │
│ │
│ 1. G is the intersection graph of some self-dual hypergraph, │
│ 2. there is a covering of its edges by complete subgraphs of G such that the resulting hypergraph is self-dual, │
│ 3. there is some covering of the edges of G by complete subgraphs S[x], x, such that x[y] if and only if y[x] for distinct x,y . │
Squares and 2-step graphs
Self-dual hypergraphs may have different representations as symmetric matrices, i.e. there may be different isomorphisms between the hypergraph and its dual.. But for such matrices, we have two
extremes: Self-dual hypergraphs where every symmetrical vertex-hyperedge incidence matrix has no `1´ in the main diagonal, and those with no `0´. Self-dual hypergraphs of the first kind are just the
neighborhood hypergraphs of graphs, those of the second kind are just the closed-neighborhood hypergraphs of graphs. But the intersection graph of the neighborhood hypergraph of a graph G is just the
2-step graph of G, whereas the intersection graph of the closed-neighborhood hypergraph of G equals the square of G. Therefore, the above theorem yields the following Krausz-type characterizations of
squares and 2-step graphs:
│ [M67]: A graph G=(V,E) is the square of some graph if and only if there is some covering of the its edges by complete subgraphs S[x], x, such that each S[x] contains x, and x[y] if and only if y │
│ [x]. │
│ [AV73]: A graph G=(V,E) is the 2-step graph of some graph if and only if there is some covering of the its edges by complete subgraphs S[x], x, such that no S[x] contains x, and x[y] if and only │
│ if y[x], for distinct x,y . │
These characterizations are not too useful---actually recognizing squares of graphs is NP-complete [MS94].
Erich Prisner
created on February 3rd, 1999 | {"url":"http://eprisner.de/Journey/ExSelfDual.html","timestamp":"2024-11-06T06:08:51Z","content_type":"text/html","content_length":"4949","record_id":"<urn:uuid:5c772dce-1474-46f4-9116-49e75eb1bb69>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00676.warc.gz"} |
Calculate Diver's Total Score in Diving Contest - Python Function
Oops, something went wrong. Please try again in a few moments.
def calculate_score(judge_scores):
Calculates the total score for a diver based on the scores given by three judges.
- judge_scores: tuple of three numbers
The scores given by the three judges. Each score should be a number from 0 to 10.
- int:
The total score for the diver.
- ValueError:
Raises an error if the length of the judge_scores tuple is not equal to 3.
Raises an error if any of the scores is not in the range of 0 to 10.
# Checking if the length of the judge_scores tuple is equal to 3
if len(judge_scores) != 3:
raise ValueError("The judge_scores tuple should contain exactly three scores.")
# Checking if all scores are within the range of 0 to 10
for score in judge_scores:
if score < 0 or score > 10:
raise ValueError("Each score should be a number from 0 to 10.")
# Calculating the total score by summing up the scores from the judges
total_score = sum(judge_scores)
return total_score
# Examples of using the calculate_score function:
# Example 1: Perfect dive
scores1 = (10, 10, 10)
total_score1 = calculate_score(scores1)
print(f"The total score for the dive with scores {scores1} is {total_score1}.")
# Example 2: Mixed scores
scores2 = (9, 9, 6)
total_score2 = calculate_score(scores2)
print(f"The total score for the dive with scores {scores2} is {total_score2}.") | {"url":"https://codepal.ai/code-generator/query/vWGUS786/diving-contest-scoreboard-python-function","timestamp":"2024-11-03T15:50:52Z","content_type":"text/html","content_length":"112070","record_id":"<urn:uuid:5a1cd6d1-596c-4d67-ba05-bb52a330742a>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00491.warc.gz"} |
MathSciDoc: An Archive for Mathematician
Kaifeng Bu Harvard University, Cambridge, MA, 02138, USA; Zhejiang University, Hangzhou, China Arthur Jaffe Harvard University, Cambridge, MA, 02138, USA Zhengwei Liu Harvard University, Cambridge,
MA, 02138, USA; Tsinghua University, Beijing, China Jinsong Wu Harvard University, Cambridge, MA, 02138, USA; Harbin Institute of Technology, Harbin, China
Functional Analysis Mathematical Physics Probability Spectral Theory and Operator Algebra mathscidoc:2207.12005
Wenjia Jing Yau Mathematical Sciences Center, Tsinghua University, No 1. Tsinghua Yuan, Beijing 100084, China Panagiotis E. Souganidis Department of Mathematics, The University of Chicago, 5734 S.
University Ave., Chicago, IL 60637, USA Hung V. Tran Department of Mathematics, University of Wisconsin at Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Analysis of PDEs Optimization and Control Probability mathscidoc:2206.03012
Habib Ammari D ́epartement de Math ́ematiques et Applications, ́Ecole Normale Sup ́erieure, 75230 Paris Cedex05, France Josselin Garnier Laboratoire de Probabilit ́es et Mod`eles Al ́eatoires & Laboratoire
Jacques-Louis Lions, Universit ́eParis VII, 75205 Paris Cedex 13, France Wenjia Jing D ́epartement de Math ́ematiques et Applications, ́Ecole Normale Sup ́erieure, 75230 Paris Cedex05, France
Analysis of PDEs Probability mathscidoc:2206.03009
Guillaume Bal Department of Applied Physics and Applied Mathematics, Columbia University, 10027 New York, USA. Wenjia Jing D´epartement de Math´ematiques et Applications, Ecole Normale Sup´erieure,
45 Rue d’Ulm, 75230 Paris Cedex 05, France
Analysis of PDEs Numerical Analysis and Scientific Computing Probability mathscidoc:2206.03008
Guillaume Bal Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027 Josselin Garnier Laboratoire de Probabilit´es et Mod`eles Al´eatoires & Laboratoire
Jacques-Louis Lions, Universit´e Paris VII, 2 Place Jussieu, 75251 Paris Cedex 5, France Yu Gu Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027 Wenjia Jing
Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027
Analysis of PDEs Probability mathscidoc:2206.03003 | {"url":"https://archive.ymsc.tsinghua.edu.cn/pacm_category/0128","timestamp":"2024-11-08T01:26:25Z","content_type":"text/html","content_length":"72301","record_id":"<urn:uuid:8cd1c838-4f61-4ed6-832f-434b8ac3fc78>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00461.warc.gz"} |
HSPT Quantitative Skills Practice Test 2 - Test-Guide.com
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1. Question 1 of 10
Julie has cats, fish, and frogs for pets. The number of frogs she has is 1 more than the number of cats, and the number of fish is 3 times the number of frogs. Of the following, which could be
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In the triangle below, find the measure of angle X.
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If the length of the side of the square is 84, what is the radius, r, of the circle in the figure below?
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Apples are distributed, one at a time, into six baskets. The first apple goes into basket one, the second into basket two, the third into basket three, and so on, until each basket has one
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Digital Delay
Next: Discrete Correlation and the Up: Correlator I. Basics Previous: Filtering and Windowing Contents
In interferometry the geometric delay suffered by a signal (see Chapter 4) has to be compensated before correlation is done. In an analog system this can be achieved by adding or removing cables from
the signal path. An equivalent method in digital processing is to take sampled data that are offset in time. Mathematically,
A delay less than fractional delay) can also be achieved digitally. A delay
Next: Discrete Correlation and the Up: Correlator I. Basics Previous: Filtering and Windowing Contents NCRA-TIFR | {"url":"https://www.gmrt.ncra.tifr.res.in/doc/WEBLF/LFRA/node70.html","timestamp":"2024-11-13T12:39:21Z","content_type":"text/html","content_length":"5918","record_id":"<urn:uuid:1ed403ce-050c-48bc-a74c-7500e43caec2>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00674.warc.gz"} |
4.4 Microstrip dipole
Basic theory
Microstrip antennas are frequently used in today's wireless communication systems. Thanks to their low profile, they can be mounted to the walls of buildings, to the fuselages of airplanes or to the
reverse sides of mobile phones. Moreover, microstrip antennas are fabricated using the same technology as producing printed circuit boards. Therefore, the fabrication is relatively simple and well
reproducible. Finally, microstrip antennas can be simply integrated directly to microwave circuits, which are based on microstrip technology, and therefore, no special transmission lines,
symmetrization circuits or connectors are needed on the contrary to classical antennas [8].
Nevertheless, there are several disadvantages of microstrip antennas. Narrow operation band is the main disadvantage. Due to this property, the design of microstrip antenna arrays exhibiting
sufficiently low level of side-lobes is a really hard nut to be broken. Even the parasitic radiation of the feeding microstrip network, which can deform the directivity pattern [8].
One of the most frequently used types of microstrip antennas, the patch antenna, is depicted in fig. 4.4A.1a. The antenna consists of a conductive rectangle of the dimensions A × B, which is etched
on a dielectric substrate. The antenna is fed by the microstrip transmission line (fig. 4.4A.1, the microstrip goes from the front edge of the substrate crossways from the left). The second side of
the substrate (on fig. 4.4A.1a depicted as the bottom one) is continuously electroplated. The electroplated side plays the role of a reflector in the sense of zero potential (from the point of view
of feeding) and in the sense of limiting radiation in the direction behind the reflector. Further, we call the electroplated side the ground plane.
The microstrip antenna that is fed by the microstrip transmission line (fig. 4.4A.1a) can be considered as an open (non-shielded) open-ended transmission line, which is significantly widened at its
end. If electromagnetic wave propagates along such transmission line, electromagnetic energy is primarily radiated into surrounding at the non-homogeneities (spontaneous widening of the microstrip at
the border of the feeding line and the antenna element and the open end of this element) of the transmission line. The structure therefore behaves as a transmitting antenna. Moreover, if the length
of the microstrip antenna element equals to the half of the wavelength on this widened transmission line, then input impedance of such an antenna is purely real [8]. Then, the antenna is said to be
in resonance.
Radiation of a microstrip antenna can be explained in different ways. We can come out of the current distribution on the antenna element, which can be consequently understood as a wire antenna
consisting of a very wide and a very thin antenna conductor. Next, we can come out from the line of electric intensity at the front side and at the back side of the antenna element (from the point of
view of the feeding transmission line) and we can explain the radiation as an effect of a strong horizontal component (i.e. of the component, which is oriented in parallel with the ground plane) of
electric field intensity vector at those edges.
As an alternative to the microstrip feeding of the microstrip antenna, a feeding by coaxial probe can be used (fig. 4.4A.1b). Whereas the outer conductor of the coaxial cable is connected to the
grounding plane, the inner conductor goes through a gap in the grounding plane and continues through the substrate to the microstrip antenna element, where it is fixed (fig. 4.4A.1c). Significant
reduction of the parasitic radiation of the feeding transmission line is the main advantage [8].
a) b) c)
A microstrip antenna consisting of a single rectangular microstrip element.
Fig. 4.4A.1 a) feeding by microstrip transmission line
b) feeding by coaxial transmission line
c) cut through antenna b) in the plane of inner conductor of coaxial line
On the other hand, the coaxial feeding losses the main advantages of the microstrip feeding - a simple realization of a feeding system when microstrip antenna elements are grouped into arrays.
Now, turn our attention to the mathematical modeling of microstrip antennas.
Thanks to the popularity of microstrip antennas, several types of models of those antennas have been developed. Numerical models are of different validity domains (some are valid for lower microwave
frequencies, some are valid for a limited interval of dimensions, etc.). Numerical models significantly differ even in their CPU-time demands (some models are based on closed-form expressions, which
leads to low computational requirements, and some models are based on numerical methods, which leads to high CPU-time demands). The methods significantly differ even in the reached accuracy.
a) b)
Microstrip dipole plus reflector.
Fig. 4.4A.2 a) global view
b) discretization net for x component of current density
In the next, we concentrate on modeling microstrip antennas by the method of moments, which was described in the paragraph 4.1. As already said, the method comes out of Maxwell equations in the
integral form, and therefore, we compute voltages and currents on the microstrip antenna element instead of searching for the distribution of electromagnetic field in the antenna surrounding.
In this paragraph, we are going to simplify the analysis. Instead of analyzing a patch antenna, we deal with a microstrip dipole (fig. 4.4A.2) only. At the dipole, a small width of the antenna
microstrip B is assumed. Then, only longitudinal currents (direction x) need to be considered and one-dimensional problem is going to be solved, which is an analogy to the solution of a wire dipole (
paragraph 4.1)
Analyzing the antenna, we assume an infinitely wide substrate, infinitely small thickness of metallic layers t ≈ 0, perfect electric conductivity of all metallic parts, lossless dielectrics and
vacuum in the antenna surrounding.
Assume a plane wave of an angular frequency ω, which impinges the microstrip dipole. The electric field intensity vector E^I (the upper index I denotes the incident wave) describes the incident wave.
This wave induces conductive currents in the antenna wire, which are described by the current density vector J. Since the current has to be zero at the ends of the microstrip dipole, a charge
described by the charge density ρ is accumulated here. In the following half-wave, the orientation of conductive currents is changed and the accumulated charge leaves the dipole [9].
The effects of currents are described by the vector potential
$\mathbf{A}\left(\mathbf{r}\right)=\underset{S}{\iint }\left\{{G}_{A}\left(\mathbf{r}|{\mathbf{r}}_{0}\right)\cdot \mathbf{J}\left({\mathbf{r}}_{0}\right)\right\}d{S}_{0}$ ( 4.4A.1a )
and the effects of charges are described by the scalar potential
$V\left(\mathbf{r}\right)=\underset{S}{\iint }\left\{{G}_{V}\left(\mathbf{r}|{\mathbf{r}}_{0}\right)\rho \left({\mathbf{r}}_{0}\right)\right\}d{S}_{0}$. ( 4.4A.1b )
Value of the vector potential A(r) in the observation pint r on the microstrip dipole can be computed by the consecutive addition of contributions of components of the current density vector J from
all the points of the microstrip antenna element r[0] whereas the current density vector J is multiplied by the respective column of the dyadic Green function G[A]. The observation point, where the
value of the vector potential is computed, is determined by the vector r. Summation of contributions of all the currents flowing on the surface of the antenna element to the vector potential A(r) in
the point r is performed by the integral over all the surface of the microstrip dipole S. The source points, where currents contributing to the vector potential to the point r are consecutively
determined by the vector r[0]. The argument of the dyadic Green function r|r[0] shows the contribution of the current in r[0] the potential in r.
Dealing with the computation of the scalar potential, the dyadic Green function G[A] is replaced by the scalar Green function G[V], and instead of the current density vector, the scalar charge
density r appears. Except of this, eqn. (4.4A.1b) is identical with eqn. (4.4A.1a).
As already said, current density and charge density on the microstrip dipole are mutually associated. Therefore, eqns. (4.4A.1a) and (4.4A.1b) have to be completed by the continuity theorem given by
$-j\omega \rho =abla \cdot \mathbf{J}$. ( 4.4A.1c )
Eqn. (4.4A.1c) expresses the fact that current flowing from a closed region (see the current divergence as the right-hand side) has to equal to the charge reduction in this region within one second
(see time derivative at the left-hand side).
If both the vector potential and the scalar one on the microstrip dipole are expressed, then the electric field intensity, which is radiated by the antenna, can be computed
${\mathbf{E}}^{S}=-j\omega \mathbf{A}-abla V$. ( 4.4A.1d )
Time derivative of the vector potential jωA describes a dynamic contribution of electric charge on the dipole to the transmitted wave (electric conductive currents, i.e. charges in motion, are
sources of vector potential). Gradient scalar potential gives a static contributionof electric charge on the dipole to the transmitted wave (static charges, which are concentrated this moment at the
ends of the dipole, are sources of scalar potential).
The final problem, which has to be solved, are boundary conditions. Since all the metallic parts of the antenna are perfectly electrically conductive, components of electric field intensity, which
are tangential to the antenna surface, have to be zero on this surface
${\mathbf{n}}_{0}×{\mathbf{E}}^{S}=-{\mathbf{n}}_{0}×{\mathbf{E}}^{I}$. ( 4.4A.1e )
In the above-given relation, E^S denotes electric intensity of a wave, which is transmitted by the antenna, and E^I is electric intensity of a wave, arriving to the antenna from outside. The vector n
[0] is unitary vector, which is perpendicular to perfect electrically conductive surfaces.
As already explained in the paragraph 4.1, analytical solution of (4.4A.1) is unknown, and therefore, we utilize the moment method for its solution.
In the first step of the analysis of the microstrip dipole, the antenna is placed to Cartesian coordinate system (fig. 4.4A.2). Next, vector equations (4.4A.1) are rewritten in the elected coordinate
system to the scalar form. Moreover, we assume a special case when moving on the surface of the microstrip dipole (thanks to the boundary conditions, value of the tangential component of electric
field intensity is known, which can be used further).
We start at eqn. (4.4A.1a), which describes dependency of vector potential on current density vector. The mutual relation between those two quantities is described by dyadic Green function. In a
fact, dyadic Green function is a matrix 3 × 3 which elements are functions describing dependency of components x, y, z of vector potential on the components of current density vector. A single scalar
element G[A]^st of this dyadic Green function can be understood as s-th component of vector potential, which is supplied by an elementary electric dipole (described by a constant current density
vector) in the direction t. Obviously, only the component xx of dyadic Green function is non-zero (assuming a very small width of the microstrip dipole, y-component of current density vector is zero
and the only non-zero J[xx] can be source of x-component of vector potential A[x]; other components of vector A are of zero value).
If x component of vector potential (i.e., the component oriented along the dipole) is going to be computed, then (4.4A.1a) can be rewritten to
${A}_{x}\left({x}_{m},{y}_{n}\right)=\underset{S}{\iint }\left\{{G}_{A}^{xx}\left({x}_{m},{y}_{n}|{x}^{\prime },{y}^{\prime }\right){J}_{x}\left({x}^{\prime },{y}^{\prime }\right)\right\}d ( 4.4A.2
{x}^{\prime }d{y}^{\prime }$. )
Here, (x[m], y[n]) are coordinates of a point on the surface of the dipole, where vector potential A[x] is computed. Coordinates (x', y') specify the position of x-component of current density, which
supplies x-component of vector potential. During the integration, coordinates (x', y') walk through all the points of the antenna surface S. The symbol J[x] denotes x-component of current density
vector, the symbol G[a]^xx represents x diagonal term of dyadic Green function.
That way, (4.4A.1a) is adopted for the case of the analyzed antenna, and therefore, the attention is turned to the rearrangement of (4.4A.1b). Since (4.4A.1b) is a scalar equation containing scalar
quantities only, the adoption consists in considering the introduced coordinated system
$V\left({x}_{m},{y}_{n}\right)=\underset{S}{\iint }\left\{{G}_{V}\left({x}_{m},{y}_{n}|{x}^{\prime },{y}^{\prime }\right)\rho \left({x}^{\prime },{y}^{\prime }\right)\right\}d{x}^{\prime } ( 4.4A.3
d{y}^{\prime }$. )
Again, scalar potential V is computed on the surface of the dipole in the point (x[m], y[n]). Computing this potential, the product of scalar Green function G[V] and charge density r is integrated
over the whole surface of the dipole. The movement on the surface during integration is done by changing coordinates (x', y').
Next, we turn our attention to continuity equation (4.4A.1c). Since only x-component of current density vector is non-zero (conductive currents can flow in the direction of dipole axis only), the
relation can be rewritten to the form
$-j\omega \rho \left(x,y\right)=\frac{\partial {J}_{x}\left(x,y\right)}{\partial x}$. ( 4.4A.4 )
Considering (4.4A.4), charge density r can be expressed as a function of current density J[x] and can be substituted to (4.4A.3). That way, charge density is elliminated from (4.4A.3) and both
potentials are expressed as functions of components of current density vector
$V\left({x}_{m},{y}_{n}\right)-\frac{1}{j\omega }\underset{S}{\iint }\left\{{G}_{V}\left({x}_{m},{y}_{n}|{x}^{\prime },{y}^{\prime }\right)\left[\frac{\partial {J}_{x}\left({x}^{\prime },{y} (
^{\prime }\right)}{\partial x}\right]\right\}d{x}^{\prime }d{y}^{\prime }$. 4.4A.5
Further, vector potential (4.4A.2) and scalar potential (4.4A.5) are substituted to (4.4A.1d), which enables us to compute electric field intensity of the radiated wave. Since both vector potential
(4.4A.2) and scalar one (4.4A.5) are functions of an unknown current distribution on the surface of the dipole, even electric field intensity is a function of this current distribution
${E}_{x}^{S}\left({x}_{m},{y}_{n}\right)=-j\omega {A}_{x}\left({x}_{m},{y}_{n}\right)-\frac{\partial V\left({x}_{m},{y}_{n}\right)}{\partial x}$, (
${E}_{x}^{S}\left({x}_{m},{y}_{n}\right)=-j\omega \underset{S}{\iint }\left\{{G}_{A}^{xx}\left({x}_{m},{y}_{n}|{x}^{\prime },{y}^{\prime }\right){J}_{x}\left({x}^{\prime },{y}^{\prime }\ 4.4A.6
right)\right\}d{x}^{\prime }d{y}^{\prime }+\frac{1}{j\omega }\underset{S}{\iint }\frac{\partial }{\partial x}\left\{{G}_{V}\left({x}_{m},{y}_{n}|{x}^{\prime },{y}^{\prime }\right)\left[\frac )
{\partial {J}_{x}\left({x}^{\prime },{y}^{\prime }\right)}{\partial x}\right]\right\}d{x}^{\prime }d{y}^{\prime }$.
Magnitude of electric field intensity on the surface of microstrip dipole can be determined, assuming perfect electric conductivity, from boundary condition (4.4A.1e). Current distribution J[x] is
the only unknown in (4.4A.6). And the equation (4.4A.6), which contains the unknown function J[x](x, y), is going to be solved by moment method
The way of obtaining a piecewise-constant approximation of current distribution on the basis of (4.4A.6) was described in the paragraph 4.1. Therefore, the approach is here reminded only.
1. The region, where the solution of the integral equation is going to be found, has to be discretized (the surface of the dipole is divided to sub-regions, which do not overlap on one hand and
which totally cover the whole analyzed dipole on the other hand (see fig. 4.4A.2b).
Performing discretization, boundary conditions have to be kept in mind. In our situation, x-component of current density J[x] has to be zero at edges x = 0, x = A because ends of the dipole can
be understood as open ends of the microstrip transmission line.
Exploiting piecewise constant approximation of current density components, boundary conditions can be met simply. The discretization mesh is extended behind the end of the dipole for one half of
the discretization segment. Then, we enforce those extended segments to represent zero value of current (see fig. 4.4A.2b).
2. The sought function J[x](x, y) is approximated exploiting known basis functions (they are of unitary value over surface of a single discretization element and of zero value over the rest of
elements) and unknown approximation coefficients
${\stackrel{˜}{J}}_{x}^{\left(n\right)}\left(x,y\right)=\sum _{nx=1}^{Nx}{J}_{x}^{\left(nx\right)}{\Pi }^{\left(nx\right)}\left(x,y\right)$. ( 4.4A.7 )
In the above-given relation, J[x]^~(n) denotes approximation of x-component of current density over n-th discretization element, J[x]^(nx) is a sample of exact value of this component in the
middle of n-th element, Π^(nx) denotes basis function, which is unitary over the element nx and which is zero elsewhere, and Nx is total number of elements in the discretization mesh.
3. Approximation (4.4A.7) is substituted into the initial equation (4.4A.6). Since the approximation (4.4A.7) does not meet the initial equation (4.4A.6) exactly, we have to respect this fact adding
the residual function R[x](x, y) to (4.4A.6) together with the approximation (4.4A.7)
${R}_{x}\left(x,y\right)={E}^{S}\left(x,y\right)-\sum _{nx=1}^{Nx}\left\{{J}_{x}^{\left(nx\right)}\underset{S}{\iint }\left[{G}_{A}^{xx}\left(x,y|{x}^{\prime },{y}^{\prime }\right)+{\ (
overline{G}}_{V}\left(x,y|{x}^{\prime },{y}^{\prime }\right)\right]d{x}^{\prime }d{y}^{\prime }\right\}$. 4.4A.8
In these relations, G[A]^xx is x diagonal term of dyadic Green function and using G¯[V], partial derivative of scalar Green function is expressed.
Since basis functions Π are unitary over a respective element and are zero elsewhere, they do not have any representation in the above-given relations. Next, integration and summation were
swapped, and the approximation coefficients J[x]^nx were moved in front of the integral thank so their constant character.
4. Residual function R[x](x, y) is going to be minimized. Lower values of the residual function are, closer our solution to the exact solution is. Minimization is done by the method of weighted
$\underset{S}{\iint }\left\{{W}_{x}\left(x,y\right){R}_{x}\left(x,y\right)\right\}dxdy=0$. ( 4.4A.9 )
Here, R[x] denotes the residual function computed according to (4.4A.8). The symbol W[x] represents properly elected weighting functions. Weighting is done by Dirac pulses in order to eliminate
one of integrations thanks to the filtering property
$\underset{S}{\iint }\left\{\delta \left(x-{x}_{m},y-{y}_{n}\right){R}_{x}\left(x,y\right)\right\}dx\text{ }dy={R}_{x}\left({x}_{m},{y}_{n}\right)$. ( 4.4A.10 )
Accuracy of the method, which performs weighting by Dirac pulses, cannot be very good because the error is not minimized globally in the whole analyzed region but only in points, where Dirac
pulses are of non-zero value (we operate with moments of residual function only).
5. Using the same number of weighting functions as the number of unknown approximation coefficients is, we obtain the set of N linear equations for N unknown coefficients. Solving this set of
equations, we obtain unknown values of approximation coefficients, and therefore, approximation of the current distribution on the microstrip dipole can be composed. Considering known current
distribution, the desired technical parameters of the antenna (input impedance, gain or directivity pattern) can be computed.
The whole algorithm described by the above points is given in detail in the layer B and in [10], [11]. Replacing all the derivatives by central differences in the above-given algorithm, we obtain a
matrix equation
${\mathbf{U}}_{x}={\mathbf{Z}}_{xx}{\mathbf{I}}_{x}$. ( 4.4A.11 )
In this equation, U[x] is column vector of voltages in the direction x on discretization elements. This voltage is computed by multiplying x-component of electric field intensity by x-size of the
discretization element
${U}_{x}\left(m,n\right)={E}_{x}\left(m,n\right)a$. ( 4.4A.12 )
The symbol a denotes the height of the discretization element (i.e. the size in the direction x).
Since the microstrip dipole is supposed to be fabricated from perfect electric conductor (voltage on this conductor is zero), the vector of voltages is filled in by zeros only except of elements
relating to the excitation gap.
Next, I[x] is column vector of currents in the direction x, which is unknown this moment for us. Elements of I[x] are related to the component of current density J[x] as follows
${I}_{x}\left(m,n\right)={J}_{x}\left(m,n\right)B$ ( 4.4A.13 )
(B is the width of the dipole, and consequently the width of the discretization element). Impedance matrix Z[xx] describes contribution of currents I[x] and contribution of charge densities ρ
(expressed from continuity equation (4.4A.4) using x-components of current density J[x] on elements) to voltages U[x] on those elements. Elements of the impedance matrix Z[xx] are known (see the
layer B)
${Z}_{xx}\left(m,n\right)=\frac{j\omega a}{B}{\Gamma }_{A}^{xx}\left(m,n\right)+\frac{1}{j\omega aB}\left[{\Gamma }_{V}\left({m}^{+},{n}^{+}\right)-{\Gamma }_{V}\left({m}^{-},{n}^{+}\right) (4.4A.14
-{\Gamma }_{V}\left({m}^{+},{n}^{-}\right)+{\Gamma }_{V}\left({m}^{-},{n}^{-}\right)\right]$. )
In order to evaluate the impedance matrix Z[xx], values of integrals of Green functions Γ[A]^xx and Γ[V] over the surface of the discretization element have to be computed for various distances
between the source elements (over its surface, current distribution and charge one are integrated) and the observation one (on its surface, electric intensity is computed. Description of the numeric
computation of those integrals in Matlab is given in the layer D.
Matlab program, which performs analysis of the microstrip dipole by the moment method, is described from user's point of view in the layer C. Here, we provide only illustration results obtained by
the program.
Fig. 4.4A.3 Current distribution on half-wavelength symmetric microstrip dipole with planar reflector in the distance of one quarter of wavelength. Distribution computed for 40 cells.
For simplicity, the substrate between the dipole and the reflector is assumed to be of the same parameters as vacuum. If the dipole length equals to one half of wavelength and if the dipole width is
B = λ/1000, following values of input impedance are obtained (in the first row of the table, the number of discretization elements to which antenna is subdivided is given):
Tab. 4.4A.1 Input impedance for different number of discretization elements
N 10 20 30 40
R[vst] [Ω] 95.2 97.3 98.0 98.4
X[vst] [Ω] 72.4 73.6 74.7 75.5
The results show the method to exhibit good stability with respect to the number of discretization elements. Moreover, the input impedance of the microstrip dipole with the reflector computed by
moment method, is close to the results of analytical computations (sinusoidal current distribution J[x] a wire dipole over an infinite planar reflector assumed). For a single wire dipole, radiation
resistance of the antenna (related to the input) equals to R[Σ]= 85.6 Ω.
Investigating approximation of the distribution of x-component of the current along the microstrip dipole, we can show that depicting approximation coefficients from the vector I[x] into a chart.
That way, we obtain a course which is close to a sinusoidal current distribution (see fig. 4.4A.3). | {"url":"https://www.radio.feec.vutbr.cz/raida/multimedia/index.php?nav=4-4-A&lang=en","timestamp":"2024-11-03T01:15:08Z","content_type":"application/xhtml+xml","content_length":"102670","record_id":"<urn:uuid:7ea2334f-0309-4e10-9196-ce86e8fd5b04>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00052.warc.gz"} |
Can you find the next number in this sequence? - Kidpid
Can you find the next number in this sequence?
These are the first five elements of a number sequence. Can you figure out what comes next? Pause here if you want to figure it out for yourself. Answer in: 3 Answer in: 2 Answer in: 1 There is a
pattern here, but it may not be the kind of pattern you think it is. Look at the sequence again and try reading it aloud. Now, look at the next number in the sequence. 3, 1, 2, 2, 1, 1. Pause again
if you’d like to think about it some more. Answer in: 3 Answer in: 2 Answer in: 1 This is what’s known as a look and say sequence.
Unlike many number sequences, this relies not on some mathematical property of the numbers themselves, but on their notation. Start with the left-most digit of the initial number. Now, read out how
many times it repeats in succession followed by the name of the digit itself. Then move on to the next distinct digit and repeat until you reach the end. So the number 1 is read as “one one” written
down the same way we write eleven.
Of course, as part of this sequence, it’s not actually the number eleven, but 2 ones, which we then write as 2 1. That number is then read out as 1 2 1 1, which written out we’d read as one one, one
two, two ones, and so on. These kinds of sequences were first analyzed by mathematician John Conway, who noted they have some interesting properties. For instance, starting with the number 22, yields
an infinite loop of two twos.
But when seeded with any other number, the sequence grows in some very specific ways. Notice that although the number of digits keeps increasing, the increase doesn’t seem to be either linear or
random. In fact, if you extend the sequence infinitely, a pattern emerges. The ratio between the amount of digits in two consecutive terms gradually converges to a single number known as Conway’s
Constant. This is equal to a little over 1.3, meaning that the amount of digits increases by about 30% with every step in the sequence.
What about the numbers themselves? That gets even more interesting. Except for the repeating sequence of 22, every possible sequence eventually breaks down into distinct strings of digits. No matter
what order these strings show up in, each appears unbroken in its entirety every time it occurs.
Conway identified 92 of these elements, all composed only of digits 1, 2, and 3, as well as two additional elements whose variations can end with any digit of 4 or greater. No matter what number the
sequence is seeded with, eventually, it’ll just consist of these combinations, with digits 4 or higher only appearing at the end of the two extra elements, if at all.
Beyond being a neat puzzle, the look and say sequence has some practical applications. For example, run-length encoding, a data compression that was once used for television signals and digital
graphics, is based on a similar concept. The amount of times a data value repeats within the code is recorded as a data value itself. Sequences like this are a good example of how numbers and other
symbols can convey meaning on multiple levels.
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Interpretation of logarithms in a regression
If you do not see the menu on the left please click here
Taken from Introduction to Econometrics from Stock and Watson, 2003, p. 215:
Y=B0 + B1*ln(X) + u ~ A 1% change in X is associated with a change in Y of 0.01*B1
ln(Y)=B0 + B1*X + u ~ A change in X by one unit (∆X=1) is associated with a (exp(B1) - 1)*100 % change in Y
ln(Y)=B0 + B1*ln(X) + u ~ A 1% change in X is associated with a B1% change in Y, so B1 is the elasticity of Y with respect to X.
Out-of sample test
SOURCE: http://www.stata.com/help.cgi?predict
“predict can be used to make in-sample or out-of-sample predictions:
6) predict calculates the requested statistic for all possible
observations, whether they were used in fitting the model or not.
predict does this for the standard options (1) through (3) and
generally does this for estimator-specific options (4).
7) predict newvar if e(sample), ... restricts the prediction to the
estimation subsample.
8) Some statistics make sense only with respect to the estimation
subsample. In such cases, the calculation is automatically
restricted to the estimation subsample, and the documentation for
the specific option states this. Even so, you can still specify
if e(sample) if you are uncertain.
9) predict can make out-of-sample predictions even using other
datasets. In particular, you can
. use ds1
(fit a model)
. use two /* another dataset */
. predict yhat, ... /* fill in the predictions */” | {"url":"https://www.princeton.edu/~otorres/Stata/inference.htm","timestamp":"2024-11-04T13:34:42Z","content_type":"text/html","content_length":"31965","record_id":"<urn:uuid:2f7ab0d2-c6a7-4c91-ae40-c9bd2283554f>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00485.warc.gz"} |
Players being "clutch" when targeting 20 wins -- a follow-up
Players being "clutch" when targeting 20 wins -- a follow-up
In his 2007 essay, "The Targeting Phenomenon," (subscription required), Bill James discussed how there are more single-season 20-game winners than 19-game winners. That's the only time that happens,
that the higher number happens more frequently than the lower number.
This is obviously a case of pitchers targeting the 20-win milestone, but Bill didn't speculate on the actual mechanisms for how the target gets hit. In 2008, I tried to figure it out. But, this past
June, Bill pointed out that my conclusion didn't fit with the evidence:
"... the Birnbaum thesis is that the effect was caused one-half by pitchers with 19 wins getting extra starts, and one-half by poor offensive support by pitchers going for their 21st win, thus
leaving them stuck at 20. But that argument doesn't explain the real life data.
"[If you look closely at the pattern in the numbers,] the bulge in the data is exactly what it should be if 20 is borrowing from 19 -- and is NOT what it should be if 20 is borrowing both from 19
and 21."
(Here's the link. Scroll down to OldBackstop's comment on 6/6/2014.)
So, I rechecked the data, and rethought the analysis, and ... Bill is right, as usual. The basic data was correct, but I didn't do the adjustments properly.
My original study covered 1940 to 2007. This study, though, will cover only 1956 to 2000. That's because I couldn't find my original code and data. The "1956" is what I happened to have handy, and I
decided to stop at 2000 because Bill did.
First, here are the raw numbers of seasons with X wins:
17 wins: 159 18 wins: 132 19 wins: 92 20 wins: 113 21 wins: 56 22 wins: 35 23 wins: 20 24 wins: 20
You can see the bulge we're dealing with: there are way too many 20-win pitchers. And it can't be that the excess comes from the 21-win bucket, because, then, the average of 20 and 21 would stay the
same, and wouldn't be much lower than 19. That can't be right. And, as Bill pointed out, even if only *half* the excess came from the 21 bucket, 20 would still be too big relative to 19.
So, let me try fixing the problem.
In the other study, I checked four ways in which 20 wins could get targeted:
1. Extra starts for pitchers getting close 2. Starters left in the game longer when getting close 3. Extra relief apparances for pitchers getting close 4. Better performance or luck when shooting for
20 than when shooting for 21.
I'll take those one at a time.
1. Extra starts
The old study found that pitchers who eventually wound up at 19 or 20 wins did, in fact, get more late-season starts than others -- about 23 more overall. In this smaller study (1956-2000 instead of
1940-2007), that translates down to maybe 18 extra starts.
That's about 9 extra wins. Let's allocate four of them to pitchers who wound up at 19 instead of 18, and the other five to pitchers who wound up at 20 instead of 19. If we back that out of the actual
data, we get:
18 wins: [S:132:S] 136 19 wins: [S:92:S] 93 20 wins: [S:113:S] 108 21 wins: [S:56:S] 56 (If you're reading this on a newsfeed that doesn't support font variations: the first column is the old values,
which should be struck out.) What happens is: the 18 bucket gets back the four pitchers who won 19 instead. The 19 bucket loses those four pitchers, but gains back the five pitchers who won 20
instead of 19. The 20 bucket loses those five pitchers.
(In the other study, I didn't bother doing this, backing out the effects when I found them, so I wound up taking some of them from the wrong place, which caused the problem Bill found.)
So, we've closed the gap from 21 down to 15.
2. Starters left in the game longer
After I had posted the original study, Dan Rosenheck commented,
"You didn't look at innings per start. I bet managers leave guys with 19 W's in longer if they are tied or trailing in the hope that the lineup will get them a lead before they depart."
I checked, and Dan was right. In a subsequent comment, I figured Dan's explanation accounted for about 10 extra twenty-game winners. Those are all taken from the 19-game bucket, because the effect
occurred only for starters currently pitching with 19 wins.
For this smaller dataset, I'll reduce the effect from 10 seasons to 7.
18 wins: [S:136:S] 136 19 wins: [S:93:S] 100 20 wins: [S:108:S] 101 21 wins: [S:56:S] 56
Now, the bulge is down to 1. We still have a ways to go, if the 19 is to be significantly higher than the 20, but we're getting there.
3. Extra Relief Apparances
The other study listed every pitcher who got a win in relief while nearing 20 wins. Counting only the ones from 1956 to 2000, we get:
3 pitchers winding up at 19 5 pitchers winding up at 20 2 pitchers winding up at 21
Backing those out:
18 wins: [S:136:S] 139 19 wins: [S:100:S] 102 20 wins: [S:101:S] 98 21 wins: [S:56:S] 54
The gap now goes the proper direction, but only slightly.
4. Luck
This was the most surprising finding, and the one responsible for the "getting stuck at 20" phenomenon. Pitchers who already had 20 wins were unusually unlikely to get to 21 in a subsequent start.
Not because they pitched any worse, but because they got poor run support from their offense.
When Bill pointed out the problem, I wondered if the run-support finding was just a programming mistake. It wasn't -- or, at least, when I rewrote the program, from scratch, I got the same result.
For every current starter win level, here are the pitchers' W-L records in those starts, along with the team's average runs scored and allowed:
17 wins: 483-311 .557 4.30-3.61 18 wins: 350-250 .608 4.30-3.61 19 wins: 260-182 .588 4.24-3.56 20 wins: 150-136 .524 3.81-3.54 21 wins: 94- 61 .606 4.49-3.44 22 wins: 59- 23 .720 4.26-2.80
The run support numbers are remarkably consistent -- except at 20 wins. Absent any other explanation, I assume that's just a random fluke.
If we assume that the 20-win starters "should have" gone 171-115 (.598) instead of 150-136 (.524), that makes a difference of 21 wins.
The mistake I made in the previous study was to assume that those wins were all stolen from the "21-win" bucket. Some were, but not all. Some of the unlucky pitchers eventually got past the 20-win
mark; a few, for instance, went on to post 23 wins. In their case, it becomes the 23-win bucket stealing a player from the 24-win bucket.
I checked the breakdown. For every starter who tried for his 21st win but didn't achieve it that game, I calculated where he eventually finished the season. From there, I scaled the totals down to
21, the number of wins lost to bad luck. The result:
20 wins: 9 pitchers 21 wins: 5 pitchers 22 wins: 3 pitchers 23 wins: 1 pitcher 24 wins: 2 pitchers 25+ wins: less than 1 pitcher
So: the 20-win bucket stole 9 pitchers from the 21-win bucket. The 21-win bucket stole 5 pitchers from the 22-win bucket. And so on.
Adjusting the overall numbers gives this:
18 wins: [S:139:S] 139 19 wins: [S:102:S] 102 20 wins: [S:98:S] 89 21 wins: [S:54:S] 50 22 wins: [S:35:S] 33
And that's where we wind up. It's still not quite enough, to judge by Bill's formula and even just the eyeball test. It still looks like there's a little bulge at 20, by maybe five pitchers. If 20
could steal five more pitchers from 19, we'd be at 107/84, which would look about right.
But, we've done OK. We started with a difference of +21 -- that is, 21 more twenty-game winners than nineteen-game winners -- and finished with a difference of -13. That means we found an explanation
for 34 games, out of what looks like a 39-game discrepancy.
Where would the other five come from? I don't know. It could be luck and rounding errors. It could also be that the years 1956-2000 aren't a representative sample of the original study, so we lost a
bit of accuracy when I scaled down. Or, it could be some fifth real factor I haven't thought of.
In any case, here's the final breakdown of the number of "excess" 20-game winners:
-- 5 from getting extra starts; -- 7 from being left in games longer than usual; -- 3 from getting extra relief appearances; -- 9 from bad run support getting them stuck at 20; -- 5 from luck/
rounding/sources unknown.
By the way, one important finding still stands through both studies. Starters didn't seem to pitch any better than normal with their 20th win on the line, so you can't accuse them of trying harder in
the service of a selfish personal goal.
Labels: 20-win, Bill James, clutch, targeting
3 Comments:
At Thursday, December 04, 2014 2:26:00 PM, said...
At Thursday, December 04, 2014 2:30:00 PM, Phil Birnbaum said...
There's no model. There's randomness where I took the larger sample and scaled it down to the smaller sample.
Other than that, the results are based on observing the full population.
At Friday, December 05, 2014 12:08:00 PM, said... | {"url":"http://blog.philbirnbaum.com/2014/12/players-being-clutch-when-targeting-20.html","timestamp":"2024-11-13T21:22:16Z","content_type":"application/xhtml+xml","content_length":"43658","record_id":"<urn:uuid:da8f5dba-8483-4d40-b8e7-5a253d6f04b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00131.warc.gz"} |
Random without repeating numbers
On Thu, 28 Nov 2002, Enrique Franco wrote:
Does anyone knows how to get the random object working without repeating numbers? I mean, for instance, if the random range is 5 I want to get a different number each time.
If you want sequences of five different numbers from 0 to 4, then you can start with the list "0 1 2 3 4", pick a random valid position in that list and remove the number at that position. That gives
you equal access to the 5*4*3*2*1=120 possibilities.
If instead you just want that two consecutive numbers never be the same; then you always keep the previous number (P), and you use 4 as the range, you pick a single random number (R), and you compute
R+(R>=P). The trick is that >= is an indicatrix, which means it's worth 1 if the condition is true, and 0 if it's false; and adding that indicatrix to the number itself will cause the previous number
to be skipped as a possibility.
e.g. P=2, R's might be 0,1,2,3, then R>=P might be 0,0,1,1, then R+(R>=P) might be 0,1,3,4, and there you go, 2 is skipped.
You will also need the [t] object to route R into >= before +.
Mathieu Bouchard http://artengine.ca/matju | {"url":"https://lists.iem.at/hyperkitty/list/pd-list@lists.iem.at/thread/N6PYPRIX7EE4PIFTBW2XPMRKTIYXCNT3/","timestamp":"2024-11-07T00:16:37Z","content_type":"text/html","content_length":"154867","record_id":"<urn:uuid:fd24c636-3806-4fd9-8b85-dea7d3cdd15d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00663.warc.gz"} |
The mythical Bell Curve in Human Resources
Suppose a company's HR department wishes to establish a statistical model for their personnel performance in order to set up outcome predictions for the company's bonus system. It is all too easy to
assume that performance will follow a normal distribution such as this:
Fig. 1: Performance distribution under the Bell Curve assumption.
The assumption stems from the deeply held custom in Psychology and Sociology of using the Bell Curve to model any a priori unknown human trait. Actually, if the company's hiring process is indeed
efficient in selecting better than average people, this assumption is a complete contradiction.
Consider a simplistic hiring process in which performance is assesed by means of a test, so that applicants scoring some fixed minimum at the test are hired, and let us concede for the sake of the
argument that the test is a perfect predictor of performance. The resulting performance distribution is depicted at the figure.
Fig. 2: Performance distribution with a test-based selection process.
The distribution has positive skewness, i.e. its right tail is longer than the left tail. So, the normal approximation with the same mean and variance (shown in dotted line) both overestimates low
performers and underestimates high performers. It also underestimates low-to-normal performers and overestimates normal-to-high performers.
In other hiring process scenario, the best among N candidates is selected. The resulting distribution is depicted at the following figure for N = 10.
Fig. 3: Performance distribution with best-of-10 selection process.
We have positive skewness again, though not as marked as in the prvious case. Skewness grows as N does. Again, the normal approximation results in overestimation of low performers and underestimation
of high performers.
Finally, we consider a two-stage hiring process where applicants are first filtered by a test and then the best candidate out of N is selected.
Fig. 4: Performance distribution with test prefiltering and best-of-10 selection process.
The test filter results in a slightly larger positive skewness. As in previous cases, normal approximation predicts more low performers and less high performers than the real case.
To summarize: hiring process not only results in a personnel performance distribution with a higher than average mean (which is the primary purpose of any hiring process); the distribution will also
have positive skewness, with more excellent and less deficient people than predicted by the Bell Curve.
2 comments :
1. In other words: if HHRR department tries to impose a gaussian distribution to establish staff's yearly perfomance, it (mathematically and quantitatively) means that their hiring process
definitively sucks. The Gaussian distribution demonstrates that the hiring staff underperformes. Hence, a) either most of the workers at HHRR should be at the lower end of the Gaussian
distribution or b) the model should be changed, or c) someone should be fired (no need to say from where).
My vote goes to d): Nothing's gonna change.
2. The Gaussian distribution demonstrates that the hiring staff underperformes.
Yep. In fact, the optimum performance distribution (in terms of talent selection) is a truncated gaussian, with the truncation point as far to the right as possible. | {"url":"https://bannalia.blogspot.com/2009/03/mythical-bell-curve-in-human-resources.html","timestamp":"2024-11-01T20:42:14Z","content_type":"application/xhtml+xml","content_length":"84284","record_id":"<urn:uuid:403fa04e-276e-4096-beab-776f107b24e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00652.warc.gz"} |
why do ships float
Running schedule
Why do ships float?
Archimedes' Principle of Buoyancy Why Do Ships Float?
The Greek Mathematician and inventor Archimedes lived during the 3rd century BC. According to history he was in the bath one day when he discovered the principle of buoyancy which is the reason why
huge Greek ships weighing thousands of pounds could float on water. He noticed that as he lowered himself into the bath, the water displaced by his body overflowed the sides and he realised that
there was a relationship between his weight and the volume of water displaced. It is said that he ran naked into the street yelling "heurEka" which is where we get our word "eureka!" (I found it),
Greek heurEka I have found, from heuriskein to find.
Archimedes was not thinking about ships at the time, he was on a mission to solve a question that was asked of him by King Hieron II of Syracuse, the home of Archimedes which was a Greek city at the
time. The question that the king had asked was about his crown. Was it pure gold or partly silver? Archimedes reasoned that if the crown had any silver in it, it would take up more space than a pure
gold crown of the same weight because silver is not as dense as gold. He compared the crown's volume (measured by the amount of water displaced) with the volume of equal weights of gold and then
silver, he found the answer. He had to inform his king that the crown was not pure gold.
The Buoyancy Principle
Archimedes continued to do more experiments and came up with a buoyancy principle, that a ship will float when the weight of the water it displaces equals the weight of the ship and anything will
float if it is shaped to displace its own weight of water before it reaches the point where it will submerge.This is kind of a technical way of looking at it. A ship that is launched sinks into the
sea until the weight of the water it displaces is equal to its own weight. As the ship is loaded, it sinks deeper, displacing more water, and so the magnitude of the buoyant force continuously
matches the weight of the ship and its cargo.
The Metacenter
Archimedes figured out that the metacenter had to be determined which is a point where an imaginary vertical line (through the center of buoyancy) intersects another imaginary vertical line (through
a new centre of buoyancy) created after the ship is displaced, or tilted, in the water.The center of buoyancy in a floating ship is the point in which all the body parts exactly balance each other
and make each other float. In other words, the metacenter remains directly above the center of buoyancy regardless of the tilt of the floating ship. When a ship tilts, one side displaces more water
than the other side, and the center of buoyancy moves and is no longer directly under the center of gravity; but regardless of the amount of the tilt, the center of buoyancy remains directly below
the metacenter. If the metacenter is above the center of gravity, buoyancy restores stability when the ship tilts. If the metacenter is below the center of gravity, the boat is unstable and capsizes.
Written by Rusty Russell | {"url":"http://www.pmpba.org/pmpba_2024update_009.htm","timestamp":"2024-11-13T14:50:24Z","content_type":"text/html","content_length":"9573","record_id":"<urn:uuid:82ebb7aa-715e-42f5-8578-f7fd64dc8081>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00454.warc.gz"} |
E. Özsoy Et Al. , "Is testosterone deficiency a predictive factor for recurrence of urethral stricture?," Andrology , 2023
Özsoy, E. Et Al. 2023. Is testosterone deficiency a predictive factor for recurrence of urethral stricture?. Andrology .
Özsoy, E., KUTLUHAN, M. A., Tokuç, E., Kayar, R., Demir, S., Akyüz, M., ... Öztürk, M. İ.(2023). Is testosterone deficiency a predictive factor for recurrence of urethral stricture?. Andrology .
Özsoy, Emrah Et Al. "Is testosterone deficiency a predictive factor for recurrence of urethral stricture?," Andrology , 2023
Özsoy, Emrah Et Al. "Is testosterone deficiency a predictive factor for recurrence of urethral stricture?." Andrology , 2023
Özsoy, E. Et Al. (2023) . "Is testosterone deficiency a predictive factor for recurrence of urethral stricture?." Andrology .
@article{article, author={Emrah Özsoy Et Al. }, title={Is testosterone deficiency a predictive factor for recurrence of urethral stricture?}, journal={Andrology}, year=2023} | {"url":"https://avesis.aybu.edu.tr/activitycitation/index/1/5a7d7331-02b7-43ae-b970-a5e9e44a95c6","timestamp":"2024-11-11T00:32:07Z","content_type":"text/html","content_length":"10711","record_id":"<urn:uuid:36744f38-cb45-4206-8868-70795defb1f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00655.warc.gz"} |
An investigator compares the durability of two different
compounds used in the manufacture of a certain...
An investigator compares the durability of two different compounds used in the manufacture of a certain...
An investigator compares the durability of two different compounds used in the manufacture of a certain automobile brake lining. A sample of 124 brakes using Compound 1 yields an average brake life
of 40,704 miles. A sample of 163 brakes using Compound 2 yields an average brake life of 43,525 miles. Assume that the population standard deviation for Compound 1 is 2022 miles, while the population
standard deviation for Compound 2 is 4792 miles. Determine the 90% confidence interval for the true difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2.
Given that,
For Compound 1 :
For Compound 2 :
A 90% confidence level has significance level of 0.10 and critical value is,
The 90% confidence interval for the (μ1 - μ2) is
Therefore, the 90% confidence interval for the true difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2 is (-3507, -2135) miles. | {"url":"https://justaaa.com/statistics-and-probability/410887-an-investigator-compares-the-durability-of-two","timestamp":"2024-11-04T17:07:27Z","content_type":"text/html","content_length":"44183","record_id":"<urn:uuid:f0345d1f-8270-43dd-b15e-2605974aa7ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00137.warc.gz"} |
How to Convert Milliseconds to Seconds: Simple and Efficient Calculation - [Updated November 2024 ]
How to Convert Milliseconds to Seconds: Simple and Efficient Calculation
convert milliseconds, convert milliseconds seconds, milliseconds seconds, number milliseconds, value seconds
Milliseconds (ms) and seconds are two common units of measurement when it comes to timekeeping in a digital world. Whether you are clocking the speed of a computer program or need to convert a
certain time interval from milliseconds to seconds, it is important to understand the conversion process.
In mathematics, the equation for converting milliseconds to seconds is quite simple. It involves dividing the number of milliseconds by 1000, since there are 1000 milliseconds in one second. This
straightforward formula allows you to quantify time in a more manageable and easily understandable unit.
Calculating milliseconds to seconds can be done manually, but for efficiency and accuracy, it is recommended to use a calculator or an online converter. These tools take care of the math for you,
allowing you to focus on other important tasks.
Understanding the conversion between milliseconds and seconds is essential in many fields, particularly in the realm of technology where precise timing is crucial. Being able to convert between these
units allows you to accurately measure and compute time intervals in various applications, from programming to data analysis.
What are Milliseconds?
The millisecond is a common unit of measurement in timekeeping. It is a decimal subdivision of the second, where 1 millisecond (ms) is equal to 0.001 seconds. The prefix “milli” represents
one-thousandth, making the millisecond a small unit for quantifying time.
Millisecond is widely used in various fields that require precise timing, such as timers, clocks, and scientific experiments. In mathematics and physics, milliseconds are often used to compute
durations or measure the speed of processes with high accuracy.
To convert milliseconds to seconds, you can use a simple equation or formula. Since there are 1000 milliseconds in a second, you can divide the number of milliseconds by 1000 to calculate the
equivalent value in seconds. For example, if you have 5000 milliseconds, the calculation would be: 5000 ms ÷ 1000 = 5 seconds.
Calculating milliseconds to seconds can be done manually or with the help of digital tools like calculators or programming scripts. The process involves dividing the number of milliseconds by 1000,
which moves the decimal point three places to the left, effectively converting milliseconds to seconds.
In summary, milliseconds are a unit of measurement in timekeeping that represent a small fraction of a second. They are commonly used in various fields to measure and quantify time. By converting
milliseconds to seconds, you can accurately compute durations and perform time-related calculations.
What are Seconds?
Seconds are a unit of time measurement in the field of mathematics and timekeeping. They are used to quantify short periods of time and are an essential component of the standard unit of time, which
includes hours, minutes, and milliseconds.
In the world of timekeeping, a second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the
cesium-133 atom. This definition ensures that the length of a second remains consistent and accurate across different types of clocks and timekeeping devices.
Computing and calculating time durations often involve the conversion of milliseconds to seconds. This conversion can be done using a simple formula or by using an online calculator or software that
automates the process. By knowing the number of milliseconds, one can easily determine the equivalent duration in seconds.
A common equation to convert milliseconds to seconds is:
1. Divide the number of milliseconds by 1000 to get the equivalent value in seconds.
2. For example, if there are 5000 milliseconds, dividing it by 1000 would yield 5 seconds.
This conversion is particularly useful in various fields such as programming, sports timing, scientific experiments, and everyday tasks that require precise time measurements. Whether it’s a
countdown timer, a digital clock, or a stopwatch, milliseconds and seconds play a crucial role in accurately measuring and keeping track of time.
Method 1: Manual Calculation
To convert milliseconds to seconds manually, you can follow a simple calculation process. By definition, one second is equal to 1000 milliseconds. Therefore, to convert a given measurement of
milliseconds into seconds, you need to divide the value by 1000.
Let’s say you have a timer that measures time in milliseconds and you want to convert that to seconds. You can use a calculator or do the math manually. For example, if your timer shows 5000
milliseconds, dividing this value by 1000 will give you 5 seconds.
The manual conversion method involves basic mathematics. The equation or formula you need to use for converting milliseconds to seconds is:
Seconds = Milliseconds ÷ 1000
You can apply this formula to any given measurement of milliseconds to calculate the equivalent time in seconds. Simply divide the number of milliseconds by 1000, and you will have the result in
This manual calculation method is useful for various purposes, especially in the field of timekeeping and clocking. It allows you to quantify time and convert measurements from one unit to another,
providing a practical way to express time in a different format.
Step 1: Divide Milliseconds by 1000
To convert milliseconds to seconds, you need to quantify the number of milliseconds you have and divide it by 1000. This step is essential as it helps you convert the millisecond measurement into its
equivalent in seconds. By dividing the value by 1000, you are essentially converting it from the millisecond unit to the second unit.
Mathematically, this step can be represented by the equation:
seconds = milliseconds / 1000
For example, if you have 3000 milliseconds, you can compute its equivalent value in seconds using this step. Dividing 3000 by 1000 gives you 3 seconds.
This conversion is commonly used in various fields such as timekeeping, where precise measurements of time are crucial. By converting milliseconds to seconds, you can easily calculate durations or
interval timings displayed on devices like timers or clocks.
Understanding and applying this step is essential for anyone who needs to convert millisecond measurements to seconds accurately. By knowing the formula and the necessary computations involved, you
can quickly and easily calculate the conversion without the need for a specialized calculator or tool.
Step 2: Round the Result
After quantifying the time in milliseconds, the next step is to convert it into seconds. Since seconds are a smaller unit of time compared to milliseconds, it is important to round the result
appropriately to ensure accuracy.
When rounding the result, it is crucial to consider the number of digits after the decimal point. In mathematics, rounding involves adjusting a number to its closest whole number or a specified
number of decimal places.
To compute the conversion from milliseconds to seconds, the formula is simply dividing the number of milliseconds by 1000. The resulting quotient represents the equivalent time in seconds.
However, when calculating timekeeping or clocking measurements, it is common to round the seconds to two decimal places. This provides a more precise representation of the time, especially for timers
or clocks that require accurate tracking of seconds.
Here is an example to illustrate the rounding process:
1. Suppose we have a measurement of 3250 milliseconds (ms).
2. To convert this to seconds, we divide 3250 by 1000, resulting in 3.25 seconds.
3. Rounding the result to two decimal places, we get 3.25 seconds.
By rounding the result, we ensure that the seconds value is represented in a more concise and accurate manner. This facilitates easy understanding and precise time measurements in various scenarios
where milliseconds need to be converted to seconds.
Step 3: Add the Result to Seconds
Now that we have calculated the milliseconds to seconds conversion, we can add the result to our current value of seconds. This step allows us to obtain the total time in seconds, combining the
measurements from both the timer and the milliseconds.
To add the result to seconds, we can use a simple mathematical equation. For example, if our original measure of time was 5 seconds and we converted 500 milliseconds to seconds (0.5 seconds), the
total time would be 5.5 seconds. We add the converted milliseconds to our initial measurement to obtain the final result.
In terms of coding or using a calculator specifically designed for timekeeping, we can use a formula or function to compute this addition. For instance, we can write a code snippet in a programming
language to perform the calculation automatically. Additionally, online conversion tools often have built-in functions to add the converted milliseconds to the seconds.
It is important to note that when adding decimals to a whole number of seconds, we have to consider the decimal place. In our example, the converted milliseconds result in a decimal of 0.5. When
adding the decimals, we need to make sure they align correctly. In this case, we add the 0.5 to the rightmost digit of the original seconds, resulting in 5.5 seconds.
By adding the converted milliseconds to the seconds, we can accurately quantify the time in a unified unit of measurement. This step is crucial in time calculations, as it allows us to account for
the precision provided by milliseconds and obtain a more precise clocking of time.
Method 2: Using an Online Converter
If you find yourself often needing to convert milliseconds to seconds and you don’t want to bother with manual calculations, there is a convenient method available: using an online converter. Online
converters are tools that allow you to input a value in one unit and get the equivalent value in another unit. In this case, you can input the number of milliseconds and get the corresponding value
in seconds.
Online converters are particularly useful for anyone who needs to convert milliseconds to seconds quickly and accurately. They eliminate the need for complicated math equations, allowing you to get
the desired result in just a few simple steps. All you have to do is enter the number of milliseconds you want to convert, and the converter will handle the rest.
Using an online converter is straightforward. Simply find a reputable converter, such as one provided by a reliable source or a well-known website. Then, input the number of milliseconds you want to
convert into the designated field. Click on the “Convert” button, and the converter will immediately display the equivalent value in seconds.
Online converters allow you to convert any number of milliseconds to seconds, whether it’s a small or large value. They can handle multiple digits and decimal places, ensuring accurate conversions.
The conversion process is quick and reliable, making it a convenient option for anyone who frequently needs to convert measurements of time.
Overall, using an online converter is an efficient and reliable way to convert milliseconds to seconds. It eliminates the need for manual calculations and time-consuming math equations. By taking
advantage of the available tools, you can save time and obtain accurate conversions with just a few clicks.
Step 1: Find a Reliable Online Converter
When it comes to converting measurements and units in math and science, accuracy and reliability are key. Before you start calculating and converting milliseconds to seconds, it is important to find
a reliable online converter that you can trust to provide accurate results.
There are several websites and tools available that specialize in unit conversions and mathematical calculations. These online converters are designed to compute equations and formulas quickly and
accurately, making it easy to convert milliseconds to seconds with just a few clicks.
By using a reliable online converter, you can ensure that your conversion is precise and free from errors. These tools usually have user-friendly interfaces and clear instructions that make the
conversion process straightforward even for those who are not familiar with the intricacies of mathematics.
Whether you need to convert milliseconds to seconds for a timer, clock, or any other timekeeping device, finding a trustworthy online converter will streamline the process and provide you with
accurate results. So, instead of manually calculating and converting the milliseconds to seconds, rely on the convenience and accuracy of an online converter to quantify the time in seconds.
Step 2: Input the Value in Milliseconds
Once you have understood the concept of milliseconds and seconds and have decided to convert the value, the next step is to input the value in milliseconds. Before proceeding with this step, make
sure you have a clear understanding of the units and the numerical value you want to convert.
To input the value in milliseconds, you will typically use a calculator or a mathematical equation. The calculator or equation will allow you to compute the conversion and quantify the value in
seconds. This step is crucial as it helps you convert the measurement from milliseconds to seconds, making it easier to understand and work with.
When inputting the value in milliseconds, it is important to consider the number of digits and the accuracy required. Depending on the timekeeping device or timer you are using, you may need to round
the value or provide it to a specific number of decimal places.
There is a simple formula for converting milliseconds to seconds: divide the value in milliseconds by 1000. This will give you the equivalent value in seconds. For example, if you have 3000
milliseconds, dividing it by 1000 will give you 3 seconds. This equation makes the conversion process quick and easy, allowing you to quickly get the desired result.
Step 3: Convert to Seconds
In order to convert milliseconds to seconds, you can use a simple formula. By dividing the number of milliseconds by 1000, you can calculate the equivalent in seconds. This conversion is essential in
various fields where timekeeping and measurement are critical, such as in digital clocks, timers, and scientific experiments.
To convert milliseconds to seconds, simply divide the number of milliseconds by 1000:
seconds = milliseconds / 1000
For example, if you have 5000 milliseconds, you would compute:
seconds = 5000 / 1000 = 5
This equation allows you to convert milliseconds to seconds easily, as it quantifies the time in units that are more commonly used and understood. It’s a fundamental mathematical operation that can
be used in various applications requiring precise time calculations.
Method 3: Using a Programming Language
One of the most efficient ways to convert milliseconds to seconds is by using a programming language. Many programming languages, such as Python, Java, or C++, offer built-in functions or libraries
that make this conversion quick and easy.
Using a programming language allows you to write a formula or code that takes the number of milliseconds as input and computes the equivalent value in seconds. This method is particularly useful when
dealing with large sets of data or when you need to convert measurements in real-time.
Programming languages provide powerful tools for handling numeric operations and performing mathematical computations. By using the appropriate language-specific functions or methods, you can easily
convert milliseconds to seconds without the need for manual calculation.
For example, in Python, you can use the following code to convert milliseconds to seconds:
milliseconds = 2000
seconds = milliseconds / 1000
print("The equivalent value in seconds is:", seconds)
This code calculates and prints the equivalent value in seconds based on the number of milliseconds provided. It divides the number of milliseconds by 1000 to get the conversion factor, which is then
used to compute the value in seconds.
Using a programming language for this conversion also offers additional advantages such as the ability to automate the process, handle different units of measurement, and quantify time durations
accurately. These languages are designed to handle various aspects of timekeeping and clocking, making them ideal for precise mathematical calculations.
Step 1: Choose a Programming Language
When it comes to converting milliseconds to seconds, the first step is to choose a programming language that can handle the task efficiently and accurately. There are several popular programming
languages to choose from, including Python, Java, JavaScript, C++, and more. Each language has its own set of features and capabilities that make it suitable for different tasks, so it’s important to
select a language that you are familiar with or comfortable learning.
Once you have chosen a programming language, you can begin the process of writing the necessary code to convert milliseconds to seconds. This code will typically involve using variables to store the
value of the milliseconds and performing a mathematical calculation to convert them to seconds.
For example, in Python, you can use the following equation to convert milliseconds to seconds:
seconds = milliseconds / 1000
Here, the variable “milliseconds” represents the value you want to convert, and the variable “seconds” will store the converted value. The division operator “/” is used to divide the number of
milliseconds by 1000, as there are 1000 milliseconds in one second.
Other programming languages may have slightly different syntax or functions for performing this calculation, but the underlying principle remains the same. By using the appropriate formula or
equation, you can accurately convert milliseconds to seconds in any programming language.
It’s important to note that the formula for converting milliseconds to seconds is based on simple mathematics and can be easily understood and implemented. This makes it a good starting point for
beginners who are learning how to work with time measurements and perform basic calculations in their chosen programming language.
In conclusion, the first step in converting milliseconds to seconds is to choose a programming language that you are comfortable with. Once you have selected a language, you can use the appropriate
formula or equation to perform the conversion accurately. With a timer and some basic math skills, you can easily quantify and compute the equivalent value in seconds from a given number of
milliseconds in no time!
Step 2: Declare Variables
1. To start the conversion process, you’ll need to set up the necessary variables in your code.
First, declare a variable to store the value of the milliseconds you want to convert. This variable, which we’ll call ms, will hold the measurement of time that you want to convert into seconds.
For example, if you have a timer or clocking mechanism that is measuring the passage of time in milliseconds, you can assign the current reading to the ms variable.
2. Next, you need to declare another variable to store the converted value in seconds.
We can call this variable seconds. This variable will be used to hold the resulting value after the conversion calculation is performed.
3. Once you have declared these variables, you can move on to the next step of the conversion process.
The variables ms and seconds will be used in the subsequent steps to compute the conversion from milliseconds to seconds.
By using these variables, you can quantify the measurement of time and perform the necessary mathematical operations to calculate the equivalent value in seconds.
With the variables set up, you are now ready to proceed to the next step and begin the conversion process.
Step 3: Write the Calculation Code
To convert milliseconds to seconds, we need to write the calculation code that will perform the conversion. In this step, we will use basic mathematical operations to compute the conversion.
First, let’s define the variables:
• milliseconds: This variable will store the value of milliseconds that we want to convert.
• seconds: This variable will store the calculated value of seconds after the conversion.
Next, let’s write the formula:
The formula to convert milliseconds to seconds is:
seconds = milliseconds / 1000
• milliseconds represents the number of milliseconds we want to convert.
• 1000 is the conversion factor that converts milliseconds to seconds. There are 1000 milliseconds in one second.
Now, let’s write the code:
1. Declare the variables:
int milliseconds;
int seconds;
2. Prompt the user to enter the value of milliseconds:
printf("Enter the value of milliseconds: ");
3. Read the input value from the user:
scanf("%d", &milliseconds);
4. Perform the calculation:
seconds = milliseconds / 1000;
5. Print the result:
printf("%d milliseconds is equal to %d seconds.\n", milliseconds, seconds);
That’s it! The code above will take a value in milliseconds, convert it to seconds using the formula, and print the result.
Summary of Methods
There are several methods available to convert milliseconds to seconds for various timekeeping applications. These methods utilize mathematical equations to compute the conversion and quantify the
measurement in seconds.
1. Direct Conversion
The most straightforward method to convert milliseconds to seconds is to divide the number of milliseconds by 1000. This method is based on the fact that there are 1000 milliseconds in one second.
For example, to convert 5000 milliseconds to seconds, you divide 5000 by 1000, resulting in 5 seconds.
2. Formula-Based Conversion
An alternative method is to use a formula-based approach. This method involves multiplying the number of milliseconds by a conversion factor of 0.001. The formula to calculate seconds from
milliseconds is: seconds = milliseconds * 0.001. For example, to convert 2500 milliseconds to seconds, you multiply 2500 by 0.001, resulting in 2.5 seconds.
3. Calculator or Online Tools
For quick and easy conversion, one can use a calculator or online tools specifically designed for converting milliseconds to seconds. These tools typically require you to enter the number of
milliseconds, and they instantly provide the equivalent value in seconds. This method is convenient when dealing with larger numbers or when multiple conversions need to be done.
4. Conversion Table
Another approach is to refer to a conversion table that lists common values for milliseconds and their corresponding values in seconds. This method can be useful when dealing with specific time
intervals or when you need to quickly look up the conversion without performing any calculations.
Overall, whether you use a direct conversion, a formula-based approach, calculators, or reference tables, these methods provide efficient and accurate means to convert milliseconds to seconds in
various timekeeping applications.
Choose the Right Method for Your Needs
If you need to calculate time in milliseconds and convert it into seconds, there are several methods you can choose from. Whether you are working with measurements, using a timer, or simply need to
convert time between units, it is important to find the method that suits your needs best.
One of the most common methods is using the basic mathematical formula to convert milliseconds to seconds. This formula involves dividing the number of milliseconds by 1000, since there are 1000
milliseconds in a second. It is a straightforward calculation that can be easily done by hand or with the help of a calculator.
If you are more comfortable with technology, there are various online converters available that can quickly and accurately convert milliseconds to seconds. These converters often provide a
user-friendly interface, where you can simply enter the number of milliseconds and receive the corresponding value in seconds. They can be especially helpful when dealing with large numbers or when
precision is crucial.
Another option is to use a programming language or software to perform the conversion. Many programming languages, such as Python or JavaScript, have built-in functions or methods that allow you to
convert milliseconds to seconds without having to write the formula yourself. This can be beneficial if you are working with a large amount of data or need to automate the conversion process.
Ultimately, the method you choose will depend on your specific needs and preferences. Whether you prefer to do the math manually, use an online converter, or rely on programming, it is important to
select the method that will allow you to accurately and efficiently convert milliseconds to seconds.
FAQ about topic “How to Convert Milliseconds to Seconds: Simple and Efficient Calculation”
Why would I need to convert milliseconds to seconds?
There are several reasons why you may need to convert milliseconds to seconds. One common reason is when working with time-related data in programming or data analysis. Milliseconds are often used to
represent precise time intervals, but for many applications, it is more convenient to work with seconds. Converting milliseconds to seconds allows for easier calculations and comparisons.
What is the formula for converting milliseconds to seconds?
The formula for converting milliseconds to seconds is simple. You just need to divide the number of milliseconds by 1000. Since there are 1000 milliseconds in a second, dividing the value by 1000
gives you the equivalent in seconds. For example, if you have 5000 milliseconds, you would divide 5000 by 1000 to get 5 seconds.
Can I convert milliseconds to seconds using a calculator?
Yes, you can definitely use a calculator to convert milliseconds to seconds. To do this, you would input the number of milliseconds, press the division button, and then enter 1000. The calculator
will give you the result in seconds. This method is especially useful for converting large or complex values.
Are there any online tools or websites that can convert milliseconds to seconds?
Yes, there are many online tools and websites available that can convert milliseconds to seconds for you. Simply search for “milliseconds to seconds converter” on your favorite search engine, and you
will find multiple options. These tools usually have a simple interface where you can input the number of milliseconds and instantly get the result in seconds.
Can I convert milliseconds to seconds manually without using a calculator?
Yes, you can definitely convert milliseconds to seconds manually without using a calculator. To do this, you need to divide the number of milliseconds by 1000. If the result is a whole number, that
is the equivalent in seconds. If the result has a decimal part, you can round it to the desired number of decimal places. It may take a bit longer than using a calculator, but it is a good exercise
for practicing your math skills.
Leave a Comment | {"url":"https://digitalgadgetwave.com/how-to-convert-milliseconds-to-seconds-simple-and/","timestamp":"2024-11-10T15:53:19Z","content_type":"text/html","content_length":"141786","record_id":"<urn:uuid:76554ca4-5d70-43f8-a9c8-adfd6fec19d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00750.warc.gz"} |
The INDIRECT formula is used to return a cell reference specified by a string. This can be useful when you need to dynamically reference a cell based on the value in another cell. The INDIRECT
formula can also be used to reference cells in other sheets or workbooks.
Use the INDIRECT formula with the syntax shown below, it has 1 required parameter and 1 optional parameter:
=INDIRECT(cell_reference_as_string, [is_A1_notation])
1. cell_reference_as_string (required):
The cell reference as a string, which can be in A1 or R1C1 notation. If the cell reference refers to a cell in another sheet or workbook, the sheet or workbook name must be included in the
2. is_A1_notation (optional):
A boolean value that specifies whether the cell reference is in A1 notation (TRUE) or R1C1 notation (FALSE). If omitted, the formula assumes A1 notation.
Here are a few example use cases that explain how to use the INDIRECT formula in Google Sheets.
Dynamic referencing
You can use INDIRECT to dynamically reference a cell based on the value in another cell. For example, if you have a dropdown list in cell A1 and want to display the value in the corresponding cell in
column B, you can use INDIRECT to reference the cell in column B based on the value in A1.
Referencing cells in other sheets
You can use INDIRECT to reference cells in other sheets. For example, if you have a sheet named 'Sheet2' and want to reference cell A1 in that sheet, you can use INDIRECT with the string 'Sheet2!A1'.
Referencing cells in other workbooks
You can use INDIRECT to reference cells in other workbooks. For example, if you have a workbook named 'Workbook2' and want to reference cell A1 in a sheet named 'Sheet1' in that workbook, you can use
INDIRECT with the string '[Workbook2]Sheet1!A1'.
Common Mistakes
INDIRECT not working? Here are some common mistakes people make when using the INDIRECT Google Sheets Formula:
Incorrect cell reference format
If the cell reference provided is not in the correct format (i.e. A1 notation or R1C1 notation), the formula will return an error. Make sure to use the correct format for the reference.
Missing quotation marks
If the cell reference provided is not enclosed in quotation marks, the formula will return an error. Make sure to enclose the reference in quotation marks.
Invalid sheet name
If the sheet name provided in the cell reference is not valid, the formula will return an error. Make sure to use the correct sheet name.
Using a range instead of a single cell reference
If a range of cells is provided as the cell reference, the formula will return an error. Make sure to use a single cell reference.
Using a cell reference in a different sheet
If the cell reference provided is in a different sheet, the formula will return an error. Use the sheet name followed by an exclamation mark before the cell reference to reference a cell in a
different sheet.
Related Formulas
The following functions are similar to INDIRECT or are often used with it in a formula:
• ADDRESS
The ADDRESS formula in Google Sheets returns a cell reference as a string, given the row and column numbers. It can also return an absolute or relative reference, and can include sheet name in
the reference. This formula is most commonly used to create dynamic references in other formulas or to create a cell reference based on certain criteria.
• CELL
The CELL formula returns information about the formatting, location, or contents of a cell. The type of information returned depends on the value of the info_type argument.
• OFFSET
The OFFSET formula in Google Sheets returns a cell or range of cells that is a specified number of rows and columns from a starting cell reference. This formula is most commonly used to create
dynamic ranges that can expand or contract as data is added or removed from a sheet.
Learn More
You can learn more about the INDIRECT Google Sheets function on Google Support. | {"url":"https://checksheet.app/google-sheets-formulas/indirect/","timestamp":"2024-11-10T06:11:55Z","content_type":"text/html","content_length":"45966","record_id":"<urn:uuid:6f3ca24a-83ff-463d-b4ef-376119138338>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00817.warc.gz"} |
Why not adopt me?
This distribution is up for adoption! If you're interested then please contact the PAUSE module admins via
Math::FFT - Perl module to calculate Fast Fourier Transforms
version 1.36
use Math::FFT;
my $PI = 3.1415926539;
my $N = 64;
my $series = [map { sin(4*$_*$PI/$N) + cos(6*$_*$PI/$N) } 0 .. $N-1];
my $fft = Math::FFT->new($series);
my $coeff = $fft->rdft();
my $spectrum = $fft->spctrm;
my $original_data = $fft->invrdft($coeff);
my $other_series =
[map { sin(16*$_*$PI/$N) + cos(8*$_*$PI/$N) } 0 .. $N-1];
my $other_fft = $fft->clone($other_series);
my $other_coeff = $other_fft->rdft();
my $correlation = $fft->correl($other_fft);
This module implements some algorithms for calculating Fast Fourier Transforms for one-dimensional data sets of size 2^n. The data, assumed to arise from a constant sampling rate, is represented by
an array reference $data (as described in the methods below), which is then used to create a Math::FFT object as
my $fft = Math::FFT->new($data);
The methods available include the following.
The constructor. Pass it an array of numbers, with a length that is a power of 2.
This calculates the complex discrete Fourier transform for a data set x[j]. Here, $data is a reference to an array data[0...2*n-1] holding the data
data[2*j] = Re(x[j]),
data[2*j+1] = Im(x[j]), 0<=j<n
An array reference $coeff is returned consisting of
coeff[2*k] = Re(X[k]),
coeff[2*k+1] = Im(X[k]), 0<=k<n
X[k] = sum_j=0^n-1 x[j]*exp(2*pi*i*j*k/n), 0<=k<n
Calculates the inverse complex discrete Fourier transform on a data set x[j]. If $coeff is not given, it will be set equal to an earlier call to $fft->cdft(). $coeff is a reference to an array
coeff[0...2*n-1] holding the data
coeff[2*j] = Re(x[j]),
coeff[2*j+1] = Im(x[j]), 0<=j<n
An array reference $orig_data is returned consisting of
orig_data[2*k] = Re(X[k]),
orig_data[2*k+1] = Im(X[k]), 0<=k<n
where, excluding the scale,
X[k] = sum_j=0^n-1 x[j]*exp(-2*pi*i*j*k/n), 0<=k<n
A scaling $orig_data->[$i] *= 2.0/$n is then done so that $orig_data coincides with the original $data.
This calculates the real discrete Fourier transform for a data set x[j]. On input, $data is a reference to an array data[0...n-1] holding the data. An array reference $coeff is returned
consisting of
coeff[2*k] = R[k], 0<=k<n/2
coeff[2*k+1] = I[k], 0<k<n/2
coeff[1] = R[n/2]
R[k] = sum_j=0^n-1 data[j]*cos(2*pi*j*k/n), 0<=k<=n/2
I[k] = sum_j=0^n-1 data[j]*sin(2*pi*j*k/n), 0<k<n/2
Calculates the inverse real discrete Fourier transform on a data set coeff[j]. If $coeff is not given, it will be set equal to an earlier call to $fft->rdft(). $coeff is a reference to an array
coeff[0...n-1] holding the data
coeff[2*j] = R[j], 0<=j<n/2
coeff[2*j+1] = I[j], 0<j<n/2
coeff[1] = R[n/2]
An array reference $orig_data is returned where, excluding the scale,
orig_data[k] = (R[0] + R[n/2]*cos(pi*k))/2 +
sum_j=1^n/2-1 R[j]*cos(2*pi*j*k/n) +
sum_j=1^n/2-1 I[j]*sin(2*pi*j*k/n), 0<=k<n
A scaling $orig_data->[$i] *= 2.0/$n is then done so that $orig_data coincides with the original $data.
Computes the discrete cosine transform on a data set data[0...n-1] contained in an array reference $data. An array reference $coeff is returned consisting of
coeff[k] = C[k], 0<=k<n
C[k] = sum_j=0^n-1 data[j]*cos(pi*(j+1/2)*k/n), 0<=k<n
Computes the inverse discrete cosine transform on a data set coeff[0...n-1] contained in an array reference $coeff. If $coeff is not given, it will be set equal to an earlier call to $fft->ddct
(). An array reference $orig_data is returned consisting of
orig_data[k] = C[k], 0<=k<n
where, excluding the scale,
C[k] = sum_j=0^n-1 coeff[j]*cos(pi*j*(k+1/2)/n), 0<=k<n
A scaling $orig_data->[$i] *= 2.0/$n is then done so that $orig_data coincides with the original $data.
Computes the discrete sine transform of a data set data[0...n-1] contained in an array reference $data. An array reference $coeff is returned consisting of
coeff[k] = S[k], 0<k<n
coeff[0] = S[n]
S[k] = sum_j=0^n-1 data[j]*sin(pi*(j+1/2)*k/n), 0<k<=n
Computes the inverse discrete sine transform of a data set coeff[0...n-1] contained in an array reference $coeff, arranged as
coeff[j] = A[j], 0<j<n
coeff[0] = A[n]
If $coeff is not given, it will be set equal to an earlier call to $fft->ddst(). An array reference $orig_data is returned consisting of
orig_data[k] = S[k], 0<=k<n
where, excluding a scale,
S[k] = sum_j=1^n A[j]*sin(pi*j*(k+1/2)/n), 0<=k<n
The scaling $a->[$i] *= 2.0/$n is then done so that $orig_data coincides with the original $data.
Computes the real symmetric discrete Fourier transform of a data set data[0...n] contained in the array reference $data. An array reference $coeff is returned consisting of
coeff[k] = C[k], 0<=k<=n
C[k] = sum_j=0^n data[j]*cos(pi*j*k/n), 0<=k<=n
Computes the inverse real symmetric discrete Fourier transform of a data set coeff[0...n] contained in the array reference $coeff. If $coeff is not given, it will be set equal to an earlier call
to $fft->dfct(). An array reference $orig_data is returned consisting of
orig_data[k] = C[k], 0<=k<=n
where, excluding the scale,
C[k] = sum_j=0^n coeff[j]*cos(pi*j*k/n), 0<=k<=n
A scaling $coeff->[0] *= 0.5, $coeff->[$n] *= 0.5, and $orig_data->[$i] *= 2.0/$n is then done so that $orig_data coincides with the original $data.
Computes the real anti-symmetric discrete Fourier transform of a data set data[0...n-1] contained in the array reference $data. An array reference $coeff is returned consisting of
coeff[k] = C[k], 0<k<n
C[k] = sum_j=0^n data[j]*sin(pi*j*k/n), 0<k<n
(coeff[0] is used for a work area)
Computes the inverse real anti-symmetric discrete Fourier transform of a data set coeff[0...n-1] contained in the array reference $coeff. If $coeff is not given, it will be set equal to an
earlier call to $fft->dfst(). An array reference $orig_data is returned consisting of
orig_data[k] = C[k], 0<k<n
where, excluding the scale,
C[k] = sum_j=0^n coeff[j]*sin(pi*j*k/n), 0<k<n
A scaling $orig_data->[$i] *= 2.0/$n is then done so that $orig_data coincides with the original $data.
See "CLONING" below.
See "Convolution" below.
See "Correlation" below.
See "Deconvolution" below.
For internal use. Don't use directly.
For internal use. Don't use directly.
See "Power Spectrum" below.
The algorithm used in the transforms makes use of arrays for a work area and for a cos/sin lookup table dependent only on the size of the data set. These arrays are initialized when the Math::FFT
object is created and then are populated when a transform method is first invoked. After this, they persist for the lifetime of the object.
This aspect is exploited in a cloning method; if a Math::FFT object is created for a data set $data1 of size N:
$fft1 = Math::FFT->new($data1);
then a new Math::FFT object can be created for a second data set $data2 of the same size N by
$fft2 = $fft1->clone($data2);
The $fft2 object will copy the reuseable work area and lookup table calculated from $fft1.
This module includes some common applications - correlation, convolution and deconvolution, and power spectrum - that arise with real data sets. The conventions used here follow that of Numerical
Recipes in C, by Press, Teukolsky, Vetterling, and Flannery, in which further details of the algorithms are given. Note in particular the treatment of end effects by zero padding, which is assumed to
be done by the user, if required.
The correlation between two functions is defined as
Corr(t) = | ds g(s+t) h(s)
This may be calculated, for two array references $data1 and $data2 of the same size $n, as either
$fft1 = Math::FFT->new($data1);
$fft2 = Math::FFT->new($data2);
$corr = $fft1->correl($fft2);
or as
$fft1 = Math::FFT->new($data1);
$corr = $fft1->correl($data2);
The array reference $corr is returned in wrap-around order - correlations at increasingly positive lags are in $corr->[0] (zero lag) on up to $corr->[$n/2-1], while correlations at increasingly
negative lags are in $corr->[$n-1] on down to $corr->[$n/2]. The sign convention used is such that if $data1 lags $data2 (that is, is shifted to the right), then $corr will show a peak at
positive lags.
The convolution of two functions is defined as
Convlv(t) = | ds g(s) h(t-s)
This is similar to calculating the correlation between the two functions, but typically the functions here have a quite different physical interpretation - one is a signal which persists
indefinitely in time, and the other is a response function of limited duration. The convolution may be calculated, for two array references $data and $respn, as
$fft = Math::FFT->new($data);
$convlv = $fft->convlv($respn);
with the returned $convlv being an array reference. The method assumes that the response function $respn has an odd number of elements $m less than or equal to the number of elements $n of $data.
$respn is assumed to be stored in wrap-around order - the first half contains the response at positive times, while the second half, counting down from $respn->[$m-1], contains the response at
negative times.
Deconvolution undoes the effects of convoluting a signal with a known response function. In other words, in the relation
Convlv(t) = | ds g(s) h(t-s)
deconvolution reconstructs the original signal, given the convolution and the response function. The method is implemented, for two array references $data and $respn, as
$fft = Math::FFT->new($data);
$deconvlv = $fft->deconvlv($respn);
As a result, if the convolution of a data set $data with a response function $respn is calculated as
$fft1 = Math::FFT->new($data);
$convlv = $fft1->convlv($respn);
then the deconvolution
$fft2 = Math::FFT->new($convlv);
$deconvlv = $fft2->deconvlv($respn);
will give an array reference $deconvlv containing the same elements as the original data $data.
If the FFT of a real function of N elements is calculated, the N/2+1 elements of the power spectrum are defined, in terms of the (complex) Fourier coefficients C[k], as
P[0] = |C[0]|^2 / N^2
P[k] = 2 |C[k]|^2 / N^2 (k = 1, 2 ,..., N/2-1)
P[N/2] = |C[N/2]|^2 / N^2
Often for these purposes the data is partitioned into K segments, each containing 2M elements. The power spectrum for each segment is calculated, and the net power spectrum is the average of all
of these segmented spectra.
Partitioning may be done in one of two ways: non-overlapping and overlapping. Non-overlapping is useful when the data set is gathered in real time, where the number of data points can be varied
at will. Overlapping is useful where there is a fixed number of data points. In non-overlapping, the first <2M> elements constitute segment 1, the next 2M elements are segment 2, and so on up to
segment K, for a total of 2KM sampled points. In overlapping, the first and second M elements are segment 1, the second and third M elements are segment 2, and so on, for a total of (K+1)M
sampled points.
A problem that may arise in this procedure is leakage: the power spectrum calculated for one bin contains contributions from nearby bins. To lessen this effect data windowing is often used:
multiply the original data d[j] by a window function w[j], where j = 0, 1, ..., N-1. Some popular choices of such functions are
| j - N/2 |
w[j] = 1 - | ------- | ... Bartlett
| N/2 |
/ j - N/2 \ 2
w[j] = 1 - | ------- | ... Welch
\ N/2 /
1 / \
w[j] = --- |1 - cos(2 pi j / N) | ... Hann
2 \ /
The spctrm method, used as
$fft = Math::FFT->new($data);
$spectrum = $fft->spctrm(%options);
returns an array reference $spectrum representing the power spectrum for a data set represented by an array reference $data. The options available are
This specifies the window function; if not given, no such function is used. Accepted values (see above) are "bartlett", "welch", "hann", and \&my_window, where my_window is a user specified
subroutine which must be of the form, for example,
sub my_window {
my ($j, $n) = @_;
return 1 - abs(2*($j-$n/2)/$n);
which implements the Bartlett window.
This specifies whether overlapping should be done; if true (1), overlapping will be used, whereas if false (0), or not specified, no overlapping is used.
This specifies that the data will be partitioned into n segments. If not specified, no segmentation will be done.
This specifies that 2m data points will be used for each segment, and must be a power of 2. The power spectrum returned will consist of m+1 elements.
For convenience, a number of common statistical functions are included for analyzing real data. After creating the object as
my $fft = Math::FFT->new($data);
for a data set represented by the array reference $data of size N, these methods may be called as follows.
This returns the mean
1/N * sum_j=0^N-1 data[j]
If an array reference $data is not given, the data set used in creating $fft will be used.
This returns the standard deviation
sqrt{ 1/(N-1) * sum_j=0^N-1 (data[j] - mean)**2 }
If an array reference $data is not given, the data set used in creating $fft will be used.
This returns the root mean square
sqrt{ 1/N * sum_j=0^N-1 (data[j])**2 }
If an array reference $data is not given, the data set used in creating $fft will be used.
This returns the minimum and maximum values of the data set. If an array reference $data is not given, the data set used in creating $fft will be used.
This returns the median of a data set. The median is defined, for the sorted data set, as either the middle element, if the number of elements is odd, or as the interpolated value of the the two
values on either side of the middle, if the number of elements is even. If an array reference $data is not given, the data set used in creating $fft will be used.
version 1.36
Please report any to Randy Kobes <randy@theoryx5.uwinnipeg.ca>
Math::Pari and PDL
The algorithm used in this module to calculate the Fourier transforms is based on the C routine of fft4g.c available at http://momonga.t.u-tokyo.ac.jp/~ooura/fft.html, which is copyrighted 1996-99 by
Takuya OOURA. The file arrays.c included here to handle passing arrays to and from C comes from the PGPLOT module of Karl Glazebrook <kgb@aaoepp.aao.gov.au>. The perl code of Math::FFT is copyright
2000,2005 by Randy Kobes <r.kobes@uwinnipeg.ca>, and is distributed under the same terms as Perl itself.
The following websites have more information about this module, and may be of help to you. As always, in addition to those websites please use your favorite search engine to discover more resources.
Bugs / Feature Requests
Please report any bugs or feature requests by email to bug-math-fft at rt.cpan.org, or through the web interface at https://rt.cpan.org/Public/Bug/Report.html?Queue=Math-FFT. You will be
automatically notified of any progress on the request by the system.
Source Code
The code is open to the world, and available for you to hack on. Please feel free to browse it and play with it, or whatever. If you want to contribute patches, please send me a diff or prod me to
pull from your repository :)
git clone git://github.com/shlomif/perl-Math-FFT.git
Shlomi Fish <shlomif@cpan.org>
Please report any bugs or feature requests on the bugtracker website https://github.com/shlomif/perl-Math-FFT/issues
When submitting a bug or request, please include a test-file or a patch to an existing test-file that illustrates the bug or desired feature.
This software is copyright (c) 2000 by Randy Kobes.
This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself.
The following websites have more information about this module, and may be of help to you. As always, in addition to those websites please use your favorite search engine to discover more resources.
Bugs / Feature Requests
Please report any bugs or feature requests by email to bug-math-fft at rt.cpan.org, or through the web interface at https://rt.cpan.org/Public/Bug/Report.html?Queue=Math-FFT. You will be
automatically notified of any progress on the request by the system.
Source Code
The code is open to the world, and available for you to hack on. Please feel free to browse it and play with it, or whatever. If you want to contribute patches, please send me a diff or prod me to
pull from your repository :)
git clone git://github.com/shlomif/perl-Math-FFT.git
Shlomi Fish <shlomif@cpan.org>
Please report any bugs or feature requests on the bugtracker website https://github.com/shlomif/perl-Math-FFT/issues
When submitting a bug or request, please include a test-file or a patch to an existing test-file that illustrates the bug or desired feature.
This software is copyright (c) 2000 by Randy Kobes.
This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself. | {"url":"https://metacpan.org/pod/Math::FFT","timestamp":"2024-11-14T11:15:15Z","content_type":"text/html","content_length":"61291","record_id":"<urn:uuid:e92f4562-ba54-493b-a2e0-3d815332fe89>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00436.warc.gz"} |
THE GODEL FILE: A Platonist was hiding among them!
THURSDAY, SEPTEMBER 6, 2018
But was this a crazy idea?
Many people have heard of Einstein, who enjoyed his walks with Godel.
Aristotle's name is also well known. That said, we'll guess most people wouldn't know that Aristotle is said to have been our greatest logician, at least until Godel came along.
(For our previous Godel File report, click here.)
Godel is hailed as our
greatest logician because of his incompleteness theorems. We'll take a first peek at those theorems tomorrow—but for today, we tip our hat to Rebecca Goldstein.
Goldstein wrote the book on Godel; she did so in 2005. Along the way, she outlined Aristotle's work as a logician, in a footnote on page 55:
GOLDSTEIN (page 55): Aristotle is commonly acknowledged as the father of logic. His work in logic is laid out in the Prior Analytics, which is part of the posthumous consortium known as the
Organon. The philosopher had the seminal insight that in a deductive logical argument, some words are logically relevant while others are not. The irrelevant words can be dispensed with by making
them variables...
Goldstein goes on to offer a "stock syllogism" as an example of what she's talking about. She then says, "The move toward denoting logically irrelevant words with variables was a move toward
generality and thus toward the science of logic."
At this point, we start in surprise. Has someone actually made a science of logic? If you read American newspapers, or if your watch our "cable news" programs, you might be greatly surprised by any
such allegation.
As everyone must understand by now, "the science of logic" plays almost no role in our modern American discourse. Within our failing culture's pseudo-discussions, basic information and basic facts
are almost wholly verboten. But anything resembling logic is typically frowned upon too.
This destructive state of affairs can't be blamed on Aristotle. Because Godel lived in modern times, we'll be a bit less forgiving when it comes to him.
Sad! Modern logicians have created a system in which their highly abstruse ruminations have nothing to do with the bungled logic of our public discussions. Godel lived until 1978. At some point, he
probably should have stepped in.
At any rate, our modern logicians walk to work in the clouds, where they engage in high-brow endeavors which even they may not be equipped to explain. This brings us to a fascinating part of Godel's
horrifically tragic story, as told by Goldstein in 2005 and then by Jim Holt in the derivative title essay of his well-received new book.
We refer to a possibly crazy idea to which Godel was apparently devoted throughout the course of his life. We refer to Godel's alleged belief in "Platonism," an alleged doctrine with which Goldstein
says Godel "fell in love" when he was in his late teens.
In our previous report, we learned an unfortunate fact. Godel, who died of self-starvation, exhibited signs of severe mental illness throughout the course of his highly unusual life.
"At the age of five, he seems to have suffered a mild anxiety neurosis," Holt writes in his new book, drawing on Goldsetin's work. "At eight, he had a terrifying bout of rheumatic fever, which left
him with the lifelong conviction that his heart had been fatally damaged."
By normal reckoning, Godel's apparent paranoia eventually led to his death. But then, as noted yesterday, he had always struck the Princeton community as being extremely odd.
In her book, Goldstein cites an array of peculiar ideas Godel is said to have expressed. Our continuing question is this:
Should it seem strange to think that the western world's second greatest logician was apparently in the grip of serious mental illness? Asking a slightly different question, should it seem strange
that "Aristotle's successor" was in the grip, throughout his life, of some extremely peculiar ideas?
In theory, the mental illness, however tragic, shouldn't matter at all. If a person achieves some great intellectual breakthrough—in the physical sciences, let's say—that achievement isn't negated by
the tragedy of a severe mental illness.
That said, Godel's achievements
come in the realm of physical science. Indeed, Goldstein quotes him telling one Princeton-based scholar, "I don't believe in natural science." She quotes him telling Thomas Nagel that he doesn't
believe in evolution, offering Stalin's similar disbelief as supporting evidence.
She quotes him saying this to Noam Chomsky: "I am trying to prove that the laws of nature are a priori." Should it seem strange to hear that history's second greatest logician was bruiting such
notions around?
In fairness, these peculiar-sounding statements were made during Godel's adult life. He formulated his "incompleteness theorems" when he was just 23.
If those theorems make sense and are actually important in some way, later delusions and mental illness can't vitiate the achievement. That said, we might want to consider the lifelong belief in
"Platonism" which, according to Goldstein's book, played the key role in Godel's thinking from his teenage years on.
What the heck is Platonism? According to Goldstein, Godel "fell in love" with the doctrine while he was still a teen.
"First exposure to Plato can be an extremely heady experience for those with a passion for abstraction," Goldstein writes. "It can amount to a sort of ecstasy."
Parenthetically, Goldstein says that she can remember
her own
first exposure to Plato. But what is the doctrine which may result? What is this "Platonism?"
In the opening essay of Holt's new book, he defines this doctrine, or he at least pretends or attempts to do so. As critics robotically praise the clarity of Holt's writing, he sketches the doctrine
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that
abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas,
has always been popular among mathematicians...
Say what? According to Holt, Godel, who was still in his teens, was "seduced by" a certain notion. More specifically (or possibly less so), he was seduced by "the notion that abstractions like
numbers and circles had a perfect, timeless existence independent of the human mind."
According to Holt, this "doctrine," which is called Platonism, "has always been popular among mathematicians." In fairness, this suggests that it isn't something the teen-aged Godel somehow dreamed
up by himself.
At this point, our review of the Godel file reached a critical turn. We're told that the western world's second greatest logician was swept away by a seductive idea, an idea which lay at the heart of
his subsequent work.
What did this greatest logician believe? According to Holt, he believed that numbers have a perfect, timeless existence independent of the human mind. So Kurt Godel believed!
Readers, how about it? Do
believe that the number 3 "has a perfect existence independent of the human mind?" Do you believe that the number 3 has a perfect existence at all?
Indeed, let's ask the more relevant question: Do you have even the slightest idea what Holt is talking about? Granting the fact that you can memorize Teacher's words and reproduce those words on the
test, would you have even the slightest idea what you were talking about?
Readers may be inclined to suppose that Holt proceeds from there to explain this murky bit of word salad. In our view, such a supposition would be badly mistaken.
(To peruse his original New Yorker essay,
you can just click here
Does Holt clarify this description? We'll set that question aside for another day. Instead, we'll turn to Goldstein, to the start of
attempt to explicate "Platonism."
Godel fell in love with the doctrine, she says. She lays out the doctrine as follows:
GOLDSTEIN (page 44): Godel's commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in
honor of the ancient Greek philosopher...
Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and
proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up
that reality. The structure of, say, the natural numbers (which are the regular old counting numbers: 1, 2, 3, etc.) exists independent of us, according to the mathematical realist...and the
properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical
properties of light and gravity.
Goldstein muddies her account with a fair amount of technical language. (How many general interest readers are likely to know what a "formal system" is?)
Beyond that, she seems to think, all through her book, that the terms "objective" and "subjective" explain
in contexts like this. Sadly, they very much don't.
Whatever! In this passage, Goldstein starts describing the doctrine with which Godel "fell in love" as a teen. According to Goldstein, the ardor Godel felt for this doctrine impelled him to devise
the "incompleteness theorems" which made him famous among certain academics, while leaving him completely unknown to everyone else on the face of the earth.
Godel's ardor for this alleged doctrine forms the core of Goldstein's work. And how does
describe the doctrine? She describes it like this:
According to Goldstein, Godel fell in love with the view that "the truths of mathematics are determined...by the reality of mathematics."
So far, so completely no good! Explicating further, she says Godel adopted the view that "the truths of mathematics are determined by the nature of the entities that make up" the reality of
As an overview, she starts by saying that Godel developed a "commitment to the objective existence of mathematical reality!"
Readers, tell the truth. Would you say that the essence of this key doctrine has begun to come clear? We're going to say that we're munching on salad—that it's word salad all the way down!
At this point, we'll raise two questions, after which we'll adjourn for the day. One of our questions will concerns Kurt Godel. The other question will concern writers like Goldstein and Holt, who
are robotically praised, by scripted journalists, for the clarity of their work.
Regarding Godel, we'll ask this question: Is it possible that the western world's second greatest logician fell in love at an early age with a crackpot idea?
He believed many other crazy things in the course of his tragic life. Could this idea, which drove the work for which he's lauded, possibly have been the
of his crackpot ideas?
That's a question about our second greatest logician. Our second question concerns Holt and Goldstein—and it's built on a basic concept:
Here's the basic concept. Some claims or ideas are right or wrong. But
claims, and
ideas, don't rise to that level.
Such claims and ideas aren't right
wrong—they're simply incoherent!
This idea is stunningly basic. To what extent are general interest writers like Goldstein and Holt conversant with this idea?
A Platonist in their midst (we didn't get there today)
28 comments:
1. Suppose you were sending a message to an alien race on some other planet. You could include arithmetic statements like "1 + 1 = 2" because these numbers and concepts or addition and equality have
truth independent of human activity.
1. Only 1 in 10 programmers understands binary; the other one has no idea...
2. Numbers are products of the human brain. Whether alien intelligences have numbers at all remains to be seen.
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2. This artist died young because he had rheumatic fever as a child:
This artist died young from self-starvation:
1. Caesar, your post is completely illogical. ; )
I will say that whenever I hear a Karen Carpenter song, which is rare nowadays (though I did hear one just a couple of days ago on one of our local, eclectic stations), I’m literally floored.
Everything else falls away when I hear that voice.
3. I'd recommend Bob study some logic, as he'd then realize that his trope here rests on an equivocation over the term "logic." Godel worked in formal logic, Bob's complaint about the absence of
logic in modern discourse relates to informal logic. What informal logic is (or whether one should call it that, rather than something else) is a much-debated question nowadays (one thing not
included in this conversation is Godel--not because Godel isn't actually a great logician, but because it's practically a different subject).
4. "Could this idea, which drove the work for which he's lauded, possibly have been the first of his crackpot ideas?"
What is Somerby driving at? He ridicules Holt and Goldstein for their attempts to explain Gödel's so-called views, but then he implies that Gödel's own views were (*possibly*) "crackpot." And
yet, "later delusions and mental illness can't vitiate the achievement" "if those theorems make sense and are actually important in some way." Say what? Is Somerby going to attempt to debunk
Gödel? Is Gödel's alleged "severe mental illness" somehow important to Somerby's assessment of Gödel's work, at the same time he asserts that a true achievement, such as that in the "physical
sciences", cannot be vitiated by "mental illness?"
1. I'm with you, @2:18. I don't know were Somerby is headed. Is he trying to refute Godel's incompleteness or, at least, to disparage its importance? Well, maybe future posts will make Bob's
position more clear.
5. Condemning Gödel's embrace of aspects of Platonism as possibly "crackpot" betrays a possible misrepresentation, misunderstanding, or oversimplification by Holt and Goldstein, and Somerby as well.
Somerby seems to accept or promote the notion that Gödel was mentally ill to call Gödel's Platonism into question, but Somerby can't refrain from a flippant disregard for the questions Gödel was
thinking about that led him to a Platonist view of mathematics.
Aside from studying logic, as one commenter above suggested, Somerby probably needs to study some mathematics if he wants to grapple intelligently with Gödel.
6. Not being an expert, I'm not entirely, totally, completely convinced that there isn't some foundational flaw in the work on mathematical logic of either Russell or Gödel. Are the questions that
they addressed in some sense ill-posed? I have some nagging concerns, although I'd have a hard time articulating them with any clarity. That said, if we want to put mathematics on a rigorous
foundation, we would at least have to understand how the questions addressed by these thinkers are ill-posed.
Might Plato also have been asking ill-posed questions? I know even less, but I'm always very leery of any question that asks whether something "exists"---that seems to be the most slippery word
in all of philosophy, so, for myself, I'd be much more open to the idea that Plato barking in the wrong cave.
However, if Plato's questions were ill-posed, and if Gödel was much taken with Plato, does that imply that Gödel's work on incompleteness was also ill-posed? Not necessarily....I'd even venture
to say NO. There is no clear line that I can see from Plato to incompleteness...I don't see how the latter is contingent on the former.
Similarly, Einstein was much taken with the ideas of Ernst Mach. I'd go so far as to say that Mach's ideas were wrong, but that in no way implies that General Relativity is wrong. An wrong idea
can play a role in encouraging a thinker to do work that is correct and even great.
7. This comment has been removed by the author.
8. Hey Bob, when you retire why not take a few college classes -- on any serious subject! (No, psychology and journalism don't count.) How about some math beyond calculus; maybe some classes dealing
with "algebra" (which you may be surprised to find out isn't something you already learned in high school.) The physical sciences -- even the "old" stuff from before the 20th century -- require
some training in mathematics in order to be understood. No, it isn't fair. Sorry, not everything can be grasped by amateurs and the self-educated, people who always quit and move on to something
else when the going gets hard. I know this must sound weird in an age when everyone feels entitled to know the Mind of God -- but please learn something about what you're criticizing!
9. Readers, tell the truth. Would you say that the essence of this key doctrine has begun to come clear?
To tell the truth, for this reader, yes. But perhaps because this reader has familiarized himself with the terminology and doesn’t expect every book to start with first principles. Also because
the topic isn’t that difficult.
We’re all familiar with three-sided figures. We can draw them by putting pen to paper or arrange them out of sticks. At some point it’s natural to wonder what all these physical figures have in
common, what’s the essence of a three-sided figure. We can think of the perfect triangle: three non-collinear, co-planar straight lines of no width (call them sides) that intersect pairwise in
distinct points of no width and no length (call them vertices). All the dross three-sided figures of our experience are imperfect approximations: no matter how narrow our nib, our drawn lines
always have some thickness, and no matter how true our lumber, our carpentered sides aren’t perfectly straight.
We can prove exact statements about the perfect three-sided figure, which we’ll call a triangle. The sum of the lengths of any two sides of any triangle is strictly larger than the length of the
third side. The sum of the angle measures of the angles in a triangle is exactly 180 degrees. The area enclosed by any triangle is exactly one half the length of any side multiplied by the length
of a perpendicular from the vertex opposite that side to that side.
These properties for three-sided objects are only approximate, in part because different subjects who measure these objects will record slightly different measurements and in part because the
objects cannot be perfect. Not so for triangles: the essence of these objects does not vary, and they are not constrained by the imperfect tools of those subjects who contemplate them.
Most people are willing to concede the existence of a world outside ourselves. (Except the solipsists, and they don’t believe the rest of us are here anyway.) So most people would be willing to
agree that if three trees in a Permian forest had fallen in such a way so as to create a three-sided figure, then that figure would have existed. Even though it would have been millions of years
before any consciousness could have considered the scene.
But what about triangles? Did human minds invent them or discover them? If the former, then they have no existence beyond our minds. If the latter, then these are objects they don’t depend on our
necessarily-subjective consciousness, and their properties (like the area formula mentioned above) also exist independently. In that case, the truth of the area formula is a truth of mathematics
(or here, plane geometry) and is determined by the reality of triangles. That truth existed at the Big Bang and will exist after the heat death of this universe.
Now, you may reasonably decide that whether a reified abstraction exists is merely a matter of semantics, but that doesn’t mean you have to be a crackpot to contemplate the matter. Some ideas are
right or wrong, and some of those will be incoherent to an ignoramus.
Gödel considered the abstraction not of geometric figures, but of systems of logic, i.e., the language and assumptions that we use to decide whether statements of the language are true or false).
Gödel wanted to know whether we can use logical machinery to prove that the system is consistent (i.e, that it cannot lead to a contradiction) and whether we can use that machinery to prove any
true statement of the system.
I can’t wait to see what the slowest boy in the class has to say about all this tomorrow.
1. Triangles, shmiangles deadrat. As they exist in plane geometry is a far cry from their existence in the physical world, which we perceive through our imperfect perceptions, and grasp upon to
make into avatars – that is, perfect objects, which have certainly existed in the form of art long before their mathematical properties were ever realized. Hey, maybe minerals can form
perfect triangles in the form of crystals, but they just aren’t the same as triangles in plane geometry. Rather, they would rely on the abstractions of molecular or atomic means of
mathematical theory to explain how they came into being.
And the numbers that the great theorists manipulate to explain phenomena at that level have a certain elegance which can predict with an amazing degree of accuracy. But even today, Newton’s
theory of gravity is still the go-to for NASA when calculating the trajectory of heavenly bodies and spacecraft, despite the refinements of general relativity to his theory.
“Gödel considered the abstraction not of geometric figures, but of systems of logic, i.e., the language and assumptions that we use to decide whether statements of the language are true or
But wasn’t that in mathematical systems of logic? Because when it comes to language, all bets are off, far as I’m concerned. You might be able to prove the statement “Whatever I say is true”
is nonsense, but only if you can disprove it. And I’m not sure mathematics provides that toolbox.
And anyways, the slowest boy in class is responding today. : )
2. Fun fact: any three points are coplanar. So a triangle is a plane figure -- we don't have to assume that up front.
3. Triangles, shmiangles deadrat. As they exist in plane geometry is a far cry from their existence in the physical world
That sounds awfully familiar. Almost if I’d read it a short time ago. Or written it.
and grasp upon to make into avatars – that is, perfect objects
A Platonist would say you have it backwards. It’s not that we abstract our real-world experience to make avatars, but rather it’s that the avatars have always existed in the Platonic realm
beyond our direct perception. The avatars project themselves into our world as imperfect shadows of their glorious, perfect selves.
Rather, they would rely on the abstractions of molecular or atomic means of mathematical theory to explain how they came into being.
I’m not sure of the antecedent of “they”, but Einstein put paid to the claim that molecules and atoms are mere abstractions, useful only to compute things in statistical mechanics. That was
in 1905.
And the numbers that the great theorists manipulate to explain phenomena at that level have a certain elegance which can predict with an amazing degree of accuracy.
Sure, but do they exist a priori or did we make them up to make predictions?
But wasn’t that in mathematical systems of logic?
You might be able to prove the statement “Whatever I say is true” is nonsense, but only if you can disprove it
Well, I can prove false the statement “Whatever I say is true” if David in Cal says it. Will that do?
Don’t bogart that doobie, dude. It’s only polite to pass it on.
4. "Whatever I say is true" isn't a problem in logic. But, "Whatever I say is false" leads you into the Liar's paradox.
5. This comment has been removed by the author.
6. I’m busted. I did smoke some doobies, but quite a long time ago. Luckily, I’m not running for the SCOTUS.
It seems I’ve been the victim of unclear perceptions in the realm of reader comprehension, which is a genetic aberration (laziness genes). As a result, I read too fast, without sussing out
the entire point you were trying to make, and only glommed on to a reason to debate.
I was clumsily trying to point out that logic in mathematics does not translate very well to the emergent properties of the universe, of which we’re a part. And it’s only through agreement,
through reason between us, as human beings, that the truth can be accepted as such.
Perhaps mathematics can play a role in predicting how masses of individually sentient beings will react to any given situation. With individuals, another matter. But it’s a sure bet that when
such beings consume the same information, they will act in concert to the stimulus, though how that manifests itself is always uncertain.
Oxygen, meet hydrogen!
Sorry, it’s raining here, and I had a flashback to my doobie days, in which I encountered the postulation that even atoms may be, in a way, sentient. They always “know” how to react.
To get on with my long-winded point, I did not consider your observations on Platonism. I can only say that certain things, abstract or not, certainly existed before we ever emerged from the
matter which defines us. In that sense, Plato was surely on point. My best spiritual estimate is that we are the universe trying to understand itself. And we seem to think we know how it will
go. With our limited window, we only see beginnings and endings, so we’re a bit biased in that direction of thinking.
For we creatures, the evidence does not lie. Did Plato see death as an abstract truth? Is there an eventual heat death of the universe? Who knows!
“Some ideas are right or wrong, and some of those will be incoherent to an ignoramus.”
Curse you for calling me out!
10. Slowly to some it will dawn on them that TDH blog posts do not define whatever is the subject of the post, and TDH has no such burden. If only these blinded by ego types could attract someone's
affection perhaps they wouldn't bother with their inane comments. Either way is ok, they are great for a good hearty laugh ending with the joy of muttering fake pity like "bless their little
soulless minds."
11. To the issue of 'insanity' (which really is slavish obedience to the globalist finance) of your lib-zombie-neocon death-cult:
"U.S. job growth accelerated in August, with wages notching their largest annual increase in nine years".
1. The Trump (Mis)Administration is proving that you don't need to look globally to rip-off American citizens.
Have you seen Betsy DeVos installed a rip-off artist to be head of the student loan fraud division? No need to go outside our borders (globalism) to continue the con.
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[GIF] Riemann Hypothesis video from Quanta Magazine
15400 Views
11 Replies
38 Total Likes
[GIF] Riemann Hypothesis video from Quanta Magazine
I just wanted to share Quanta Magazine's new explainer video on the Riemann Hypothesis, to which I contributed a number of animations: https://youtu.be/zlm1aajH6gY The full Quanta Magazine article
can be found here: How I Learned to Love and Fear the Riemann Hypothesis.
Here's a GIF of part of the spiral animation :
And here's a Manipulate version of the spiral animation:
zetazeros = Table[N[Im[ZetaZero[i]]], {i, 1, 20}];
DynamicModule[{diffs, pos, xaxesLength = 5, yaxesLength = 9/16*5,
range = 3, tmax = 55, axesColor,
cols = RGBColor /@ {"#07617d", "#f9a828", "#2e383f", "#ececeb"},
n = Length[zetazeros]},
axesColor = cols[[3]];
diffs = Table[a - zetazeros[[i]], {i, 1, n}];
pos = Position[diffs, x_ /; 0 < x < 1];
Show[Graphics[{axesColor, Thickness[.002],
Line[{{-xaxesLength, 0}, {xaxesLength, 0}}],
Line[{{0, -yaxesLength}, {0, yaxesLength}}],
Reverse@Table[{Opacity[(8 - E^r)/48], Blend[cols[[;; 2]], E^r/2],
Disk[{0, 0}, E^r]}, {r, -5, 5, 1/4}]}],
ParametricPlot[ReIm[Zeta[1/2 + I t]], {t, 0, a},
PlotStyle -> Directive[Thickness[.005], CapForm["Round"]],
ColorFunctionScaling -> False,
ColorFunction ->
Function[{x, y, t}, Blend[cols[[;; 2]], Norm[{x, y}]/2]]],
Graphics[{Blend[cols[[;; 2]], Abs[Zeta[1/2 + I a]]/2],
PointSize[.015], Point[ReIm[Zeta[1/2 + I a]]], cols[[1]],
If[Length[pos] >= 1, {Opacity[1 - diffs[[pos[[1, 1]]]]],
Disk[{0, 0}, 2 diffs[[pos[[1, 1]]]]]}]}],
ImageSize -> 50 {16, 9}, Background -> cols[[-1]],
PlotRange -> range {{-16/9, 16/9}, {-1, 1}}],
{a, .0001, tmax, (tmax - .0001)/840}]
11 Replies
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The segments I animated are probably not hard to guess. They are:
The other animations were done by an actual professional motion graphics designer, Guan-Huei Wu.
Fixed that, I could see it myself, but not others. Now removed the music and reuploaded. link updated.
Hi Sander,
Looks like the video is no longer available on YouTube
Video unavailable This video contains content from WMG, who has blocked it on copyright grounds.
No worries, even just knowing which ones you created would be interesting, and would open my mind of the possibilities of what Mathematica can do… (I, for example, have made this in Mathematica and
also ffmpeg).
Unfortunately, the others were all produced in multiple parts, which were stitched together at the end with FFmpeg (for example, the sequence from 12:59–14:36 in the video consists of 11 separate
parts). In the end I think it's something like 40 different parts spread over 7 different notebooks, so the effort of putting it all together into something comprehensible will be substantial.
Not to say I won't do it at some point, but, given that our semester is starting soon and I have a lot else going on, it's not at the top of my priority list at the moment.
Fantastic! Are you gonna share the others? They are really nice!
A very interesting thing to look at with the "zeta spiral" is Lehmer's phenomenon, which manifests itself as a near-cusp at the origin. Stripping down Clayton's code from above, here's how to
visualize the first Lehmer pair:
ParametricPlot[ReIm[Zeta[1/2 + I t]], {t, 7005, 7005 + 1/8},
Background -> RGBColor["#ececeb"], ColorFunctionScaling -> False,
ColorFunction -> Function[{x, y, t},
Norm[{x, y}]/2]],
Frame -> True, PlotRange -> {-0.01, 0.01},
PlotStyle -> Directive[Thickness[0.005], CapForm["Round"]]]
If the spiral did not in fact hit the origin, then the hypothesis would be false there. The MathWorld page I linked to has examples of other Lehmer pairs.
Another nice thing to look at using the spiral would be the Gram points.
Wow, such an exquisite work, thanks for sharing! Just finished watching the video and reading the article, -- the animations are greatly educational. Besides the elegance of maths in your animations
I also always enjoy your color schemes :-) Congrats on Quanta Magazine collaboration!
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CMO Workflow
In general, the CMO workflow is:
1. Calculate underlying mortgage cash flows.
2. Define CMO tranches
3. If using a PAC or TAC CMO, calculate the principal schedule.
4. Calculate cash flows for each tranche.
5. Analyze the CMO by computing price, yield, spread of CMO cash flows.
Calculate Underlying Mortgage Cash Flows
Underlying mortgage pool pass-through cash flows are calculated by the existing function mbspassthrough. The CMO cash flow functions require the principal payments (including prepayments) calculated
from existing functions mbspassthrough or mbscfamounts.
principal = 10000000;
coupon = 0.06;
terms = 360;
psa = 150;
[principal_balance, monthly_payments, sched_principal_payments,...
interest_payments, prepayments] = mbspassthrough(principal,...
coupon, terms, terms, psa, []);
principal_payments = sched_principal_payments.' + prepayments.';
After determining principal payments for the underlying mortgage collateral, you can generate cash flows for a sequential CMO, with or without a Z-bond, by using cmoseqcf. For a PAC or TAC CMO, the
cash flows are generated using cmoschedcf
Define CMO Tranches
Define CMO tranche; for example, define a CMO with two tranches:
TranchePrincipals = [500000; 500000];
TrancheCoupons = [0.06; 0.06];
If Using a PAC or TAC CMO, Calculate Principal Schedule
Calculate the PAC/TAC principal balance schedule based on a band of PSA speeds. For scheduled CMOs (PAC/TAC), the CMO cash flow functions additionally take in the principal balance schedule
calculated by the CMO schedule function cmosched.
terms = 360;
coupon = 0.06;
principal = 10000000;
speed = [100 300];
[balanceSchedule, initialBalance] = cmosched(principal, coupon,...
terms, terms, speed, TranchePrincipals(1));
Calculate Cash Flows for Each Tranche
You can reuse the output from the cash flow generation functions to further divide the cash flows into tranches. For example, the output from cmoschedcf for a PAC tranche can be divided into
sequential tranches by passing the principal cash flows of the PAC tranche into the cmoschedcf function. The outputs of the CMO cash flow functions are the principal and interest cash flows, and the
principal balance.
[principal_balances, principal_cashflows, interest_cashflows] = cmoschedcf(principal_payments,...
TranchePrincipals, TrancheCoupons, balanceSchedule);
Analyze CMO by Computing Price, Yield, and Spread of CMO Cash Flows
The outputs from the CMO functions (cmoseqcf and cmoschedcf) are cash flows. The functions used to analyze a CMO are based on these cash flows. To that end, you can use cfbyzero, cfspread, cfyield,
and cfprice to compute prices, yield, and spreads for the CMO cash flows. In addition, using the following, you can calculate a weighted average life (WAL) for each tranche in the CMO:
$WAL=\sum _{i=1}^{n}\frac{{P}_{i}}{P}{t}_{i}$
P is the total principal.
P[i] is the principal repayment of the coupon i.
$\frac{{P}_{i}}{P}$ is the fraction of the principal repaid in coupon i.
t[i] is the time in years from the start to coupon i.
See Also
cmoseqcf | cmosched | cmoschedcf | mbscfamounts | mbspassthrough
Related Examples
More About | {"url":"https://it.mathworks.com/help/fininst/cmo-workflow.html","timestamp":"2024-11-11T01:30:09Z","content_type":"text/html","content_length":"73658","record_id":"<urn:uuid:52a6f5e6-40d2-497d-ab5c-30ae1d90f6f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00099.warc.gz"} |
In physics, spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i.e., as ${\displaystyle {\tfrac {1}{R}}.}$
Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.
For clarity, we illustrate the expansion for a point charge,^[1] then generalize to an arbitrary charge density ${\displaystyle \rho (\mathbf {r} ').}$ Through this article, the primed coordinates
such as ${\displaystyle \mathbf {r} '}$ refer to the position of charge(s), whereas the unprimed coordinates such as ${\displaystyle \mathbf {r} }$ refer to the point at which the potential is being
observed. We also use spherical coordinates throughout, e.g., the vector ${\displaystyle \mathbf {r} '}$ has coordinates ${\displaystyle (r',\theta ',\phi ')}$ where ${\displaystyle r'}$ is the
radius, ${\displaystyle \theta '}$ is the colatitude and ${\displaystyle \phi '}$ is the azimuthal angle.
Spherical multipole moments of a point charge
Figure 1: Definitions for the spherical multipole expansion
The electric potential due to a point charge located at ${\displaystyle \mathbf {r'} }$ is given by ${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}{\frac {1}{R}}={\frac {q}{4\pi \
varepsilon }}{\frac {1}{\sqrt {r^{2}+r^{\prime 2}-2r'r\cos \gamma }}}.}$ where ${\displaystyle R\ {\stackrel {\mathrm {def} }{=}}\ \left|\mathbf {r} -\mathbf {r'} \right|}$ is the distance between
the charge position and the observation point and ${\displaystyle \gamma }$ is the angle between the vectors ${\displaystyle \mathbf {r} }$ and ${\displaystyle \mathbf {r'} }$ . If the radius ${\
displaystyle r}$ of the observation point is greater than the radius ${\displaystyle r'}$ of the charge, we may factor out 1/r and expand the square root in powers of ${\displaystyle (r'/r)<1}$ using
Legendre polynomials ${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{\ell =0}^{\infty }\left({\frac {r'}{r}}\right)^{\ell }P_{\ell }(\cos \gamma )}$ This is exactly
analogous to the axial multipole expansion.
We may express ${\displaystyle \cos \gamma }$ in terms of the coordinates of the observation point and charge position using the spherical law of cosines (Fig. 2) ${\displaystyle \cos \gamma =\cos \
theta \cos \theta '+\sin \theta \sin \theta '\cos(\phi -\phi ')}$
Figure 2: Angles between the unit vectors ${\displaystyle \mathbf {\hat {z}} }$ (the coordinate axis), ${\displaystyle \mathbf {\hat {r}} }$ (the observation point) and ${\displaystyle \mathbf {{\hat
{r}}'} }$ (the charge position).
Substituting this equation for ${\displaystyle \cos \gamma }$ into the Legendre polynomials and factoring the primed and unprimed coordinates yields the important formula known as the spherical
harmonic addition theorem ${\displaystyle P_{\ell }(\cos \gamma )={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$ where the ${\displaystyle
Y_{\ell m}}$ functions are the spherical harmonics. Substitution of this formula into the potential yields ${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{\ell =0}^{\infty }
\left({\frac {r'}{r}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$
which can be written as ${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {Q_{\ell m}}{r^{\ell +1}}}\right){\sqrt {\frac
{4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )}$ where the multipole moments are defined ${\displaystyle Q_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ q\left(r'\right)^{\ell }{\sqrt {\frac {4\pi }{2\ell
+1}}}Y_{\ell m}^{*}(\theta ',\phi ').}$
As with axial multipole moments, we may also consider the case when the radius ${\displaystyle r}$ of the observation point is less than the radius ${\displaystyle r'}$ of the charge. In that case,
we may write ${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r'}}\sum _{\ell =0}^{\infty }\left({\frac {r}{r'}}\right)^{\ell }\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\
ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$ which can be written as ${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\
ell }I_{\ell m}r^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )}$ where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics $
{\displaystyle I_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {q}{\left(r'\right)^{\ell +1}}}{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ')}$
The two cases can be subsumed in a single expression if ${\displaystyle r_{<}}$ and ${\displaystyle r_{>}}$ are defined to be the lesser and greater, respectively, of the two radii ${\displaystyle r}
$ and ${\displaystyle r'}$ ; the potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion ${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon
}}\sum _{\ell =0}^{\infty }{\frac {r_{<}^{\ell }}{r_{>}^{\ell +1}}}\left({\frac {4\pi }{2\ell +1}}\right)\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\phi )Y_{\ell m}^{*}(\theta ',\phi ')}$
Exterior spherical multipole moments
It is straightforward to generalize these formulae by replacing the point charge ${\displaystyle q}$ with an infinitesimal charge element ${\displaystyle \rho (\mathbf {r} ')d\mathbf {r} '}$ and
integrating. The functional form of the expansion is the same. In the exterior case, where ${\displaystyle r>r'}$ , the result is: ${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\
sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {Q_{\ell m}}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )\,,}$ where the general multipole moments are
defined ${\displaystyle Q_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '\rho (\mathbf {r} ')\left(r'\right)^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi ').}$
The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to Y
[ℓm], not to its complex conjugate. This is a common convention, see molecular multipoles for more on this.
Interior spherical multipole moments
Similarly, the interior multipole expansion has the same functional form. In the interior case, where ${\displaystyle r'>r}$ , the result is: ${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \
varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }I_{\ell m}r^{\ell }{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi ),}$ with the interior multipole moments defined as ${\
displaystyle I_{\ell m}\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '{\frac {\rho (\mathbf {r} ')}{\left(r'\right)^{\ell +1}}}{\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}^{*}(\theta ',\phi
Interaction energies of spherical multipoles
A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. Let the first charge distribution ${\displaystyle \rho _{1}(\mathbf {r} ')}$ be
centered on the origin and lie entirely within the second charge distribution ${\displaystyle \rho _{2}(\mathbf {r} ')}$ . The interaction energy between any two static charge distributions is
defined by ${\displaystyle U\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} \rho _{2}(\mathbf {r} )\Phi _{1}(\mathbf {r} ).}$
The potential ${\displaystyle \Phi _{1}(\mathbf {r} )}$ of the first (central) charge distribution may be expanded in exterior multipoles ${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \
varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}\left({\frac {1}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\theta ,\phi )}$ where ${\displaystyle Q_{1\
ell m}}$ represents the ${\displaystyle \ell m}$ exterior multipole moment of the first charge distribution. Substitution of this expansion yields the formula ${\displaystyle U={\frac {1}{4\pi \
varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}\int d\mathbf {r} \ \rho _{2}(\mathbf {r} )\left({\frac {1}{r^{\ell +1}}}\right){\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell m}(\
theta ,\phi )}$
Since the integral equals the complex conjugate of the interior multipole moments ${\displaystyle I_{2\ell m}}$ of the second (peripheral) charge distribution, the energy formula reduces to the
simple form ${\displaystyle U={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }Q_{1\ell m}I_{2\ell m}^{*}}$
For example, this formula may be used to determine the electrostatic interaction energies of the atomic nucleus with its surrounding electronic orbitals. Conversely, given the interaction energies
and the interior multipole moments of the electronic orbitals, one may find the exterior multipole moments (and, hence, shape) of the atomic nucleus.
Special case of axial symmetry
The spherical multipole expansion takes a simple form if the charge distribution is axially symmetric (i.e., is independent of the azimuthal angle ${\displaystyle \phi '}$ ). By carrying out the ${\
displaystyle \phi '}$ integrations that define ${\displaystyle Q_{\ell m}}$ and ${\displaystyle I_{\ell m}}$ , it can be shown the multipole moments are all zero except when ${\displaystyle m=0}$ .
Using the mathematical identity ${\displaystyle P_{\ell }(\cos \theta )\ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell 0}(\theta ,\phi )}$ the exterior multipole expansion
becomes ${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }\left({\frac {Q_{\ell }}{r^{\ell +1}}}\right)P_{\ell }(\cos \theta )}$ where the axially symmetric
multipole moments are defined ${\displaystyle Q_{\ell }\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '\rho (\mathbf {r} ')\left(r'\right)^{\ell }P_{\ell }(\cos \theta ')}$ In the limit that
the charge is confined to the ${\displaystyle z}$ -axis, we recover the exterior axial multipole moments.
Similarly the interior multipole expansion becomes ${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{\ell =0}^{\infty }I_{\ell }r^{\ell }P_{\ell }(\cos \theta )}$ where the
axially symmetric interior multipole moments are defined ${\displaystyle I_{\ell }\ {\stackrel {\mathrm {def} }{=}}\ \int d\mathbf {r} '{\frac {\rho (\mathbf {r} ')}{\left(r'\right)^{\ell +1}}}P_{\
ell }(\cos \theta ')}$ In the limit that the charge is confined to the ${\displaystyle z}$ -axis, we recover the interior axial multipole moments.
See also
1. ^ Jackson, John David (1999). Classical electrodynamics. New York: Wiley. ISBN 978-0-471-30932-1. | {"url":"https://www.knowpia.com/knowpedia/Spherical_multipole_moments","timestamp":"2024-11-12T00:36:27Z","content_type":"text/html","content_length":"221878","record_id":"<urn:uuid:af692fec-701c-44bf-b3fe-15385fd197b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00086.warc.gz"} |
Exercise: Moorings and time series
We go on scientific expeditions to the Amundsen Sea every other year or so, and then we do a lot of measurements and learn what things are like right then, when we are there. But we’d off course want
to know also what happens when we are not around. To do so we deploy “moorings”. A mooring is and anchor, a line on which we attach instruments and floating elements to keep it upright in the water.
The moorings are standing on the bottom and measures (normally currents, salinity and temperature but I’ll also have sensors to measure the concentration of oxygen) until we return a year or two
later. What the mooring looks like and what instrument we put on it will depend on what we want to measure. At the bottom of the page there is a drawing of one of my moorings!
Exercise 1
Now you can design you mooring! What would it look like if you’d want to:
a) Study the evolution (salinity and temperature) of the surface layer (let’s say the upper 200 m) over a year in a area where the water is 500 m deep?
b) Study a bottom current that extends 300 m above the bottom
c) Find out how much warm water is entering the cavity below an ice shelf. The depth is 800 m and the draft of the ice shelf is 300 m deep (which means that ice bergs can reach just as deep!!)
Exercise 2
Do the following calculations for you mooring from above or for a mooring which has
1 x acoustic release (25 mab); 3 X SBE37 (25, 150 and 300 mab), 1 x RCM (50 mab) 1 x ADCP (300 mab) and 5 x SBE56 (50, 75, 100, 200, 250 mab). Mab = meters above bottom.
The instrument has the following dimensions (they are all roughly equal to cylinders)
a) What is the weight of all the instruments in air? In water?
The mooring also needs floating elements to stand up in the water.
b) How much can one glass sphere lift? Their diameter is 43 cm and the weigh 22 kg in air.
c) How many spheres do we need to keep the mooring up right?
Exercise 3
When the current is strong it will tend to pull the mooring down towards the bottom – that is not good. Firstly, you don’t get measurements from the depth you wanted and secondly, many of the
instrument cannot measure when they tilt too much, An ADCP, for example, cannot measure correctly if the tilt is above 15 \(^\circ\). So we need to add extra floating elements to make sure that the
mooring is not pulled down if the current is strong. The force from the water pulling the mooring down is proportional to the area that the water hits (i.e. the cross sectional area of the mooring).
a) What percentage of the mooring cross sectional area is made up by the rope?
b) By how much is the area reduced if we change for a line that is 6 mm in diameter?
c) An instrument was pulled down to 170 mab and 218 m downstream, another instrument was pulled down to 65 mab and 105 m downstream. The shape of the line can be decribed by (half) a parable. Find
the expression for the parable!
d) The pressure sensor on the ADCP shows that it now is 40 mab. How much is it tilting? Can we use the measurements?
e) How high up must it be (at least) for the measurements to be good?
f) Why do you think we prefer to have the ADCP high up looking down, rather than deep down looking up?
One of my moorings during an event with strong current. we use a special program to calculate how the moorings will perform in a current. The program will also tell us how heavy the anchor needs to
Exercise 4
When we retrieve the mooring and download the data from the instruments we get time series of current, temperature and salinity. We will now do some calciulations based on data from a mooring that
was deployed in the Amundsen Sea in 2012.
a) Import and plot the current measurements from mooring S4, 17-24 June 2012 (ENG_Riggdata_S4_1). Describe what you see!
b) What is the mean current? In what direction was the water flowing? (\(u\) is the current in the \(x\)-direction (towards the east) and \(v\) is the current in the \(y\)-direction (towards the
c) If you were to describe the observations with a mathematical function, which one would you chose?
d) Determine the coefficient of your function!
e) An iceberg is floating with the current in the vicinity of S4. Set up an expression for it movement (using vectors) and plot its trajectory. How would you describe its movement?
f) Plot the current a week forward in time using your function from (d).
g) Import the data from S4 24/6-1/7 and plot it in the same figure. Is your model doing a good job predicting the currents? Why? Why not? (The file is named ENG_Riggdata_S4_2)
Exercise 5
Import and plot the measurements of pressure from mooring C2. The file is named ENG_Riggdata_C2. In this location the currents were much stronger than we expected and we had put on too little
flotation 🙁
a) How would you interpret the record? (hint – look at exercise 2 above)
b) At what depth is the instrument located when the current is weak? (1 m \(\approx\) 1 dbar)
c) How deep is it pulled down?
d) What percentage of time is it pulled down more than 40 m? 80 m?
e) \(u\) gives the current in the \(x\)-direction (towards the east) and \(v\) gives the current in the \(y\)-direction (towards the north). Set up an expression for the current strength the
calculate and plot it! How strong is the strongest current? The mean current? How many km/h is that?
f) Is there a relation between current strength and pressure (that is, the instrument pull- down?) Describe the relation mathematically and with words.
Exercise 6
The instruments are powered by batteries, and every time we make measurements we use a small amount – or a lot if it is an ADCP – energy. We want the instruments to keep measuring until we get back
to pick them up – so we have to calculate beforehand how often we can make measurements without emptying the battery. But we also have to think about e.g. tides when determining how often to sample.
a) One of the components of the daily tides has a period of 25.8 h. The tidal current can be described by a sinus curve. The amplitude (and the phase) depends on where we are, but let’s assume the
amplitude is 10 cm/s and the phase is 0. Set up an expression for the tidal current and plot it for 30 days.
b) If we made measurements only once a day, what would or time series look like?
c) What is the period of the “oscillation” that we would then observe?
d) The tidal current tha we observe is the sum of many components with different periods, amplitudes and phases. Another component has a period of 23.93h. Set up an expression for that tidal
component if the amplitude is 9 cm/s. (You can set the phase to 0) and plot the sum of the two components. What do you see? How is the amplitude changing? How far is it between to “amplitude
maximum”? Can you explain what you see? (hint: plot the components as well as the sum of them). Compare the results with the results from exercise 3! | {"url":"https://elindarelius.no/2016/02/01/moorings-and-time-series/","timestamp":"2024-11-14T11:07:25Z","content_type":"text/html","content_length":"58343","record_id":"<urn:uuid:c567257c-40a8-4f07-b68d-93d3168f7683>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00167.warc.gz"} |
Rules for Graphs
The following rules for graphs are enforced in physics 1100. Each rule will have some examples showing bad graphs (and what exactly makes them bad) and good graphs. All of the graphs are actual
graphs that were submitted by students, with identifying information removed.
For help creating your own line graphs, I’ve created videos showing you how to do so in Microsoft Excel, Google Sheets, and by hand.
Use graph paper or a computer program such as Excel or Google Sheets
Why? If you draw a graph free-hand, it is difficult to make it accurate.
Bad graph example:
This graph is hand drawn on lined paper, not graph paper. (Another thing to note is that the percentages of the three data points don’t add to 100%!)
Another bad graph example:
This graph was plotted on a graphing calculator. Note that the graphing calculator does not include any values on the x or y axes, doesn’t label the axes (and because there are no labels there are no
units), and doesn’t have a title. If you don’t have a computer on which to make a graph, use a website like Google Sheets to do your graphs.
Good graph example:
This graph was plotted using a computer program. It is much easier to read. (In addition, the percentages of the three data points add up to 100%, as they should!)
Show each individual data point
Be clear about what each of your data points are. Why? Some graphing software makes a line that goes between data points without actually showing the points. This isn’t scientifically meaningful, and
doesn’t allow anybody to see the actual data that was collected in the experiment.
Bad graph example:
This graph doesn’t show any of the individual data points — it only shows a meaningless line that goes between the points.
Good graph example:
This graph shows all of the individual data points. The line is not necessary to see the trend of the data. It is linear and slopes upward.
The scale on both axes must be consistent
In other words, the interval between numbers must be the same everywhere along the horizontal axis, and everywhere along the vertical axis. Why? If you change the spacing between numbers, then you
can’t make an assessment of whether or not the data is linear.
Bad graph example:
Note that the x-axis scale is not consistent. The first square has a value of 0.1, then the next two squares have a value of 0.2, then the next square has a value of 0.5, then the next two squares
have values of 0.05… Note also the lack of units on both axes, and the fact that connect-the-dots was played.
Another bad graph example:
If you look at the vertical axis of this graph, it starts at -0.100, then decreases to -0.900, then jumps to 0.100 and then increases to 0.700. Remember that negative numbers get smaller when the
absolute value of the numbers get bigger! (If you can’t remember, ask yourself how much money is more money: owing somebody $100 or owing somebody $900.)
Another bad graph example:
The vertical axis of this graph is appropriately scaled. However, the x-axis has numbers that decrease, increase, decrease, increase, and decrease again. The x-axis of this graph does not make sense.
Good graph example:
The spacing on the x-axis is consistent. Each spacing has a value of 0.2 kg.
Put a title on the graph
A graph should be understood by looking at it, even if the reader is not sure what you are doing. A title allows anybody looking at the graph to understand what they are looking at.
Bad graph example:
This graph tells nothing about what is being plotted. There is no title, and the axes are not labeled and have no units. This data could be literally anything!
Good graph example:
This graph title tells me a lot more about what experiment was performed.
Label both axes and include units
The information on a graph should be scientifically meaningful. If it is not clear what is being plotted, then the graph is not meaningful. If units are not included in your graph, then it’s unclear
exactly what you are plotting. Remember that units are just as important as the quantities that they describe!
Bad graph example:
This graph has no label on the vertical axis, but it does have units. It should say: time for one oscillation (seconds). The horizontal axis has a label but no units. It should say: angle (degrees).
Note also that the student who made this graph played connect-the-dots with the data.
Good graph example:
This data has been plotted with a title, clearly labeled axes, and units. A nice smooth fit line was used between data points.
Don’t play connect-the-dots
A scientific graph is used to show a relationship between independent and dependent variables. All of the information on the graph should be meaningful. Drawing lines between each data point, like
the child’s game of connect-the-dots, is not scientifically meaningful. It does not allow us to understand the true relationship between the independent and dependent variables.
Bad graph example:
Note how there is a line drawn between every single data point. That connect-the-dot line does not add any meaningful information to this graph.
Better graph example:
This data does not include a fit line at all. It is better to not include a fit line than to play connect-the-dots, if you are not sure what type of fit line to use. A good example would have
included a linear fit line (as the data is linear, it just doesn’t change much with respect to time).
Use the correct type of fit line
If the data is linear, use a linear fit line. If the data is not linear, use the correct type of fit line. If you draw by hand, just draw a smooth line (don’t connect-the-dots) that follows the trend
of of the data. If you are using computer software, it is better not to add a fit line if you don’t know what type to use. Why? The fit line helps us to see through any variations in the data that
could be caused by inaccurate lab procedures or calculations. If the wrong type of fit line is used, then it is not scientifically meaningful. In addition, the fit line can be used to interpolate and
extrapolate data. The wrong type of fit line will lead to incorrect interpolations and extrapolations.
Bad graph example:
This graph uses a linear fit on data that is not linear. The data is actually a square root relationship, but it’s fine to just not include a fit line if you’re not sure what kind of relationship it
Good graph example:
This graph uses the correct type of fit line for the data. Notice how it doesn’t connect-the-dots, but makes a smooth curve through the data.
If there are two sets of data, there should be two fit lines
Why? Remember, the point of a fit line is for us to interpret the relationship between independent and dependent variables, and for us to interpolate and extrapolate data. If the fit line is not
correct, then it is not going to help us in our pursuit of science. Usually the “one fit line for two sets of data” is an issue with how the graphing software is used. The data needs to be entered
into the program as two separate data sets.
Bad graph example:
This graph shows two sets of data, but only has one fit line.
Good graph example:
This graph has two sets of data, and one fit line for each.
Do not use Excel’s “line mode”
In Excel, “line mode” will not create the type of graph that we are interested in. It creates a completely meaningless graph. Use “scatter plot” instead.
This graph was created using line mode in Excel. It separates the x-axis data from the y-axis data in a completely nonsensical way. Line mode might work for some things… but not physics graphs! This
graph leaves me completely unable to determine what relationship (if any) the angle of swing has with time.
Note also that there are no axes labels or units on this graph, and that the creator of this graph played “connect-the-dots” with the data.
Use the correct type of graph
The types of graphs used in physics 1100 are scatter plots (to determine if there is correlation between variables) in which there are no fit lines included. A line plot which shows each data point
with the correct type of fit line, when we know that there is a relationship between the independent and dependent variables. The third type is a bar graph, when there is a relationship between
independent and dependent variables, but when there is a qualitative independent variable that cannot be graphed numerically.
Bad graph example:
This is a bar graph used to plot quantitative data. A line graph should have been used instead.
Bad graph example:
A pie chart is not one of the approved types of graphs. This should be a bar graph.
Bad graph example:
This is the correct type of graph (bar graph) because the independent variable is qualitative. However, the question asked to plot the average percentage over each trial as a function of color. This
graph does not show that information. Instead it shows the number of cubes by color plotted as a function of trial number. Just because you are using the correct graph type doesn’t mean you are
following the instructions!
Bad graph example:
I am not sure what kind of graph this is, but it is not a line graph, as it should be for this type of data.
Bad graph example:
This graph is plotting quantitative data in a bar graph. It should be a line graph.
Unless otherwise mentioned, use the horizontal axis to plot the independent variable and the vertical axis to plot the dependent variable
Most of the time, we want to understand the relationship between the independent and dependent variables. Therefore we usually plot the IV on the horizontal axis, and see how the vertical axis
information (DV) changes. There are going to be a couple of exceptions to this rule. One of which is if we can swap the IV and DV in order to calculate the slope of a graph to get a meaningful
quantity such as mass or resistance.
Bad graph example:
This graph meant to show how the position of an object changes with respect to time. In that case, the independent variable is time and should be on the horizontal axis. Note also the lack of units
on each axis, and the connect-the-dots.
Good graph example:
This data shows the independent variable (time) on the horizontal axis, and the dependent variable (position) on the vertical axis.
Use up all available space on your graph paper, don’t make your graph so tiny that your professor can’t read it
Your professor is probably older than you are. Please make your graph easy for them to read! Also, it’s generally a good idea to make efficient use of space and make your graph easy to read for
anybody, not just your professor.
Bad graph example:
Note how all of the data is squished into a tiny section of the page. Rather than do this, expand the horizontal axis so that it takes up the entire page. (Note also that this graph does not contain
a title, and also plays connect-the-dots.)
Good graph example:
This graph makes efficient use of space in both axes. It is much easier to read without having all of the data squished into a small space.
Understand the meaning of the spacing used on the vertical axis
Note the spacing between tick marks on the vertical axis. If the spacing is very small, then there may not be a dramatic relationship between the IV and the DV. It is good practice to start the
vertical axis at 0, and then let your computer software do automatic scaling on the vertical axis. If you are drawing by hand, ask yourself what a good spacing would be. Why? It is not correct to say
that there are large changes in the DV when there is actually a very small change compared to the average value.
Bad graph example:
This graph isn’t technically wrong or bad, but the spacing on the vertical axis is just 0.2 seconds, which is about 10% of the average value. This makes the data appear to have a rather large upward
Good graph example:
This is the same data shown with the vertical axis scaled from 0. Note how it is a lot more clear now that the dependent variable essentially does not change with respect to the independent variable.
Don’t just include (0,0) as a data point
Don’t just include (0,0) in your data unless it was a data point that you collected.
Bad graph example:
This data includes (0,0) for all sets of data. It makes the data appear to be linear. Notice that it doesn’t necessarily “look wrong”, but it is wrong. It would imply that it took zero seconds for an
object to move a space of about a meter!
Good graph example:
This data does not include (0,0) as data points, because they are not data points! It’s also more clear from this data that the relationship between independent and dependent variables is not linear.
Plot all data from a single trial on a single graph
Unless your professor asks you to place two sets of data on one graph, each experiment that you conduct should have its own graph. It’s especially important not to put one set of data on two graphs…
then it becomes impossible to meaningfully compare any trends throughout the entire duration of the experiment.
This data all comes from a single experiment, but is placed on two separate graphs with differently scaled axes. This means that we cannot look at this data and determine what trend (if any) there is
between position and time. Note also that the y-axis, which has negative values, is positioned above the x-axis instead of going below the x-axis, which is where negative values belong.
Pay attention to your data and ask yourself if it makes sense
It’s always a good idea to do a “smell test” on your data. Does it look like it makes sense? Does the shape of the data appear consistent with what you expect? Sometimes just looking at a graph can
help you to see if you made an error somewhere in your calculations or measurements. If you’re not sure, ask your professor for help or clarification.
Bad graph example:
In this plot, the data forms a vertical line. Vertical lines almost never happen in real life. If you see a vertical line, ask yourself if something is incorrect. The student accidentally plotted
time on the horizontal axis, even though it is labeled as mass.
Bad graph example:
Ask yourself if it makes sense that most of the data is between 0 and 5 seconds, until the length changes to a meter, and then the data suddenly jumps to greater than 60. It’s unlikely that data is
going to have such a huge fluctuation like this.
Bad graph example:
Here is another example of the same bad graph as the one above. Does it make sense for acceleration to be almost -3000 m/s^2 and then immediately jump up to zero? No. So that data point should be
deleted. The relationship that all of the other data points have cannot be deduced because the outlier data point is preventing our ability to see any trends on this graph. | {"url":"https://doctor-pasquale.com/rules-for-graphs/","timestamp":"2024-11-02T15:52:42Z","content_type":"text/html","content_length":"142330","record_id":"<urn:uuid:8ee78c7f-c8bb-42d9-b179-c52c7c8f39d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00482.warc.gz"} |
The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: single-probe measurements from CMASS anisotropic galaxy clustering
With the largest spectroscopic galaxy survey volume drawn from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS), we can extract cosmological constraints from the measurements of redshift
and geometric distortions at quasi-linear scales (e.g. above 50 h−1 Mpc). We analyse the broad-range shape of the monopole and quadrupole correlation functions of the BOSS Data Release 12 (DR12)
CMASS galaxy sample, at the effective redshift z = 0.59, to obtain constraints on the Hubble expansion rate H(z), the angular- diameter distance DA(z), the normalized growth rate f(z)σ8(z), and the
physical matter density Ωmh2. We obtain robust measurements by including a polynomial as the model for the systematic errors, and find it works very well against the systematic effects, e.g. ones
induced by stars and seeing. We provide accurate measurements {DA(0.59)rs,fid/rs, H(0.59)rs/rs,fid, f(0.59)σ8(0.59), Ωmh2} = {1427 ± 26 Mpc, 97.3 ± 3.3 kms−1Mpc−1, 0.488 ± 0.060, 0.135 ± 0.016},
where rs is the comoving sound horizon at the drag epoch and rs,fid = 147.66 Mpc is the sound scale of the fiducial cosmology used in this study. The parameters which are not well constrained by our
galaxy clustering analysis are marginalized over with wide flat priors. Since no priors from other data sets, e.g. cosmic microwave background (CMB), are adopted and no dark energy models are
assumed, our results from BOSS CMASS galaxy clustering alone may be combined with other data sets, i.e. CMB, SNe, lensing or other galaxy clustering data to constrain the parameters of a given
cosmological model. The uncertainty on the dark energy equation of state parameter, w, from CMB+CMASS is about 8 per cent. The uncertainty on the curvature fraction, Ωk, is 0.3 per cent. We do not
find deviation from flat ΛCDM.
• astro-ph.CO
• cosmological parameters
• cosmology: observations
• distance scale
• largescale structure of Universe
Dive into the research topics of 'The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: single-probe measurements from CMASS anisotropic galaxy clustering'. Together
they form a unique fingerprint. | {"url":"https://researchportal.port.ac.uk/en/publications/the-clustering-of-galaxies-in-the-sdss-iii-baryon-oscillation-spe-19","timestamp":"2024-11-10T05:18:47Z","content_type":"text/html","content_length":"68248","record_id":"<urn:uuid:aa3fac73-f8f1-4249-8b4b-c0ef3626c243>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00690.warc.gz"} |
Theory of Evolution?
Charles Darwin, the father of the Theory of Evolution, earned a Bachelor’s Degree in Theology from Christ College at Cambridge University in 1831. He could have been the priest of a church in
England. It’s fairly safe to assume that if he had a degree in theology, he studied the Bible.
Charles Darwin based his theory of evolution on the proposition that all living things descended from a common ancestor and are the product of a mechanical process that alters every species over the
course of millions of years. Darwin waited over 20 years to publish On the Origin of Species by Means of Natural Selection in 1859 because he knew Christendom would get their panties all knotted up.
There is no shortage of controversy over when the first texts of the Hebrew Bible were written. There is nevertheless no doubt that the Dead Sea Scrolls predate all Hebrew texts that eventually
became the Hebrew Bible. Most of the scrolls were written on parchment and papyrus circa 350 BCE. They were written mostly in Hebrew but also in Aramaic and Greek. Hipparchus laid the foundation for
trigonometry circa 140 BCE, 210 years after the Dead Sea Scrolls were written. Nevertheless, in 350 BCE, there were no microscopes or telescopes. The most advanced technology of circa 350 BCE was the
wheel, which, ironically, is probably the most significant amelioration leading to the evolution of human civilization. The wheel was invented in the 4th millennium BC.
Did the Bible inspire Darwin’s Theory of Evolution?
Creationism and "Creation Science," which is religion and does not even qualify as a pseudo-science, are based upon the Hebrew Bible stories. The similarities between creationism and the theory of
evolution almost certainly suggest that Darwin's theory was influenced by the Bible
The chronological order of creation according to the Bible: Day 1: Light; Day 2: Atmosphere / Firmament; Day 3: Dry ground and plants; Day 4: Sun, moon, and stars; Day 5: Birds and sea creatures; and
Day 6: Land animals and humans.
The chronological order according to cosmology is: 1st: Big Bang (light); 2nd: matter accretes and forms stars, planets, and moons; 3rd: atmosphere suitable to form life on earth appears; 4th: life
forms in the oceans and fish evolve; 5th: sea life evolves into land animals and birds; 6th: humans evolve from land animals.
The Creation chronology simply isn't logical. How could the atmosphere on earth appear before the sun, moon, and stars appeared? So the earth existed before the entire universe? The explanation for
that could be as simple as a scribe error or that the scribe had too much beer the night before and had a hangover. A more plausible explanation for that is that the philosophers of the day lacked
the technology to test their hypotheses.
How did people with no scientific equipment or understanding of calculus and trigonometry formulate a basic understanding of cosmic evolution? Arguing about the differences in the chronological order
is balderdash. Both theories say the same thing. What fascinates me are the similarities. What makes sense is the Theory of Evolution was inspired by the bible. | {"url":"https://magabullshit.com/DidtheBibleInspireDarwinsTheoryofEvoluton.php","timestamp":"2024-11-13T05:06:28Z","content_type":"text/html","content_length":"10184","record_id":"<urn:uuid:b8ab7d65-02bd-4228-b107-370d5805e5e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00711.warc.gz"} |
Y = fft(X) computes the discrete Fourier transform (DFT) of X using a fast Fourier transform (FFT) algorithm. Y is the same size as X.
• If X is a vector, then fft(X) returns the Fourier transform of the vector.
• If X is a matrix, then fft(X) treats the columns of X as vectors and returns the Fourier transform of each column.
• If X is a multidimensional array, then fft(X) treats the values along the first array dimension whose size does not equal 1 as vectors and returns the Fourier transform of each vector.
Y = fft(X,n) returns the n-point DFT.
• If X is a vector and the length of X is less than n, then X is padded with trailing zeros to length n.
• If X is a vector and the length of X is greater than n, then X is truncated to length n.
• If X is a matrix, then each column is treated as in the vector case.
• If X is a multidimensional array, then the first array dimension whose size does not equal 1 is treated as in the vector case.
Y = fft(X,n,dim) returns the Fourier transform along the dimension dim. For example, if X is a matrix, then fft(X,n,2) returns the n-point Fourier transform of each row.
Noisy Signal
Find the frequency components of a signal buried in noise and find the amplitudes of the peak frequencies by using Fourier transform.
Specify the parameters of a signal with a sampling frequency of 1 kHz and a signal duration of 1.5 seconds.
Fs = 1000; % Sampling frequency
T = 1/Fs; % Sampling period
L = 1500; % Length of signal
t = (0:L-1)*T; % Time vector
Form a signal containing a DC offset of amplitude 0.8, a 50 Hz sinusoid of amplitude 0.7, and a 120 Hz sinusoid of amplitude 1.
S = 0.8 + 0.7*sin(2*pi*50*t) + sin(2*pi*120*t);
Corrupt the signal with zero-mean random noise with a variance of 4.
X = S + 2*randn(size(t));
Plot the noisy signal in the time domain. The frequency components are not visually apparent in the plot.
title("Signal Corrupted with Zero-Mean Random Noise")
xlabel("t (milliseconds)")
Compute the Fourier transform of the signal.
Because Fourier transforms involve complex numbers, plot the complex magnitude of the fft spectrum.
title("Complex Magnitude of fft Spectrum")
xlabel("f (Hz)")
The plot shows five frequency peaks including the peak at 0 Hz for the DC offset. In this example, the signal is expected to have three frequency peaks at 0 Hz, 50 Hz, and 120 Hz. Here, the second
half of the plot is the mirror reflection of the first half without including the peak at 0 Hz. The reason is that the discrete Fourier transform of a time-domain signal has a periodic nature, where
the first half of its spectrum is in positive frequencies and the second half is in negative frequencies, with the first element reserved for the zero frequency.
For real signals, the fft spectrum is a two-sided spectrum, where the spectrum in the positive frequencies is the complex conjugate of the spectrum in the negative frequencies. To show the fft
spectrum in the positive and negative frequencies, you can use fftshift. For an even length of L, the frequency domain starts from the negative of the Nyquist frequency -Fs/2 up to Fs/2-Fs/L with a
spacing or frequency resolution of Fs/L.
title("fft Spectrum in the Positive and Negative Frequencies")
xlabel("f (Hz)")
To find the amplitudes of the three frequency peaks, convert the fft spectrum in Y to the single-sided amplitude spectrum. Because the fft function includes a scaling factor L between the original
and the transformed signals, rescale Y by dividing by L. Take the complex magnitude of the fft spectrum. The two-sided amplitude spectrum P2, where the spectrum in the positive frequencies is the
complex conjugate of the spectrum in the negative frequencies, has half the peak amplitudes of the time-domain signal. To convert to the single-sided spectrum, take the first half of the two-sided
spectrum P2. Multiply the spectrum in the positive frequencies by 2. You do not need to multiply P1(1) and P1(end) by 2 because these amplitudes correspond to the zero and Nyquist frequencies,
respectively, and they do not have the complex conjugate pairs in the negative frequencies.
P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
Define the frequency domain f for the single-sided spectrum. Plot the single-sided amplitude spectrum P1. As expected, the amplitudes are close to 0.8, 0.7, and 1, but they are not exact because of
the added noise. In most cases, longer signals produce better frequency approximations.
f = Fs/L*(0:(L/2));
title("Single-Sided Amplitude Spectrum of X(t)")
xlabel("f (Hz)")
Now, take the Fourier transform of the original, uncorrupted signal and retrieve the exact amplitudes at 0.8, 0.7, and 1.0.
Y = fft(S);
P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
title("Single-Sided Amplitude Spectrum of S(t)")
xlabel("f (Hz)")
Gaussian Pulse
Convert a Gaussian pulse from the time domain to the frequency domain.
Specify the parameters of a signal with a sampling frequency of 44.1 kHz and a signal duration of 1 ms. Create a Gaussian pulse with a standard deviation of 0.1 ms.
Fs = 44100; % Sampling frequency
T = 1/Fs; % Sampling period
t = -0.5:T:0.5; % Time vector
L = length(t); % Signal length
X = 1/(0.4*sqrt(2*pi))*(exp(-t.^2/(2*(0.1*1e-3)^2)));
Plot the pulse in the time domain.
title("Gaussian Pulse in Time Domain")
xlabel("Time (t)")
axis([-1e-3 1e-3 0 1.1])
The execution time of fft depends on the length of the transform. Transform lengths that have only small prime factors result in significantly faster execution time than those that have large prime
In this example, the signal length L is 44,101, which is a very large prime number. To improve the performance of fft, identify an input length that is the next power of 2 from the original signal
length. Calling fft with this input length pads the pulse X with trailing zeros to the specified transform length.
Convert the Gaussian pulse to the frequency domain.
Define the frequency domain and plot the unique frequencies.
f = Fs*(0:(n/2))/n;
P = abs(Y/sqrt(n)).^2;
title("Gaussian Pulse in Frequency Domain")
xlabel("f (Hz)")
Cosine Waves
Compare cosine waves in the time domain and the frequency domain.
Specify the parameters of a signal with a sampling frequency of 1 kHz and a signal duration of 1 second.
Fs = 1000; % Sampling frequency
T = 1/Fs; % Sampling period
L = 1000; % Length of signal
t = (0:L-1)*T; % Time vector
Create a matrix where each row represents a cosine wave with scaled frequency. The result, X, is a 3-by-1000 matrix. The first row has a wave frequency of 50, the second row has a wave frequency of
150, and the third row has a wave frequency of 300.
x1 = cos(2*pi*50*t); % First row wave
x2 = cos(2*pi*150*t); % Second row wave
x3 = cos(2*pi*300*t); % Third row wave
X = [x1; x2; x3];
Plot the first 100 entries from each row of X in a single figure in order and compare their frequencies.
for i = 1:3
title("Row " + num2str(i) + " in the Time Domain")
Specify the dim argument to use fft along the rows of X, that is, for each signal.
Compute the Fourier transform of the signals.
Calculate the double-sided spectrum and single-sided spectrum of each signal.
P2 = abs(Y/L);
P1 = P2(:,1:L/2+1);
P1(:,2:end-1) = 2*P1(:,2:end-1);
In the frequency domain, plot the single-sided amplitude spectrum for each row in a single figure.
for i=1:3
title("Row " + num2str(i) + " in the Frequency Domain")
Phase of Sinusoids
Create a signal that consists of two sinusoids of frequencies 15 Hz and 40 Hz. The first sinusoid is a cosine wave with phase $-\pi /4$, and the second is a cosine wave with phase $\pi /2$. Sample
the signal at 100 Hz for 1 s.
Fs = 100;
t = 0:1/Fs:1-1/Fs;
x = cos(2*pi*15*t - pi/4) + cos(2*pi*40*t + pi/2);
Compute the Fourier transform of the signal. Plot the magnitude of the transform as a function of frequency.
y = fft(x);
z = fftshift(y);
ly = length(y);
f = (-ly/2:ly/2-1)/ly*Fs;
title("Double-Sided Amplitude Spectrum of x(t)")
xlabel("Frequency (Hz)")
Compute the phase of the transform, removing small-magnitude transform values. Plot the phase as a function of frequency.
tol = 1e-6;
z(abs(z) < tol) = 0;
theta = angle(z);
title("Phase Spectrum of x(t)")
xlabel("Frequency (Hz)")
Interpolation of FFT
Interpolate the Fourier transform of a signal by padding with zeros.
Specify the parameters of a signal with a sampling frequency of 80 Hz and a signal duration of 0.8 s.
Fs = 80;
T = 1/Fs;
L = 65;
t = (0:L-1)*T;
Create a superposition of a 2 Hz sinusoidal signal and its higher harmonics. The signal contains a 2 Hz cosine wave, a 4 Hz cosine wave, and a 6 Hz sine wave.
X = 3*cos(2*pi*2*t) + 2*cos(2*pi*4*t) + sin(2*pi*6*t);
Plot the signal in the time domain.
title("Signal superposition in time domain")
xlabel("t (ms)")
Compute the Fourier transform of the signal.
Compute the single-sided amplitude spectrum of the signal.
f = Fs*(0:(L-1)/2)/L;
P2 = abs(Y/L);
P1 = P2(1:(L+1)/2);
P1(2:end) = 2*P1(2:end);
In the frequency domain, plot the single-sided spectrum. Because the time sampling of the signal is quite short, the frequency resolution of the Fourier transform is not precise enough to show the
peak frequency near 4 Hz.
title("Single-Sided Spectrum of Original Signal")
xlabel("f (Hz)")
To better assess the peak frequencies, you can increase the length of the analysis window by padding the original signal with zeros. This method automatically interpolates the Fourier transform of
the signal with a more precise frequency resolution.
Identify a new input length that is the next power of 2 from the original signal length. Pad the signal X with trailing zeros to extend its length. Compute the Fourier transform of the zero-padded
n = 2^nextpow2(L);
Y = fft(X,n);
Compute the single-sided amplitude spectrum of the padded signal. Because the signal length n increased from 65 to 128, the frequency resolution becomes Fs/n, which is 0.625 Hz.
f = Fs*(0:(n/2))/n;
P2 = abs(Y/L);
P1 = P2(1:n/2+1);
P1(2:end-1) = 2*P1(2:end-1);
Plot the single-sided spectrum of the padded signal. This new spectrum shows the peak frequencies near 2 Hz, 4 Hz, and 6 Hz within the frequency resolution of 0.625 Hz.
title("Single-Sided Spectrum of Padded Signal")
xlabel("f (Hz)")
Input Arguments
X — Input array
vector | matrix | multidimensional array
Input array, specified as a vector, matrix, or multidimensional array.
If X is an empty 0-by-0 matrix, then fft(X) returns an empty 0-by-0 matrix.
Data Types: double | single | int8 | int16 | int32 | uint8 | uint16 | uint32 | logical
Complex Number Support: Yes
n — Transform length
[] (default) | nonnegative integer scalar
Transform length, specified as [] or a nonnegative integer scalar. Specifying a positive integer scalar for the transform length can improve the performance of fft. The length is typically specified
as a power of 2 or a value that can be factored into a product of small prime numbers (with prime factors not greater than 7). If n is less than the length of the signal, then fft ignores the
remaining signal values past the nth entry and returns the truncated result. If n is 0, then fft returns an empty matrix.
Example: n = 2^nextpow2(size(X,1))
Data Types: double | single | int8 | int16 | int32 | uint8 | uint16 | uint32 | logical
dim — Dimension to operate along
positive integer scalar
Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension whose size does not equal 1.
• fft(X,[],1) operates along the columns of X and returns the Fourier transform of each column.
• fft(X,[],2) operates along the rows of X and returns the Fourier transform of each row.
If dim is greater than ndims(X), then fft(X,[],dim) returns X. When n is specified, fft(X,n,dim) pads or truncates X to length n along dimension dim.
Data Types: double | single | int8 | int16 | int32 | uint8 | uint16 | uint32 | logical
Output Arguments
Y — Frequency domain representation
vector | matrix | multidimensional array
Frequency domain representation returned as a vector, matrix, or multidimensional array.
If X is of type single, then fft natively computes in single precision, and Y is also of type single. Otherwise, Y is returned as type double.
The size of Y is as follows:
• For Y = fft(X) or Y = fft(X,[],dim), the size of Y is equal to the size of X.
• For Y = fft(X,n,dim), the value of size(Y,dim) is equal to n, while the size of all other dimensions remains as in X.
If X is real, then Y is conjugate symmetric, and the number of unique points in Y is ceil((n+1)/2).
Data Types: double | single
More About
Discrete Fourier Transform of Vector
Y = fft(X) and X = ifft(Y) implement the Fourier transform and inverse Fourier transform, respectively. For X and Y of length n, these transforms are defined as follows:
$\begin{array}{l}Y\left(k\right)=\sum _{j=1}^{n}X\left(j\right)\text{\hspace{0.17em}}{W}_{n}^{\left(j-1\right)\text{}\left(k-1\right)}\\ X\left(j\right)=\frac{1}{n}\sum _{k=1}^{n}Y\left(k\right)\
${W}_{n}={e}^{\left(-2\pi i\right)/n}$
is one of n roots of unity.
• The execution time of fft depends on the length of the transform. Transform lengths that have only small prime factors (not greater than 7) result in significantly faster execution time than
those that are prime or have large prime factors.
• For most values of n, real-input DFTs require roughly half the computation time of complex-input DFTs. However, when n has large prime factors, there is little or no speed difference.
• You can potentially increase the speed of fft using the utility function fftw. This function controls the optimization of the algorithm used to compute an FFT of a particular size and dimension.
The FFT functions (fft, fft2, fftn, ifft, ifft2, ifftn) are based on a library called FFTW [1] [2].
[2] Frigo, M., and S. G. Johnson. “FFTW: An Adaptive Software Architecture for the FFT.” Proceedings of the International Conference on Acoustics, Speech, and Signal Processing. Vol. 3, 1998, pp.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• For limitations related to variable-size data, see Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
• For MEX output, MATLAB^® Coder™ uses the library that MATLAB uses for FFT algorithms. For standalone C/C++ code, by default, the code generator produces code for FFT algorithms instead of
producing FFT library calls. To generate calls to a specific installed FFTW library, provide an FFT library callback class. For more information about an FFT library callback class, see
coder.fftw.StandaloneFFTW3Interface (MATLAB Coder).
• For simulation of a MATLAB Function block, the simulation software uses the library that MATLAB uses for FFT algorithms. For C/C++ code generation, by default, the code generator produces code
for FFT algorithms instead of producing FFT library calls. To generate calls to a specific installed FFTW library, provide an FFT library callback class. For more information about an FFT library
callback class, see coder.fftw.StandaloneFFTW3Interface (MATLAB Coder).
• Using the Code Replacement Library (CRL), you can generate optimized code that runs on ARM^® Cortex^®-A processors with Neon extension. To generate this optimized code, you must install the
Embedded Coder^® Support Package for ARM Cortex-A Processors (Embedded Coder). The generated code for ARM Cortex-A uses the Ne10 library. For more information, see Ne10 Conditions for MATLAB
Functions to Support ARM Cortex-A Processors (Embedded Coder).
• Using the Code Replacement Library (CRL), you can generate optimized code that runs on ARM Cortex-M processors. To generate this optimized code, you must install the Embedded Coder Support
Package for ARM Cortex-M Processors (Embedded Coder). The generated code for ARM Cortex-M uses the CMSIS library. For more information, see CMSIS Conditions for MATLAB Functions to Support ARM
Cortex-M Processors (Embedded Coder).
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The fft function supports GPU array input with these usage notes and limitations:
• The output Y is always complex even if all the imaginary parts are zero.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
• For distributed arrays, instead of using a parallel FFT algorithm, fft gathers vectors on a single worker to perform prime length FFTs. For large prime-length vector FFTs, out-of-memory errors
can result.
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a | {"url":"https://ch.mathworks.com/help/matlab/ref/fft.html","timestamp":"2024-11-06T03:07:59Z","content_type":"text/html","content_length":"137265","record_id":"<urn:uuid:3cb98a82-4d7b-44ce-9383-6d71ec49a6bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00412.warc.gz"} |
Optimal state discrimination and unstructured search in nonlinear quantum mechanics
Title Optimal state discrimination and unstructured search in nonlinear quantum mechanics
Publication Type Journal Article
Year of Publication 2016
Authors Childs, AM, Young, J
Journal Physical Review A
Volume 93
Issue 2
Pages 022314
Date Published 2016/02/11
Nonlinear variants of quantum mechanics can solve tasks that are impossible
in standard quantum theory, such as perfectly distinguishing nonorthogonal
states. Here we derive the optimal protocol for distinguishing two states of a
qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum
mechanics that arises as an effective description of Bose-Einstein condensates.
Using this protocol, we present an algorithm for unstructured search in the
Abstract Gross-Pitaevskii model, obtaining an exponential improvement over a previous
algorithm of Meyer and Wong. This result establishes a limitation on the
effectiveness of the Gross-Pitaevskii approximation. More generally, we
demonstrate similar behavior under a family of related nonlinearities, giving
evidence that the ability to quickly discriminate nonorthogonal states and
thereby solve unstructured search is a generic feature of nonlinear quantum
URL http://arxiv.org/abs/1507.06334
DOI 10.1103/PhysRevA.93.022314 | {"url":"https://www.quics.umd.edu/publications/optimal-state-discrimination-and-unstructured-search-nonlinear-quantum-mechanics","timestamp":"2024-11-04T20:01:32Z","content_type":"text/html","content_length":"21836","record_id":"<urn:uuid:58cac7c4-436b-4129-b1d5-017009e189df>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00532.warc.gz"} |
Conformal Connections in Physics
881 views
For my diploma thesis in Mathematics I investigate conformal connections (as an example of Cartan connections). All in all the thesis should deal with geometric aspects (associated bundles, (pseudo)
-Riemannian manifolds, G-structures,...).
I just started my thoughts, where the concrete working assumption is not yet fixed. Besides I have a lot of freedom in writing my thesis; this is why I am looking for interesting fields to work.
Now, I'm interested in some physical applications. I think one of the fields most using this geometric object is general relativity. Is there a good reference (text book or paper)? Are there good
references for applications in other fields, i.e. QFT?
Further I'm interested in the following question: Are there any attempts to write the Yang-Mills equations in terms of conformal connections? If not: Is it ever useful to write down the YM equations
in this language?
This post imported from StackExchange Physics at 2016-05-05 11:28 (UTC), posted by SE-user Richard
If I understand what you mean by "conformal connection" correctly, then no, no gauge theory may be written in terms of it because gauge theories lack the solder forms that Cartan connections come
with. (That is, the principal bundle of gauge connections is not tied to the geometry of spacetime itsef as is the case with Cartan connections)
This post imported from StackExchange Physics at 2016-05-05 11:28 (UTC), posted by SE-user ACuriousMind
If you are studying conformal structures, the physics counterpart are the conformal field theories, which can be defined on conformal manifolds (while a general QFT requires metric structure). Much
of the work done in physics is for conformally flat manifolds, but there definitely are some works on curved setting as well.
This post imported from StackExchange Physics at 2016-05-05 11:28 (UTC), posted by SE-user Peter Kravchuk | {"url":"https://www.physicsoverflow.org/36020/conformal-connections-in-physics","timestamp":"2024-11-07T23:48:45Z","content_type":"text/html","content_length":"106337","record_id":"<urn:uuid:3dd7b17f-8673-47af-bed8-7275c9cf9886>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00566.warc.gz"} |
EQTC - Item: Magnetized Patch Plate
recipes|sources|full list
Magnetized Patch Plate
WT: 1.5 Size: MEDIUM
Class: NONE
Race: NONE
Stackable up to 1000
NPCs will not buy this item
Item lore: Still magnetic from long contact with electricity.
EQ item ID: 72008
Find recipes that use this item.
item|sources|full list
• Smithing Components: Magnetic Metal Brick, Smithy Hammer
In: Forge, Tanaan Forge, Forge - Floor (Placeable), Forge - Standing (Placeable), Guild Forge (Placeable)
Yield: 1
Also Returns: Smithy Hammer(1)
On Failure Returns: Smithy Hammer
Notes: EQRecipeID: 67062
Trivial at: 162
item|recipes|full list
No Sources found for this Item
Full Recipe List:
If you see (Restricted) click on (Restricted) to see details.
Some items have many recipes not directly related to the main item you're looking up. If you see (subcombines not shown) you will need to click on the item to get its recipes.
Items marked with (*) are vendor sold.
• To make Magnetized Patch Plate (Smithing: Yield 1, trivial 162), Combine the following in a Forge, Tanaan Forge, Forge - Floor (Placeable), Forge - Standing (Placeable), or
Guild Forge (Placeable)
□ Smithy Hammer (*)
□ Magnetic Metal Brick
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Magnetized Patch Plate
WT: 1.5 Size: MEDIUM
Class: NONE
Race: NONE
Stackable up to 1000
NPCs will not buy this item
Item lore: Still magnetic from long contact with electricity.
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machinist - [machinist] Trig Solution to Tapering Hole Diameter Problem
Subject: Machinist
List archive
• From: Lawrence London <lfljvenaura@gmail.com>
• To: machinist@lists.ibiblio.org, Lawrence London <lfljvenaura@gmail.com>, "Lawrence F. London, Jr." <lflj@bellsouth.net>, Lawrence London <venaurafarm@bellsouth.net>
• Subject: [machinist] Trig Solution to Tapering Hole Diameter Problem
• Date: Sun, 29 Dec 2013 06:28:09 -0500
Join Date
Jun 2008
Bellingham, WA
For anyone interested in the question I am posting the trig solution to the problem of spacing holes on a tapered arm with uniform edge-to-edge spacing along the axis of their centers. I decided
to start a new thread as this trig problem was not really the main point of the first thread Is there a scratch-free layout method available? . The trig solution included is applicable to any
number of holes with any taper and any spacing.
The "bottom line" solution is in red at the bottom of the page. For completeness the derivation of the solution is included above that.
There are two images attached. One is a photo of the drawing of just two circles with an exaggerated taper angle as this setup allowed a more clear separation of some of the tangent and
intersecting lines than would have been possible if the circles were drawn to the same taper as the part I was making. The second picture shows the part I intend to make drawn to scale with the
spacing of the holes and their radii indicated.
The work I did on this makes no sense in some ways as a nearly as good and practical result could have been obtained in a much shorter time by just using paper and pencil or a CAD program and
empirically fitting the circles between the lines drawn. I guess the challenge of getting a math solution sucked me in.... Maybe this math solution will be of use to someone else who doesn't care
to grind through the mental gymnastics to figure it out.
.................................................. .................................................. ..........................
The goal of this exercise to determine a method to calculate the diameters of circles of smoothly diminishing diameters and spaced at a specified distance from circumference to circumference and
with their centers falling on a straight line. In the diagram shown only two circles are considered. But, once the calculation is made for one pair of circles, subsequent calculations can be made
using the same formula.
a is the angle subtended by one line tangential to both of the circles and another center line through the center of the two circles.
R is the known radius of the larger circle.
S is the distance between the circles as measured on a line connecting their centers. Its length is arbitrary but is constant.
r is the radius of the smaller circle. Its value is unknown and we want to calculate it relative to R for a given S.
First we need to find the height of the line F-E:
A-B = R/cos(a)
E-D = (S+R)
D-B = [E-D/cos(a)] sin(a) = (S+R) tan(a)
A-D = A-B - D-B = F-E = [R/cos(a)] - [(S+R) tan(a)]
F-E = [R/cos(a)] - [(S+R) tan(a)]
Now that we have calculated F-E we can find the radius (r) of the smaller circle by determining F-E in terms of r:
H-G = H-J + J-G = r sin(a) + J-G = r sin(a) + r = r[sin(a) + 1]
H-E = H-G/cos(a)
E-G = H-E sin(a) = {r[sin(a) + 1] sin(a)}/cos(a) = r[sin(a) + 1]tan(a)
F-E = E-G + r = r([sin(a) + 1]tan(a) + 1)
So we have now determined F-E in terms of R and S and also determinedF-E in terms of r.
So: F-E = [R/cos(a)] - [(S+R) tan(a)] = r([sin(a) + 1] tan(a) + 1)
r = {[R/cos(a)] - [(S+R) tan(a)]}/({[sin(a) + 1] sin(a)} + 1)
r= [R/cos(a)] - [(S+R) tan(a)]
.......{[sin(a) + 1] tan(a)} + 1
Last edited by dgfoster; 12-28-2013 at 09:33 PM. Reason: formatting
Join Date
Sep 2005
San Diego
All I can say, is I'll keep you DGfoster, for any wild math I need help with. I'm not to bad in trig, but I'm
Stuck on formulas for an equal angular spiral, and the formulas for transformation from log-polar to Cartesian coordinates. Also known as a logarithmic spiral. I'd like to have an understandable
formula where I could spread sheet it and have it geometrically accurate where I could plug in an angle variable. I can build the excel sheet.
If thats ok,
• [machinist] Trig Solution to Tapering Hole Diameter Problem, Lawrence London, 12/29/2013
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HELP PLEASEEEEE THank You Fractions Amp Ratios Fill In Space
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HELP PLEASEEEEE THank you fractions & ratios fill in space:
Sagot :
Step-by-step explanation:
the answers is 2/3 /4/2 and 5/6
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Zaer ABU-HAMMOUR | Professor | Ph.D. | University of Jordan, Amman | UJ | Department of Mechatronics Engineering | Research profile
How we measure 'reads'
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Learn more
Research Interests..... Motion planning of robot manipulators, Inverse kinematics of robot manipulators, Control systems design (conventional and optimal), Model order reduction techniques, Numerical
differentiation and integration, Numerical solution of ordinary and partial differential equations, Numerical solution of initial and boundary value problems, Continuous and discrete optimization
problems, Continuous and conventional genetic algorithms.
September 2006 - September 2008
September 1998 - December 2002
Pakistan Institute of Engineering and Applied Sciences, Pakistan Atomic Energy Commission.
September 1996 - September 1998
Pakistan Institute of Engineering and Applied Sciences, Pakistan Atomic Energy Commission. | {"url":"https://www.researchgate.net/profile/Zaer-Abu-Hammour","timestamp":"2024-11-11T18:28:36Z","content_type":"text/html","content_length":"948738","record_id":"<urn:uuid:acda9c5f-d279-4f9a-86df-ddc43fa5e661>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00818.warc.gz"} |
The first interfacing lab covers some of the fundamental concepts of electronics and laboratory interfacing. These fundamental concepts are then used in other labs throughout the remainder of
the year.
An open inquiry lab asks the student to investigate (collect the data and find the algebraic equation) the cooling curve for hot water. The data is collected using a thermistor interfaced to a TRS-80
computer. The computer controls the experiment, makes a measurement every 1 0 minutes, stores the data on a disk file and displays a crude graph on the screen during the experiment. Later, the
students read the disk file , graph the data and find the algebraic equation.
Experiment K-1 in the lab manual investigates the kinetics of the reaction of bromocresol green and bleach. The bleach and dye are mixed and the time it takes the changing color to match a standard
solution is determined. When the two colors match, the timer is stopped. This proved difficult for some students. A more effective way is to interface a Blocktronic (a photoresistor and green LED
encased in two 2x4's bolted together) to a TRS-80. The internal clock of the computer is used to measure the elapsed time. The voltage output of the Blocktronic, while the color is changing, is
compared to the constant voltage output of the standard. When the two voltages match, the internal clock in the computer is stopped and the elapsed time is displayed on the screen.
The oxidation of persulfate ion by iodide ion is investigated by measuring the changing absorbance with time, using the Blocktronic (see laboratory module, LM024, Project SERAPHIM, Department of
Chemistry, University of WisconsinMadison, Madison, WI 53706). The computer collects the data and stores it on a disk file for future analysis. The initial slope of the line on the absorbancetime
graph is proportional to the initial rate of the reaction. Seven reaction rates at different concentrations and temperatures are measured, enabling the student to determine the order of the reaction
and the activation energy.
The molar mass of an unknown solute is determined by freezing point depression. A thermistor is interfaced to a TRS-80 computer and the cooling curves are obtained by the computer. The computer makes
the measurements, stores the data and displays
a crude graph on the screen. Later the student reads the disk file and graphs the data. The freezing points of the solvent and appropriate solutions are measured and eventually the molar mass of the
unknown solute is determined. A crude gas chromatograph is interfaced to a TRS-80 computer to separate a pentane-hexane mixture using a column of Tide detergent. This is a modification of procedures
used by several authors (Chemtrek: Small-Scale Experiments for General Chemistry, Stephen Thompson, Prentice Hall, 1990; Interfacing the High School Science Laboratory to a Computer, John Fox,
Vernier Software, Portland OR, 1988). The carrier gas is natural gas. An ir phototransistor detects changes in the temperature of the flame. The changing temperature is displayed on a line printer
connected to a TRS- 80. Various parameters, e. g. , retention times, plate thickness and the number of theoretical plates, are measured. The computer will determine the area under the curve by
numerical integration (trapezoid method). From this a crude calibration curve can be determined and the percent composition of various pentane-hexane mixtures determined.
The gas chromatograph is the last of a set of three chromatography experiments (TLC separation of washable and permanent inks, PC separation of 5 transition metal ions, and the gas chromatographic
separation of a pentane/ hexane mixture). The SERAPHIM computer program SEPARATIONS (AP808) is used to introduce the TLC experiment. During the first 45 minutes of a
two hour lab period the class views the program projected on a large screen using a LCD in the classroom. The program reviews some basic concepts of electronegativity and polarity in order to
introduce some simple ideas of chromatography. A simulation of a TLC separation of washable and permanent inks shows the considerations necessary in choosing a solvent. The students then move into
the lab and apply these concepts to separate the components of some washable and permanent inks.
the last four lab periods, the students are required to do a mini-independent research project. To introduce the students to the research process, they independently work through the SERAPHIM program
LAKE STUDY (AP804 or PC3704), a simulated investigation of the cause of a fish kill. They work through an assigned worksheet outlining the details of the lake study investigation.
To assist them in selecting a research project and to acquaint them with computer literature searching and the use of key words, the students use CHEMLAB (PC2001 ), a data base of labs from the
Journal of Chemical Education
In the Physical Chemistry Laboratory, students work with computers in a variety of ways. This course is taken in their junior I senior year. By that time they have had a programming course, either
BASIC or Pascal. They are asked to write several computer programs during this course.
The first computer assignment is to write a program to find the root of an equation (volume in van der Waals cubic equation) by four different algorithms: ( 1) successive approximations, ( 2)
bisection, (3) tangent, and (4) Newton They are given two modules: ITERATION AND COMPLJTER PROBLEM SOLVING, and ALGORITHMS FOR FINDING ZEROS OF FUNCTIONS (#478 and #264, respectively; COMAP, Inc., 60
Lowell St., Arlington, MA 0217 4 ), which describe these methods. They are asked to compare the methods for speed of convergence, accuracy, etc.
The second computer assignment requires the student to find the root of an equation (the solubility of an insoluble salt whose anion interacts with the solvent). They must write the six equations
involving the various equilibria (solubility product, two acid dissociation equations, dissociation of water, mass balance and charge balance). (See, for example, Fundamentals of Analytical
Chemistry, Douglas Skoog, Donald West and James Holler, Saunders, 6th ed., 1992, chapter 8.) The six simultaneous nonlinear equations must be reduced to one very nonlinear equation, whose root is
determined by one of the methods mentioned in the previous paragraph. involves numerical integration. The students are given Cp-T data and a module, ELEMENTARY TECHNIQUES OF NUMERICAL INTEGRATION AND
THEIR COMPUTER IMPLEMENTATION (#379, COMAP), and told to determine the entropy. The module describes three methods of numerical integration (left-rectangle method, trapezoid rule and Simpson's rule);
the students write a program using the three methods and compare the results.
The students are asked to find experimentally the Ksp of Cu(I03)2 by potentiometric titration. An ADA LAB card interfaces the electrodes to an Apple computer. The computer stores the data, displays a
graph of emf versus volume of titrant ( making it easier to follow the titration and select an appropriate volume of titrant for the next addition); when the titration is over, the computer prints
out the emf-volume data, including the average first and second derivatives of emf versus volume. The endpoint can be determined very accurately from the derivatives.
The students use a computer program that demonstrates the Monte Carlo simulation of a variety of chemical reactions. A module, MONTE CARLO: THE USE OF RANDOM NUMBERS TO SIMULATE EXPERIMENTS (#269,
COMAP), describes the Monte Carlo method. The Monte Carlo method is applied to five chemical reactions; some simple first and second order reactions so that the Monte Carlo results can be compared to
the results from the integration of the appropriate rate equation. Then more complicated reactions, that are not easily solved analytically, are "solved" by the Monte Carlo method to demonstrate its
The data from a kinetics experiment (the oxidation of SzOa- 2 by 1-) is analyzed, using two statistical techniques: analysis of variance and factorial design, to investigate the interaction of
temperature and catalyst concentration on the reaction rate. Spreadsheets area very convenient way of obtaining the required sums, sum of squares, squares of sums, etc. needed in this statistical
To illustrate the power of the computer in doing quantum chemical calculations, the students are asked to find the minimum energy of the He atom by a self-consistent-field inearcombination-
of-atomic-orbitals molecular-orbital (SCFLCAO- MO) calculation using the trail wave function C1 e·zeta-1 r + C2e-zeta·2 r. One needs to guess the values of zeta-1 and zeta-2, then find the Ci's and
the energy by iteration. The students are given the computer program that optimizes the Ci's and energy for a given set of zetas. The students spend one lab period trying to determine by whatever
method they choose the best set of zetas, i. e. , those that give the lowest energy. The following week they are given another computer program which uses a simplex optimization algorithm to find the
best set of zetas. They are then asked to compare the methods.
In Analytical Chemistry Laboratory the computer is used for record keeping and calculations. A potentiometric titration of iron is done by interfacing the electrodes to an Apple computer by an ADALAB
card. The computer stores the data, displays a graph of emf versus volume of titrant. At the conclusion of the titration, the emf-volume data is printed out, including the average first and second
derivatives of emf and volume; these derivatives enable the students to more accurately determine the endpoint of Copies of the laboratories described here are available from the author upon request. | {"url":"https://confchem.ccce.divched.org/1992FallCCCENLP10","timestamp":"2024-11-04T13:50:27Z","content_type":"application/xhtml+xml","content_length":"49581","record_id":"<urn:uuid:25e2d162-d0ac-4534-b7a4-7b2263af418f>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00796.warc.gz"} |
How do you write the partial fraction decomposition of the rational expression (x^3+x^2+x+2)/(x^4+x^2)? | HIX Tutor
How do you write the partial fraction decomposition of the rational expression #(x^3+x^2+x+2)/(x^4+x^2)#?
Answer 1
Quotient = $\left(\frac{1}{x} ^ 2\right) \left(x + 1\right)$
Remainder = 1
Denominator is #x^2(x^2+1)#. Rearrange the numerator as #x(x^2+1)+(x^2+1)+1# and divide.to get immediately the partial fraction decomposition
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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What is gross margin
What is gross margin?
The concept of gross margin refers to the direct benefit that a company obtains from a service or good. To calculate it, the difference between the sale price of the item excluding VAT and the
production cost must be taken into account. This term is also known as profit margin. It is generally calculated as a percentage of the total sales of the business.
The gross margin is the direct profit from the company's activity, without discounting impuestos, personnel expenses and others costes. The interpretation of the gross margin on sales is very
important, since thanks to it, businesses will be able to determine if they are profitable or not. In the event that the gross margin is negative, this will mean that it is impossible to cover all
other expenses. Hence, it also receives the name of gross profit.
What is gross margin for?
This term, gross margin, is a good gauge of the state of the business. If we are able to compare it with other similar businesses, we will know if we are working on the correct line based on the
benefits of other companies in the sector.
An example of gross margin could be understood in the situation of a company dedicated to the manufacture of clothing that generates sales income of 1.000 euros and the production involves an expense
of 600 euros. In this way, the gross margin would amount to 400 euros. For every euro of sale it generates 40 cents.
How to calculate gross margin?
When making investment decisions, gross margin is used as an analysis method alongside the solvency ratio or discounted cash flows.
There are three gross margin formulas to help you understand this concept a little more thoroughly:
• Gross margin formula = Sales - Cost of goods sold
• Gross margin percentage formula -% Gross margin = (Sales - Cost of sales) / Sales
• Unit gross margin formula = Selling price - Item unit cost
The latter case would imply subtracting the sale price of the item less the cost of producing it.
Calculator for profit margin
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PPT - Chapter 2 PowerPoint Presentation, free download - ID:7030097
1. Chapter 2 Combinational Systems And / Or / Not
2. TRIAD PRINCIPLE: Combinational is about And / Or / Not combinations As well as equivalent functions. It does not include memory or feedback. Input to any system is a switch. Output is an LED. The
devices are OFF (0, false) or ON (1, true)
3. Continuing Examples (CE) These are representative and will show up being implemented in various ways. CE1. A system with four inputs, A, B, C, and D, and one output, Z, such that Z=1 if three of
the inputs are 1. CE2. A single light (that can be on or off) that can be controlled by any one of three switches. One switch is the master on/off switch. If it is off, the lights are off. When
the master switch is on, a change in the position of one of the other switches (from up to down or from down to up) will cause the light to change state. CE3. A system to do 1 bit of binary
addition. It has three inputs (the 2 bits to be added plus the carry from the next lower order bit) and produces two outputs, a sum bit and a carry to the next higher order position. CE4. A
system that has as its input the code for a decimal digit, and produces as its output the signals to drive a seven-segment display, such as those on most digital watches and numeric displays
(more later). CE5. A system with nine inputs, representing two 4-bit binary numbers and a carry input, and one 5-bit output, representing the sum. (Each input number can range from 0 to 15; the
input can range from 0 to 31.)
4. SOLUTION TECHNIQUES Step 1: Represent each of the inputs and output in binary. Step 1.5: If necessary, break the problem into smaller subproblems. Step 2: Formalize the design specification
either in the form of a truth tableor of an algebraic expression. Step 3: Simplify the description. Step 4: Implement the system with the available components, subject to the design objectives
and constraints.
5. Truth table • Create column for each input • Create column for each output • Number of rows needed = 2#cols • Start with inside column, alternate 0 & 1 • Next column uses pairs • Outside column
is first half 0, second 1 • Result should give all possible combinations • Output shows result for corresponding input. • Labels are flexible, in = low characters, out=high • Each row represents
a digital value made of the o & 1’s.
6. CE1. A system with four inputs, A, B, C, and D, and one output, Z, such that Z=1 if three of the inputs are 1. • Multiple ways to interpret. • One output for each way • Only 3 have 1 • 3 or more
have 1 • >3 don’t care • Four inputs, 3 outputs • First row has value of 0000 = 0 • Second row value of 0001 = 1 • Last row value of 1111 = 15 • Outputs are independent.
7. CE2. A single light (that can be on or off) that can be controlled by any one of three switches. One switch is the master on/off switch. If it is off, the lights are off. When the master switch
is on, a change in the position of one of the other switches (from up to down or from down to up) will cause the light to change state. 1. A is master, if it =0, then out = 0 2. Two different
interpretations depend on start state. 3. If start = 0, then change makes out = 1 4. If start = 1, then change makes out = 0 Complete description is critical to get correct results or desired
8. CE3. A system to do 1 bit of binary addition. It has three inputs (the 2 bits to be added plus the carry from the next lower order bit) and produces two outputs, a sum bit and a carry to the next
higher order position. This problem was done in Chapter 1 with the illustration of a full adder. CE5. A system with nine inputs, representing two 4-bit binary numbers and a carry input, and one
5-bit output, representing the sum. (Each input number can range from 0 to 15; the input can range from 0 to 31.) This problem would require 512 rows using the techniques we have so far. Later it
will be illustrated with other processes.
9. CE4. A system that has as its input the code for a decimal digit, and produces as its output the signals to drive a seven-segment display, such as those on most digital watches and numeric
displays (more later). Inputs are a code for the decimal digit. W, X, Y, Z Outputs are 7 lines to light 7 leds. A common provides the return path. Digits 6, 7, & 9 have alternate ways to display.
10. Table 2.6 A truth table for the seven-segment display driver. This is 8421 code. 1 indicates the segment is lit. 0 is off, X is don’t care. Note DC for 6, 7, 9 Note DC for all digits >9 The other
inputs can be anything
11. Switching Algebra 3 & only 3 operators OR (written as +) a + b (read a OR b) is 1 if and only if a = 1 orb = 1 or both. AND (written as ·, * , or simply two variables catenated) a · b = ab (read
a AND b) is 1 if and only ifa = 1 andb = 1. NOT (written´) a´ (read NOT a) is 1 if and only if a = 0. The complement (not) is not distributive new
13. Equations are simply mathematical symbols or shortcuts to illustrate relationships. Diagrams are another way of representing the same information. Properties ( Inside cover) Associative
illustration using gate symbols for operators Associative property states all three symbols have same output a (b c) = (a b) c = a b c
14. Manipulation of Algebra (binary / switching)Definitions for AND A literal is the appearance of a variable or its complement. A product termis one or more literals connected by AND operators. A
standard product term, also mintermis a product term that includes each variable of the problem, either uncomplemented or complemented. A sum of products expression (often abbreviated SOP) is one
or more product terms connected by OR operators. A canonical sum or sum of standard product terms is just a sum of products expression where all of the terms are standard product terms. a (b c) =
(a b) c = a b cstandard products a b + a’ b’ canonical sum
15. A minimum sum of products expression is one of those SOP expressions for a function that has the fewest number of product terms. If there is more than one expression with the fewest number of
terms, then minimum is defined as one or more of those expressions with the fewest number of literals. (1) x´yz´ + x´yz + xy´z´ + xy´z + xyz 5 terms, 15 literals (2) x´y + xy´ + xyz 3 terms, 7
literals (3) x´y + xy´ + xz 3 terms, 6 literals (4) x´y + xy´ + yz 3 terms, 6 literals
16. Manipulation of Algebra (binary / switching)Definitions for OR A sum term is one or more literals connected by OR operators. A standard sum term, also called a maxterm, is a sum term that
includes each variable of the problem, either uncomplementedor complemented. A product of sums expression (POS) is one or more sum terms connected by AND operators. A canonical product or product
of standard sum terms is just a product of sums expression where all of the terms are standard sum terms. SOP: x´y + xy´ + xyz POS: (x + y´)(x´ + y)(x´ + z´) Both: x´ + y + z or xyz´ Neither: x
(w´ + yz) or z´ + wx´y + v(xz + w´)
17. Representation of POS Equation form: output = product of input + product of inputs Diagram form: input – product – sum = output F = x’y + xy’ + xz
18. Devices and software can use different representations of logic levels. Binary has only 0 & 1. These can be complemented to 1 & 0. (Duh) In an electrical circuit 0 volts is considered as logic 05
(or some other value) volts is considered as logic 1 Either 0 or 1 can be interpreted as active.Positive logic is conventional.Negative logic is the complement and is easier to physically build.
Complement function: leave a & b, complement only F. Demorgan’s Theorem Negative Logic is different: complement every term in the table. F = a + b F ‘= (a + b)’ = a’b’
19. From truth table to algebraic expression Each grouping is the same way of saying the info in the truth table f is 1 if a = 0 AND b = 1 OR if a = 1 AND b = 0 OR if a = 1 AND b = 1 f is 1 if a´ = 1
AND b = 1 OR if a = 1 AND b´ = 1 OR if a = 1 AND b = 1 f is 1 if a´b = 1 OR if ab´ = 1 OR if ab = 1 f = a´b + ab´ + ab
20. Minterms Given 3 inputs How many states (rows) result? 2#inputs Mintermis a product term that includes each variable of the problem, either uncomplemented or complemented. Each combination of
inputs (on a row) has an equivalent decimal number Suppose the function (output) was true for number 1,2,3,4,5. Then, F = Σ (1,2,3,4,5) F = A’B’C + A’BC’ + A’BC + AB’C’ + AB’C F’ = Σ (0,6,7) F =
(F’)’ = maxterms. Use Demorgan
21. How many different functions can come from two variables? N variable = 2n rows Number of functions = 2 rows
22. A NAND is an AND invert. It is also an invert OR A NOR is an OR invert. It is also an invert AND So, two NAND gates can represent OR, AND, NOT So, two NOR gates can represent OR, AND, NOT new
23. A NAND can be used to represent a OR, AND, or NOT by using combinations. F = x’y + xy’ + xz For the AND use a NAND then invert it. For the OR invert it then use a NAND. A NAND – NAND logically is
null. (a’)’ = a
24. Exclusive OR: one or the other, but not both f = a’b + ab’ f = Σ(1,2)
25. CONCENSUS For any two product terms where exactly one variable appears uncomplemented in one and complemented in the other, the consensus is defined as the product of the remaining literals. If
no such variable exists or if more than one such variable exists, then the consensus us undefined. If we write one term as at1 and the second as a´t2 (where t1 and t2 represent product terms),
then, if the consensus is defined. at1¢ a´t2 = t1t2 P13a.at1 + a´t2 + t1t2 = at1 + a´t2 P13b. (a + t1)(a + t2)(t1 + t2)= (a + t1)(a + t2) The consensus theorem allows you to remove the consensus
terms (t1t2) Not the two terms that formed the consensus (at1 + a’t2)
26. Consider a function with five literal inputs Implement the function using two-input NAND gates. We could factor. G = C´(A´B´ + ABE) + DE´ + CD´E G = C´(B´ + E)(B + A´) + DE´ + CD´E G = (C´ + D´
E)[C + (B´ + AE)(B + A´)] + DE´ G = DE´ + A´B´C’ + CD´E + ABC´E Remember NANE = AND invert = invert OR | {"url":"https://www.slideserve.com/lana-combs/chapter-2-powerpoint-ppt-presentation","timestamp":"2024-11-02T15:26:26Z","content_type":"text/html","content_length":"95819","record_id":"<urn:uuid:d1156132-0aa5-4685-8eb0-0d9e2e05e8bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00747.warc.gz"} |
, and
Study of
letter serial correlation (LSC)
in some English, Hebrew,
Aramaic, and Russian texts
1. Measurement and calculation
By Brendan McKay and Mark Perakh
Posted October 20, 2009
1. Introduction
2. Letter Serial Correlation effect. Measurement and calculation
3. Calculation of the expected serial correlation sum
4. Calculation of the expected serial correlation density
5. Approximate estimate of the expected correlation sum for n=1
6. Approximate calculation of the expected correlation sum for arbitrary n
Meaningful texts consist of paragraphs (or verses), sentences, words, and at the most basic level, of letters. To convey a meaningful message, all this elements of a text must be placed in a certain
order, prescribed by the language's grammatical rules and by the specific contents. As a result, each meaningful text is highly structured, comprising many levels of order superimposed upon each
other in a complex manner. The complexity of a text's structure is assured by the enormous number of possible combinations of letters, words, sentences etc.
A general measure of the text's degree of disorder vs order is its entropy. Having determined the entropy of a text provides only a generalized idea of the degree to which the text is not random.
Different types of information could be extracted from texts by unearthing specific forms of order present in a text and by trying to connect them to the semantic peculiarities or to the
meaning-bearing contents of texts. Any information obtained in that respect seems to be of interest if one wishes to understand such a complex and extremely important phenomenon of human's existence
as language.
The subject of this paper emerged as a side topic in the course of investigation of the Bible code controversy [1-4] which largely deals with the so called ELS (Equidistant Letter Sequences) found in
abundance in the Bible, as well as in any non-Biblical texts. While it is hard to indicate the direct connection of the above controversy to the effect described in this paper, the effect in question
seems to be of interest in its own right, and, moreover, some connection to ELS, which is not obvious at this time, may well be found later.
To avoid introducing any new terminology when it is not dictated by the requirement of clarity, in the following parts of this paper we will use word "text" both for meaningful texts, such as that of
the Book of Genesis, or L. Tolstoy's novel War and Peace, etc, and for any random collection of symbols, including those obtained by permuting letters of the original meaningful text, even when these
collections of symbols (in our case letters of alphabets) constitute a gibberish without conveying any meaningful contents.
One of the many types of order found in texts is what we will refer to as Letter Serial Correlation (LSC), and this paper reports on a study of that type of order in some English, Hebrew, Aramaic,
and Russian texts. Its essence is as follows.
Let us denote the total number of characters in the text by L. We divide the text into k segments of equal size n=L/k. These segments will be referred to as chunks. The total number of occurrences of
a specific letter x in the entire text will be denoted M[x] . Let the numbers of occurrences of letter x in any two adjacent chunks, identified by serial numbers i and i+1, be X[i] and X[i+1] . We
will be measuring and calculating the following sum taken over all letters of the alphabet (i.e. for x varying between 1 and z where z is the number of letters in the alphabet) and over all chunks
(i.e. for i varying between 1 and k):
In this study, the measurement of the above sum was performed using a computer program which divided the text into various numbers k of equal chunks, counted the numbers of each letter in each chunk,
and calculated expression (A). If the division of the text into k chunks resulted in the last chunk (chunk # k) to be incomplete, i.e. having less letters than the rest of the chunks, such residual
incomplete chunk was cast off and not accounted for in expression (A).
If the total number of complete chunks in the text is k, there are k-1 boundaries between the chunks, and since the pairs of adjacent chunks's overlap, there are k-1 pairs of chunks. The summation is
performed both over (k-1) pairs of chunks and over all z letters of the alphabet. If a certain letter is absent in both adjacent chunks, the term in sum (A) corresponding to that letter and to the
pair of chunks in question is zero.
Consider the particular case of chunks having size n=1. In that case the number of chunks in the text is k=L, where L is the total length of the text. Chunks of that size can contain only one letter
each. Therefore the terms in sum (A) in that case can only be either 0 or 2. The zero value happens in two possible situations. One is when some two adjacent chunks contain the same letter x. In this
case the term in sum (A), corresponding to letter x and to that pair of chunks, becomes zero. The other situation is when some letter x is absent in both of any two adjacent chunks. Then the term in
sum (A) corresponding to x and to that pair of chunks also becomes zero. If one of the adjacent chunks contains letter x, and its neighhboring chunk contains another letter y, then both x and y found
in that pair of chunks contribute to sum (A) equal terms of 1, so the total contribution to sum (A) of that pair of chunks is 2. Therefore the maximum possible value of sum in (A) is S[m]=2(L-1),
which happens if no two adjacent chunks contain identical letters. If n>1, the maximum possible value of the measured sum will be correspondingly larger, and its calculation is more complex. What is
of interest though is not the maximum possible value of sum (A) but its expected value, which we will calculate precisely for texts randomized by permutations.
If all the chunks contained exactly equal numbers of each letter, then obviously we would find that S[m]= 0. The actual behavior of S[m ], in particular in its relation to the calculated "expected"
sum, and in comparison to its behavior in randomized texts, would indicate the presence of a certain type of order in the tested texts. Unearthing the features of such order is the goal of this
To analyze the behavior of the measured sum in the real meaningful texts, we need to be able to compare it with the behavior of the expected sum S[e], calculated on the assumption of the text being a
randomized conglomerate of z letters, each letter having the frequency of its occurrence in the randomized text exactly equal to its frequency in the real, not randomized, meaningful text.
We have to distinguish between perfectly random texts and texts randomized by permutation of a specific initial text.
The text, which has been randomized by a permutation of the letters of a specific initial text, contains the same letters as the original text, with the same letters' frequency distribution. It means
that every letter x which happens M[x] times in the original text (which also may be referred to as identity permutation) will happen the same M[x] times in every random permutation of the letters of
the original text. Depending on the composition of the original text, the numbers of occurrences of each letter will be different for each original text but the same in all of its random
There can be, rarely, a situation, when a certain letter is absent in the original text, and then it will be also absent in all of its permutations. A good example is the novel titled A Story of Over
50000 Words Without Using Letter E, by E.V. Wright, published in 1939 by Wetzel Publishing Co of Los Angeles. Letter E is the most frequent one in English (as it is also in German and Spanish). E.V.
Wright managed though to write a novel 267 pages long without using letter E even a single time. Obviously, any random permutation of the text of that novel would not contain letter E either.
A perfectly random text is different. In a perfectly random text each letter of the alphabet has the same chance to appear at any location in the text, and in a sufficiently long text the letters
frequency distribution is uniform.
The following section contains the derivation of a formula for the calculation of the expected sum S[e] , based on the assumption that the text in question has been randomized by permuting its
letters. (For perfectly random texts the formula would need to be slightly modified).
Considering the distribution of values of X we have to make choice between multinomial and hypergeometric distributions [5]. The first one, being an extension of the binomial distribution, pertains
to tests with replacement, while the second one, to the tests without replacement. In our case the stock of letters available to fill up a chunk is limited to the set of letters contained in the
identity permutation. After letter x has been picked for a chunk, there is no replacement for it available in the stock of letters when the second letter is to be picked (which does not mean that the
second letter cannot be identical with the first one, but only that the choice of letters becomes more restricted with every subsequent letter to be plucked from the stock). Therefore our situation
is obviously meeting the conditions of tests without replacement. Hence, we postulate hypergeometric distribution of X, being identical for chunks i and i+1 as the chunks are of the same size.
Since the sizes of all chunks (in the same test) are identical, we have
Var(X[i]) = Var (X[i+1])...................................(1)
E(X[i])=E(X[i+1]), ...........................................(2)
where Var(X) denotes variance and E(X) denotes expected value of X [5].
Step 1. Variance is determined by the following formula of Math. statistics [5, page 175]:
Var (X) = E(X^2) - [E(X)]^2...........................(3)
The first term on the right side of eq. (3) is the expected value of squared X and the second term is the squared expected value of X.
Consider expression E[(X[i]+X[i+1])^2] i.e. the expected value of a squared sum of X[i] and X[i+1].
Applying formula (3) we have
E[(X[i]+X[i+1])^2]=Var (X[i]+X[i+1]) + [E(X[i]+X[i+1])]^2..................(4)
From Mathematical Statistics [4] the expected value of a sum equals the sum of expected values of its components. Accounting also for eq. (2), we obtain from (4):
E[(X[i]+X[i+1])^2]=Var (X[i]+X[i+1]) + 4[E(X[i])]^2...........................(5)
Now consider expression
E[(X[i]-X[i+1])^2] + E[(X[i]+X[i+1])^ 2].........................................(6)
Replacing the sum of expected values with the expected value of the sum and accounting for eq.(2) we get from (6)
E[(X[i]-X[i+1])^2] + E[(X[i]+X[i+1])^2 = E[(X[i]-X[i+1])^2 + (X[i]+X[i+1])^2] =
= E[(X[i]^2+X[i+1]^2- 2X[i]X[i+1]+ X[i]^2+X[i+1]^2+2X[i]X[i+1]]=
=E[2X[i]^2+2X[i+1]^2] =E[4X[i]^2] = 4E[X[i]^2]...................(7)
Now subtract eq (5) from eq (7):
[(X[i]-X[i+1])^2] = 4E[X[i]^2] - 4 [E(X[i])]^2 - Var (X[i]+X[i+1])..............(8)
From eq. (3) we see that the first two terms in the right side of (8) equal 4Var[X[i]]. It yields
E[(X[i]-X[i+1])^2] = 4Var[X[i]] -Var [X[i]+X[i+1]]..................(9)
Comment: 1) If the text under consideration were a perfectly random one, then X[i ]and X[i+1] would be independent variables. Our text is though not a perfectly random one, as defined earlier in this
paper, but a text randomized by permutation. In a perfectly random text, every letter of the alphabet is equally available to fill any site in that text. In a text randomized by permutation only
those letters are available to fill up the chunks which are present in the original text, and in specific numbers M[x]. Therefore, if chunk #i contains more of a letter x, it diminishes the available
stock of that letter x for chunk #(i+1). Hence, there is a certain negative correlation between X[i] and X[i+1], which means these two numbers are not independent variables. Therefore variance of the
sum X[i]+X[i+1 ]cannot be replaced with the sum of variances [5]. Var (X[i]) and Var(X[i]+X[i+1]) in formula (9) must be calculated separately and then substituted into (9). If though X[i] and X[i+1]
were independent variables, i.e if we assumed that the text was perfectly random, then the right side of equation (9) would reduce to 2Var(X[i]).
Step 2.
In the case of a hypergeometric distribution the formula for variance is as follows [6, page 219]:
where p=M[x]/L, and in our case, for the first term on the right side of (9) the sample size m[1]=n where n=L/k, k being the number of chunks in the particular text, and n being the size of a chunk.
L is the total number of all letters in the entire text, and M[x] is the total number of occurrences of character x in the entire text. For the second term on the right side of (9), the sample size
is m[2]=2n=2L/k. Then :
4Var [X[i]] = 4 (L-L/k)(1-M[x] /L)M[x]/k(L-1),
or, after an elementary algebraic operation,
Similarly, replacing L/k with 2L/k we obtain for the second term in (9)
Finally, substituting (11) and (11a) into (9) we obtain
E[(X-X[i+1])^2] = 2M[x](L-M[x])/k(L-1)........................... (12)
The next step on the way to calculating the serial sum S[e] is summing up expressions (12) for all pairs of chunks and for all letters of the alphabet. Since all chunks in the same test have the same
size and the distribution of each letter is identical for all chunks, the summation over all pairs of chunks can be effected simply by multiplying expression (12) by k-1, which is the number of pairs
of chunks in the text. Then the final formula for the calculation of the expected serial sum is as follows:
Comment: * If X[i] and X[i+1] were independent variables, i.e. if we assumed that the text was perfectly random, the distribution of any X within a chunk would be approximated by a binomial
distribution (as a marginal distribution of a multinomial one) rather than by a hypergeometric distribution, since in a perfectly random text the stock of available letters is unlimited. It would
make our case analogous to tests with replacement. The actual calculation (which we omit here) shows that using the variance for a binomial distribution yields a formula which differs from (13B) only
by a factor of (L-1)/L. Since the text's lengths in our study were typically minimum tens of thousands letter long, the quantitative difference between formula (13b) and that for a perfectly random
text turns out to be utterly negligible. *
For each value of k the summation in (13) is performed over all letters of the alphabet, accounting for the actual numbers M[x] of occurrences of each letter in the tested text.
Since k=L/n, where n is the size of a chunk, equation (13B) can be rewritten as an explicit function of chunk's size n:
Comments: a) The sum in formulas (13B) and (13C) contains as many terms as there are various letters in the text. With a very few exceptions, texts usually contain all letters of the alphabet,
although in different numbers M[x]. Therefore, the sum in (13) almost always contains z terms, where z is the number of letters in the alphabet.
b) Theoretically, equation (13C) appears to be one of a straight line in S[e]-n coordinates, with the intercept
and the negative slope
An equation in the form S[e]=A-Bn describes a straight line starting at S[e]=A when n=0 and dropping to zero at n=L. However, quantities A and B are actually not constant for the following reason. In
actual calculations, the text is divided into k chunks, each of size n. For n=1 always k=L. However, already for n=2 two different situations are possible. If the total number L of letters in the
text is even, then for n=2, k=L/2, and the total length L of the text in formula (13C) is the same L as for n=1. If, though, L happens to be an odd number, the last chunk is a residual one,
containing only one letter instead of n=2. In this case the last chunk is cast off, both when calculating S[e] by formula (13) and when measuring S[m ]in accordance with formula (A). Then in formula
(13C), instead of L, the quantity of L-1 is used. This may also change by 1 the quantity M[i] for one of the letters. Hence, in the case of an odd L, the intercept A and the slope B become slightly
different for n=2 compared to n=1.
Analogously, for each value of n, the last chunk may happen to have fewer letters than n, and such a chunk is cast off. For example, the Book of Genesis in Hebrew comprises 78064 letters. Then, if
the chunk's size is chosen to be n=1, the number of chunks will be k=78064. For chunk's size of n=2 the number of chunks will be k=78064/2=39032, and the overall length of the text is L=78064, which
is the same as for n=1. However, if the chunk's size is n=3, the number of chunks appears to be k=78064/3= 26021.333. The number of chunks cannot be fractional, therefore for n=3 the number of chunks
must be taken as k=26021, casting off the last, incomplete chunk, whose size is 0.333 of a complete chunk. This means truncating the text, whose length L in formula (13) will be replaced by L*=
26021*3=78063 instead of L=78064. This changes the values of the intercept A and slope B in equation (13).
The variations in the values of A and B are different for various values of n. When the size of a chunk is measured in thousands, the last, incomplete chunk may be substantial in size (for example,
if the size of a chunk is chosen to be 10000, the amount by which the text is truncated can be as large as 9999 letters). In Table 1, as an example, the values of L* are shown for the text of the
Book of Genesis, as a function of the chunk's size n. This table illustrates the variations in the texts' lengths, used for calculation of S[e] and for measurement of S[m], which occur because of the
text's truncation.
Larger size of the cast off chunk does not necessarily translate into a larger variation of A and B, since simultaneously with the decrease of L (due to truncation) also the values of M[i ]for some
letters decrease, thus softening the overall variation of A and B.
Table 1. Actual texts' lengths L* as a function of n and k. L=78064, Genesis, Hebrew
│> │> │> │
│ │ │ │
│n │k │L* │
│ 1│ 78064│ 78064│
│ 2│ 39032│ 78064│
│ 3│ 26021│ 78063│
│ 5│ 15612│ 78060│
│ 7│ 11152│ 78064│
│ 10│ 7806│ 78060│
│ 20│ 3903│ 78060│
│ 30│ 2602│ 78060│
│ 50│ 1561│ 78050│
│ 70│ 1115│ 78050│
│ 100│ 780│ 78064│
│ 200│ 390│ 78000│
│ 300│ 260│ 78000│
│ 500│ 156│ 78000│
│ 700│ 111│ 77700│
│ 1000│ 78│ 78000│
│ 2000│ 39│ 78000│
│ 3000│ 26│ 78000│
│ 5000│ 15│ 75000│
│ 7000│ 11│ 77000│
│ 10000│ 7│ 70000│
Now let us introduce the Letter Serial Correlation density. First we introduce the expected density d[e], and later we will likewise introduce the measured Lettter Serial Correlation density d[m]. To
calculate the expected density, we modify formula (13C) by dividing it by n, thus defining the expected Letter Serial Correlation density d[e] as the expected LSC sum per one letter in a chunk:
which is an equation of a hyperbolic curve for a quantity d[e]+T=d[t] [ ]which is
where the constants are
In log-log coordinates equation (17) is represented by a perfect straight line. It starts at n=1 where d[e]=Q-T and is dropping toward d[e]=0 at n=L (since T=Q/L). Note that curves for d[e] [ ]and d
[t] [ ]are at a distance of T from each other along d[e] [ ]axis, but in log-log coordinates both curves, for d[e ]and d[t] [ ], have the same slope. In the actual calculations the straight line for
eq. (17) in log-log coordinates will necessarily be slightly distorted because of the truncation of texts described earlier in this paper. A formal representation of the distortion in question can be
given by modifying equation (17) as follows:
d[e]=d[t]-T= Q[(1/(n^q)]-T...........................................(20)
where the power is q=1 for the ideal d[t]-n hyperbole, but q is slightly different from 1 for real, almost hyperbolic curves, the deviation of q from 1 being caused by the texts' truncation effect.
In the following sections of this paper we will see how well equation (17) is obeyed by real d[e]=d[t]-T curves. The curves for d[e ]will serve as reference measures for the measured densities d[m]
which are measured LSC sums per one letter in a chunk.
Both expected and measured Letter Serial Correlation densities are introduced in a way analogous to that commonly used in Thermodynamics for such quantities as, for example, chemical potential which
most often is chosen to be Gibbs potential per one particle (or per one mole). While Gibbs potential is an extensive quantity, the chemical potential is an intensive one. Using that intensive (as all
specific quantities are) variable often enables one to reveal some fundamental features of a phenomenon. Likewise, in our case both expected and measured sums are extensive quantities, while the
expected and measured densities are intensive. For the interpretation of experimental data, both extensive and intensive parameters have their appropriate places. As it will be demonstrated later in
this article, considering both types of quantities allows for a more compete analysis of experimental results than if discussing the total sums alone.
While the value of S[e] varies for various texts, it is possible to roughly estimate the expected value of that sum as a function of the text's total length, L, without using the precise formula
(13). This can be done in a rather simple, even if a quite approximate way, for the simplest case when the chunk's size is n=1, so that the number of chunks in the text is k=L where L is the total
length of the text. For this approximation we assume that the distribution of all letters is uniform, i.e. that M[x] , which is the number of occurences of letter x in the text, is equal for all
First note that each pair of adjacent chunks i and i+1 can contribute to the sum only one of two values, namely either 0 or 2. If the text under exploration contained spaces between words, the
following situations would be possible. 1) letter x is found neither in chunk i nor in chunk (i+1). Then the term in the sum corresponding to letter x in that pair of chunks is 0 (even though that
pair of chunks may contribute a non-zero term due to a letter other than x). 2) Chunk i contains letter x and chunk i+1 contains a space, so it is empty. In that case the term in the sum contributed
by that pair of chunks is 1. 3) Both chunks i and i+1 contain either identical letters other than x, or spaces. In that case the term in the sum corresponding to letter x in that pair of chunks
contributes 0 to the sum (even though that pair of chunks may contribute either 0, 1 or 2 due to letters other than x). 4) Chunk i contains letter x and chunk i+1 contains some other letter y. In
this case the pair of chunks in point contributes 2 to the sum, as both x and y contribute 1 each.
In our case, though, spaces between the words are ignored. Therefore each chunk contains some letter, and there are no empty chunks. Hence, case 2, and consequently contribution of 1 by any pair of
chunks with n=1 is impossible. Thus the terms in sum S[e] , for n=1, can be only either 0 or 2.
Pick an arbitrary chunk i and assume that it contains letter x. What is then the probability p[x] that in the adjacent chunk there is again the same letter x? In a random text, the probability of any
letter to occupy any location is p[x]=M[x]/L where M[x] [ ]is the number of occurrences of letter x in the entire text. Since one letter x is already occupying the chosen chunk i, the probability
that the adjacent chunk i+1 also contains the same letter x is (M[x]-1)/(L-1). The texts subjected to study all contained at least tens of thousands of letters. Since M[x] is roughly between twenty
and thirty times smaller than L, the values of M[x] in the explored texts all were at least several thousands letters large. Then a good approximation is the replacement of (M[x]-1) with M[x] and (L
-1) with L. The probability that the chunk adjacent to i contains a letter other than x is then 1-M[x]/L. Hence, there is the probability of M[x]/L that the corresponding term in the sum for S[e] is
0 and the probability of 1-M[x]/L that the term in point is 2. Now, assume that all letters of the alphabet appear in our text with the same frequency, which then equals M=L/z, where z is the total
number of letters in the alphabet. In this case, there is a probability of 1/z that the term contributed to the sum by any two adjacent chunks is 0 and the probability of 1-1/z that the term in
question is 2. In such a text the expected number of chunks of 1 containing non-identical letters is then L(1-1/z). Then the expected value of the sum is S[e]=2(L-1)(1-1/z) while its maximum possible
value is 2(L-1) which of course is the same as for the measured sum.
For example, in an English text 100000 letters long, accounting for z=26 for English, we find the expected sum, in the case of chunks having n=1, to be: S[m1]=2(100000-1)(1-1/26)=198385. Then S[e]/L=
1.903. Similar calculation for various languages and text lengths shows that the ratios of the expected sum to the text length, for n=1, usually fall between 1.6 and 1.92, their mean value being
about 1.85. More precise calculation for specific texts in English, Hebrew, Aramaic, and Russian, for n=1, using formula (13), produced numbers between 1.55L and 1.87L, their mean value being about
It is possible to reasonably estimate the value of S[e], starting from formula (13C) and assuming that all z letters in the text have the same frequency which then will be M=L/z for each letter.
{This assumption is of course wrong, as it is tantamount to the suggestion that the expected value of expression M[i](L-M[i]) in formula (13C) equals M(L-M). The expected value of a product equals
the product of expected values only for independent variables [6, page 173] while M and L-M are obviously not independent from each other}. However, as we will see, quantitatively, our assumption
that the mean value of M[i](L-M[i]) equals M(L-M) provides for the values of S[e] which are reasonably close to the actual values determined by formula (13C)}.
We rewrite formula (13C) replacing M[i] with M, and, hence replacing the sum in it with the product z.M(L-M). Accounting for M=L/z:
S[e]= (1-n/L) 2 zM(L-M)/(L-1) = (1-n/L) 2z (L/z)(L-L/z)/(L-1) =
= (1-n/L)2L^2(1-1/z)/(L-1).................(21)
This is an equation of a straight line in S[e]-n coordinates with the intercept of
and the negative slope of
That straight line drops to zero at n=L.
Let us compare the results obtained by equations (21)-(23) to the values of S[e] calculated by precise formula (B).
For example, for the Book of Genesis in Hebrew L=78064, z=22, then the intercept is
A=2 . 78064^2 (1-1/22)/(78064-1)= 149033.
The value of S[e] at n=1 is S[e](1)=A(1-1/L)= 149033(1-1/78064)=149031.
For n=10 S[e](10)=149033(1-10/78064)=149013.
Using formula (B), the corresponding values are S[e](1)=145121, and S[e](10)=145097.
The discrepancy for S[e](1) is (149033-145121)/145121=0.026 i.e about 2.6%
The discrepancy for S[e](10) is (149013-145097)/145103=0.026 i.e. also about 2.6%.
If we created artificially a text containing equal numbers of each letter (and also in absence of a text's truncation) formula (21) would be the precise one for that text. If accounting for
truncation, that formula could be made precise for the text in question by replacing the nominal text's length L with its truncated length L*. (Such a text has been created indeed for the study of
some effects not covered in this report. This topic is discussed in separate papers at Letter serial correlation (LSC) in additional languages and various types of texts and Letter serial correlation
in additional languages).
f) It follows from the derivation of formula (13C) that the expected serial sum S[e] [ ]is averaged over all possible permutations of letters in the tested text. On the other hand, the measured sum S
[m] is found in each measurement as a value for that particular text. Therefore, even if the test is performed on a version randomized by permuting letters of the original meaning-bearing text, the
measured sum S[m] [ ]will necessarily differ from the calculated, averaged expected sum S[e]. Of course we expect that for randomized texts the difference
will be limited to reasonably small fluctuations around zero value. This our expectation will be verified experimentally.
As to the non-permuted meaningful texts, finding and analyzing the difference between the expected sum S[e] [ ]and the experimentally measured sum S[m]^ is one of the specific goals of the experiment
in point.
The experimental results obtained for various texts are described in the second and in the third parts of this report (see Experimental results -- randomized texts and Experimental results -- real
meaningful texts) and their discussion and interpretation are offered in the fourth part (see http://members.cox.net/marperak/Texts/Serialcor4.htm ).
1. D. Witztum, E. Rips, and Y. Rosenberg, Statistical Science, 1994, v. 9, No 3, 429-438.
2. B. McKay et al. Web postings at http://cs.anu.edu.au/~bdm/dilugim/.
3. M. Perakh et al. Web posting at http://members.cox.net/mkarep/.
4. List of references to Bible-code-related publications at http://www.answering-islam.org/Religions/Numerics/index.html.
5. R. J. Larsen and M. L. Marx. An introduction to Mathematical Statistics and its applications. Prentice-Hall Publishers, 1986.
6. M. Dwass. First Steps in Probability. McGraw-Hill Co., 1967.
Originally posted to Mark Perakh's website on February 9, 1999. | {"url":"http://www.talkreason.org/articles/Serialcor1.cfm","timestamp":"2024-11-05T18:38:59Z","content_type":"text/html","content_length":"53065","record_id":"<urn:uuid:102b422f-e709-40e5-9d09-65b8c72baec2>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00392.warc.gz"} |
Microchannel Heat Sink Design Based on Topology Optimization
Microchannel heat sink design consists in an innovative technology which has been studied as alternative to increase cooling efficiency of small electronic devices, such as high-end microprocessors
of CPUs. These electronics devices dissipate a large amount of heat, which requires very efficient cooling systems. Microchannels constructed on a conductivity body allow obtaining an efficient heat
sink design having better thermal dissipation with small mass and volume, and large convective heat transfer coefficient, and, thus, suitable for cooling compact areas of small electronic devices.
Thus, the main objective of this work is the study of a methodology to develop a microchannel heat sink design through the application of the Topology Optimization Method, which allows the
distribution of a limited amount of material, inside a given design domain, in order to obtain an optimized system design. This method combines the Finite Element Method (FEM) and Sequential Linear
Programming (SLP) to find, systematically, an optimized layout design for microchannels in heat sinks. Essentially, the topology optimization problem applied to channel fluid flow consists of
determining which points of a given design domain (small heat sink) should be fluid, and which points should be solid to satisfy a multi-objective function that maximizes the heat dissipation, with
minimum pressure drop. In this proposed methodology, computational simulations of some optimized microchannel layouts are employed to validate the implemented topology optimization algorithm. Some
obtained results are shown to illustrate the methodology.
Asociación Argentina de Mecánica Computacional
Güemes 3450
S3000GLN Santa Fe, Argentina
Phone: 54-342-4511594 / 4511595 Int. 1006
Fax: 54-342-4511169
E-mail: amca(at)santafe-conicet.gov.ar
ISSN 2591-3522 | {"url":"https://cimec.org.ar/ojs/index.php/mc/article/view/3599","timestamp":"2024-11-08T11:04:18Z","content_type":"application/xhtml+xml","content_length":"16810","record_id":"<urn:uuid:91dc780b-12cc-4d8f-a914-e33b3ee2dff5>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00123.warc.gz"} |
Is the continuum hypothesis relevant in physics?
#006: Is the continuum hypothesis relevant in physics?
Category: Mathematical foundations - Tags: Set theory, Topology
Determine whether the continuum hypothesis can be relevant for physical theories.
Mathematical problem. As part of the general theory, we have determined that any space of experimentally distinguishable cases forms a second-countable \(T_0\) topological space. Therefore we can ask
mathematically whether there exists such a topological space where the cardinality of the set is more than countable but less of that of the continuum.
Physical significance. As part of our general theory, we have shown that the cardinality of a set of experimentally distinguishable cases is at most that of the continuum as it must allow a topology
that is \(T_0\) and second-countable. The standard formulation of set theory (ZFC) does not tell us whether sets of cardinality between countable and continuum exist, which is then a separate axiom.
This extra choice is called the “continuum hypothesis” (CH).
Given this mathematical premise, we should consider whether this choice is relevant for physics. For example, it may be that the set of all possible space-time events (i.e. the structure of
space-time) has cardinality that is in between the countable and the continuum. The first step would be to understand whether such a set could be given a suitable topology such that each open set
corresponds to an experimentally verifiable statement, as it is done in our general theory. | {"url":"https://assumptionsofphysics.org/problems/006-ContinuumHypothesis","timestamp":"2024-11-04T18:19:44Z","content_type":"text/html","content_length":"8612","record_id":"<urn:uuid:db353899-f109-44ca-a395-11c3938c4a0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00065.warc.gz"} |
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Krause, Andreas ; Brunskill, Emma ; Cho, Kyunghyun ; Engelhardt, Barbara ; Sabato, Sivan ; Scarlett, Jonathan (Ed.)
We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in R^d, on the basis of i.i.d. samples. All existing statistical analyses of this
problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards
developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in R^d. We
study a computationally efficient estimator initially proposed by Pooladian & Niles-Weed (2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the
minimax-optimal rate n^{ā 1/2}, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of
dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising
methodology for other discontinuous transport map estimation problems.
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confusion in the DeBruijn Graph
In the DeBruijn Graph the nodes(vertices) are le k-mer or the (k-1)mer?
There are works which shows that the nodes(vertices) are k-mer and other works shows that the nodes(vertices) are k-1 mer!
It's ambiguous.
Thanks for your response
Another question: the DeBruijn Graph is NP-Hard or polynomial problem??
I don't understand the question? The dBG is a combinatorial structure, not a problem. Which problem are you asking about?
sorry, I mean the assembly using de Bruijn graph ( find the eulerian path)?
If there exists an Eulerian path in a given dBG, then such a path can be found in polynomial time. However, in real assembly graphs, such path usually do not exist (even in relatively simple genomes)
see e.g. this paper. Furthermore, if such paths do exist, there can easily be an exponential number of them. This is why most dBG assemblers simply report the unambiguous contigs (after quite a bit
of "fixing" and simplification). Anyway, the answer to your original question is that Eulerian Cycle and Eulerian Path are both in P.
Entering edit mode
Rob's answer above is valid, but is only one of the existing definitions of de Bruijn graphs in the literature. I actually disagree that it is the most typical. Consider the representations of
Conway-Bromage, Minia and cascading Bloom filters, Bowe's succinct dBG. They all define nodes as k-mers, and edges as exact (k-1)-overlaps between nodes.
Anyway, this is only a matter of definitions. In the end, the important property of de Bruijn graphs is that nodes are x-mers for some x, and edges have the property that there is an exact (x-1)
-overlap. Whether x=k or x=k-1 does not matter, as k is a parameter. In some definitions, such as Conway-Bromage's, edge (x+1)-mers (formed by overlapping sequences of two nodes in an edge) also need
to be present in the data.
Entering edit mode
If we think dBG as an overlap graph, that is (k-1)-mer reads as vertices with (k-2)-mer overlaps as edges. Nonetheless, probably your view is easier for implementation. I don't have first-hand
experiences on implementing dBG.
EDIT: actually the list of (k-1)-mer is also a sufficient representation. Given a starting (k-1)-mer, you can cut 1 base from one end and then test whether a 1bp extension from the other end leads to
a present (k-1)-mer.
Entering edit mode
You're right - if we think in terms of the implied overlap, then it will be a (k-2)-mer. The k-mer actually encodes the two vertices and the implied edge between them.
Entering edit mode
Generally speaking, in graphs, is my understanding correct: nodes hold nouns (data) and edges hold verbs (the satisfied condition that two nodes are indeed connected, which in this case is a k-2
If my understanding is correct, is the (k-2)mer not stored explicitly, just deduced from the two (k-1)mers it connects? If not, what do the nodes and edges store?
Entering edit mode
So to summarize, the K-mer are the edges and (k-1)mers are the nodes?
Entering edit mode
The nodes are (k-1)-mers and edges are overlaps of length (k-2). Specifically, these overlaps have the property that the length (k-2) edge is both a suffix of the source node and a prefix of the
target node. Those are the "objects" in the graph. What I discussed in addition to this in the answer above, is that the set of k-mers is often used as a representation of all of this information.
Given a k-mer, its left and right (k-1)-mers become nodes in the dBG. Since these (k-1)-mers are drawn directly from the prefix and suffix of a given k-mer, they will always overlap in the prescribed
manner (the length k-2 suffix of the left (k-1)-mer will be a length k-2 prefix of the right (k-1)-mer).
TLDR: The nodes of the dBG are (k-1)-mers and the edges are (k-2)-mers that link them. The original set of k-mers contains the information in the resulting dBG, and is a useful representation of the
Entering edit mode
Oh. I was worried that my knowledge of dBGs had gone totally rusty :)
Entering edit mode
thank you for your explanation but in:
Compeau PEC, Pevzner PA, Tesler G (2011) How to apply de Bruijn graphs to genome assembly. Nat Biotech 29 (11):987-991. doi: 10.1038/nbt.2023
Briefly, construct a graph B (the original graph called a de Bruijn graph) for which every possible (k - 1)-mer is assigned to a node; connect one (k - 1)-mer by a directed edge to a second (k -
1)-mer if there is some k-mer whose prefix is the former and whose suffix is the latter (Fig. 2). Edges of the de Bruijn graph represent all possible k-mers, and thus an Eulerian cycle in B
represents a shortest (cyclic) superstring that contains each k-mer exactly once.
I think there are a confusions.
thanks for your response
Entering edit mode
This is the edge centric de Bruijn graph of order k -1 (See my answer below, with p = k -1) | {"url":"https://www.biostars.org/p/175058/#175066","timestamp":"2024-11-04T01:16:06Z","content_type":"text/html","content_length":"62502","record_id":"<urn:uuid:2cc80ed7-ffb4-4c62-8f28-d2b06db90f6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00713.warc.gz"} |
What is a histogram? | Definition from TechTarget
What is a histogram?
A histogram is a type of chart that shows the frequency distribution of data points across a continuous range of numerical values. The values are grouped into bin or buckets that are arranged in
consecutive order along the horizontal x-axis at the bottom of the chart. Each bin is represented by a vertical bar that sits on the x-axis and extends upward to indicate the number of data points
within that bin.
The bins provide a simple structure for mapping the distribution of data points across the range of values. A data point in this sense is simply one of the values within the overall range. This range
-- sometimes referred to as the independent variable -- might include people's ages, tree heights, exam scores, animal weights, stock prices or any number of other value types. The data points are
plotted along the vertical y-axis and are referred to as the data's frequency, which is in reference to the frequency distribution of the data.
An example of a histogram
Figure 1 shows a histogram of test scores for 200 students. The scores are grouped into six consecutive bins, with each bin represented by a vertical bar of a different color. A histogram's bins can
be evenly distributed or adjusted to accommodate specific scenarios, such as the existence of outliers. In this example, the first bin includes test scores from 0 to 50, while the remaining bins are
evenly distributed in groups of 10. The second bin contains test scores greater than 50 and up to and including 60, the third bin contains test scores greater than 60 and up to and including 70, and
so on.
Figure 1. Histogram example showing the test scores for 200 students
The data points in this case are the students' test scores. The height of each bin, as measured on the y-axes, indicates the number of students whose test scores fall within that bin's range. For
example, the histogram indicates that 40 of the 200 students received test scores greater than 90, with 100 being the highest score they could achieve. On the other hand, three students received
scores of 50 or below, and 47 students received scores greater than 70 but lower than or equal to 80.
In this histogram, the frequency is the number of student test scores per bin; however, frequency can refer to any values that are being measured, such as the ages, weights or heights of a group of
Labrador retrievers. In some cases, a histogram is presented with the independent variable along the y-axis and the frequency of the data points along the x-axis, but this format is much less common.
Typically, the independent variable is plotted along the x-axis.
How are histograms used?
Histograms are often used in statistics to visualize the shape of data distribution across a range of values. For example, the histogram above can help analysts learn about the effectiveness of the
class, instructor or even the exam itself. They might compare these results with those of other classes, either conducted by the same instructor at different times or conducted by other instructors
offering the same class and exam. From these comparisons, analysts might be able to pinpoint where potential issues exist or at least have a starting point for digging deeper into the data.
Analysts often use the physical shape of a histogram's bar pattern to gain quick insights into a data set, but this is effective only when taking into account the context in which the data is
generated. For example, Figure 2 shows three histogram patterns: symmetric, right-skewed and left-skewed.
Figure 2. A trio of histogram patterns: symmetric, right-skewed and left-skewed
Someone analyzing the test score histogram from above might expect a left-skewed pattern, indicating that the bulk of the students achieved a higher medium score. If the distribution were
right-skewed or symmetric, it could indicate an issue with the exam, the class, the instructor or something else. Histograms can produce other patterns as well, each one meaningful within the context
of the data.
A histogram is similar in appearance to a bar chart, but the two differ in important ways. The histogram provides a mechanism for visualizing the data points within a continuous range of numeric
values -- as seen in the test scores example above. In this case, the data points are distributed across the range of test scores, which are separated into bins.
In contrast, a bar chart compares data categories based on a set of numeric measures. For example, a retail company might use a bar chart to show the number of sales in each product category over the
past year, or a medical center might create a bar chart that provides a visual overview of the number of patients receiving different types of treatments over a five-year period.
See the differences among statistical mean, median, mode and range, and check out eight steps to improve data visualization literacy.
This was last updated in June 2023
Continue Reading About histogram
Dig Deeper on Software design and development | {"url":"https://www.techtarget.com/searchsoftwarequality/definition/histogram","timestamp":"2024-11-01T20:35:44Z","content_type":"text/html","content_length":"224481","record_id":"<urn:uuid:89684a43-1741-4b1b-97ac-6fe21891a25d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00433.warc.gz"} |
how to make corresponding angles model working model for BEd and Teachers - 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 corresponding angles model working model for BEd and Teachers
In this video blog post we write about making of the corresponding angles model working model for BEd and Teachers
Creating a corresponding angles model using cardboard and colored paper is a great way to visually understand the concept of corresponding angles in geometry.
corresponding angles model working model for BEd and Teachers
Here’s a step-by-step guide to making a simple corresponding angles model:
Materials you’ll need:
1. Cardboard (thick and rigid)
2. Colored paper (two or more different colors)
3. Paper Cups for Stand
4. Scissors
5. Ruler
6. Pencil
7. Glue or adhesive tape
8. Markers or colored pens (optional, for additional details)
Step-by-step process:
1. Cut out the base:
□ On the cardboard, draw and cut out a large square or rectangle to serve as the base of your model.
2. Create angle strips:
□ Take one colored paper and cut out several long, thin strips. These strips will represent the sides of the angles in your model. The width of the strips should be uniform.
3. Label the angle strips:
□ Using a pencil or marker, label each strip with an angle measurement (e.g., 30°, 45°, 60°, etc.). Make sure to have at least two strips of each angle measurement.
4. Attach the angle strips:
□ Take the angle strips and attach them to the base (cardboard) in a way that they form pairs of corresponding angles. For example, if you have a strip labeled 30°, find another strip labeled
30° and place it diagonally across from the first one so that the two strips form an “X” shape. Repeat this for all angle measurements.
5. Glue or tape the angle strips:
□ Use glue or adhesive tape to secure the angle strips onto the base. Make sure they are firmly attached.
6. Once your corresponding angles model is complete, you can use it to visually understand the concept of corresponding angles. Experiment with different angle measurements and observe how the
corresponding angles are congruent (equal) to each other.
This model will help you see how corresponding angles are formed when a transversal (a line that intersects two or more other lines) cuts through two parallel lines. Corresponding angles are located
on the same side of the transversal and in the same position relative to the two parallel lines.
#correspondingangles #workingmodel #tlm #craftpiller #mathsmodel #maths #model #diy
Video step by step instructions on corresponding angles model working model for BEd and Teachers
Leave a Comment | {"url":"https://howtofunda.com/corresponding-angles-model-working-model-for-bed-and-teachers/","timestamp":"2024-11-05T00:02:45Z","content_type":"text/html","content_length":"63244","record_id":"<urn:uuid:2419d7c1-f3bf-4cc6-90d7-98de0257bb56>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00120.warc.gz"} |
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Mcgraw, Westley Mcgraw, Marcelino Mcgraw, Emerson Mcgraw, Mary Mcgraw, Kwame Mcgraw, Toney Mcgraw, Marcelo Mcgraw, Rasheed Mcgraw, Derrell Mcgraw, Jermey Mcgraw, Russ Mcgraw, Vincenzo Mcgraw, Raheem
Mcgraw, Asa Mcgraw, Lemuel Mcgraw, Murray Mcgraw, Pasquale Mcgraw, Burt Mcgraw, Hollis Mcgraw, Valentin Mcgraw, Bernie Mcgraw, Jamin Mcgraw, Lenard Mcgraw, Roel Mcgraw, Weldon Mcgraw, Ahmed Mcgraw,
Cortney Mcgraw, Tito Mcgraw, Claudio Mcgraw, Kristin Mcgraw, Lorne Mcgraw, Enrico Mcgraw, Marlo Mcgraw, Loyd Mcgraw, Amir Mcgraw, Arnoldo Mcgraw, Harris Mcgraw, Cristobal Mcgraw, Mac Mcgraw, Corbin
Mcgraw, Cullen Mcgraw, Shedrick Mcgraw, Maria Mcgraw, Jermain Mcgraw, Ramsey Mcgraw, Torey Mcgraw, Nicole Mcgraw, Corry Mcgraw, Anson Mcgraw, Heather Mcgraw, Parker Mcgraw, Elizabeth Mcgraw, Tye
Mcgraw, Lesley Mcgraw, Lamonte Mcgraw, Sanford Mcgraw, Karim Mcgraw, Stephanie Mcgraw, Toriano Mcgraw, Bruno Mcgraw, Estevan Mcgraw, Jerel Mcgraw, Orville Mcgraw, Brion Mcgraw, Demetrios Mcgraw,
Anibal Mcgraw, Reginal Mcgraw, Lavar Mcgraw, Rosendo Mcgraw, Romeo Mcgraw, Simeon Mcgraw, Elisha Mcgraw, Stan Mcgraw, Vidal Mcgraw, Enoch Mcgraw, Prince Mcgraw, Eugenio Mcgraw, Leigh Mcgraw, Mervin
Mcgraw, Bartholomew Mcgraw, Delmar Mcgraw, Khalid Mcgraw, Kristen Mcgraw, Johnpaul Mcgraw, Carnell Mcgraw, Domenic Mcgraw, Adalberto Mcgraw, Jaron Mcgraw, Luciano Mcgraw, Kai Mcgraw, Stanford Mcgraw,
Uriah Mcgraw, Demarco Mcgraw, Jabari Mcgraw, Zackary Mcgraw, Ritchie Mcgraw, Ezequiel Mcgraw, Basil Mcgraw, Brenden Mcgraw, Garett Mcgraw, Major Mcgraw, Jerold Mcgraw, Wes Mcgraw, Yancy Mcgraw,
Benjamen Mcgraw, Fredric Mcgraw, Jerrell Mcgraw, Benji Mcgraw, Barron Mcgraw, Chip Mcgraw, Michele Mcgraw, Britton Mcgraw, Milo Mcgraw, Samir Mcgraw, Charlton Mcgraw, Royal Mcgraw, Rubin Mcgraw,
Stoney Mcgraw, Kieth Mcgraw, Javon Mcgraw, Derwin Mcgraw, Duke Mcgraw, Antone Mcgraw, Daryle Mcgraw, Demetric Mcgraw, Layne Mcgraw, Reyes Mcgraw, Walker Mcgraw, Fletcher Mcgraw, Freeman Mcgraw,
Jarret Mcgraw, Delano Mcgraw, Oren Mcgraw, Marcell Mcgraw, Trinidad Mcgraw, Luigi Mcgraw, Rustin Mcgraw, Shan Mcgraw, Sharif Mcgraw, Isiah Mcgraw, Levon Mcgraw, Brody Mcgraw, Coleman Mcgraw, Ollie
Mcgraw, Thor Mcgraw, Darell Mcgraw, Desi Mcgraw, Horacio Mcgraw, Kiley Mcgraw, Mauro Mcgraw, Vinson Mcgraw, Dimitrios Mcgraw, Kevan Mcgraw, Merlin Mcgraw, Monroe Mcgraw, Rebecca Mcgraw, Rashawn
Mcgraw, Tyrell Mcgraw, Zackery Mcgraw, Samson Mcgraw, Darcy Mcgraw, Donavon Mcgraw, Jessica Mcgraw, Napoleon Mcgraw, Muhammad Mcgraw, Trevis Mcgraw, Berry Mcgraw, Antwain Mcgraw, Eliezer Mcgraw,
Kendell Mcgraw, Lamarr Mcgraw, Micky Mcgraw, Demian Mcgraw, Seneca Mcgraw, Tucker Mcgraw, Cheyenne Mcgraw, Davon Mcgraw, Norbert Mcgraw, Regan Mcgraw, Adolph Mcgraw, Jerimiah Mcgraw, Benton Mcgraw,
Jeb Mcgraw, Rian Mcgraw, Corwin Mcgraw, Jameel Mcgraw, Tyrus Mcgraw, Alexandro Mcgraw, Benson Mcgraw, Darien Mcgraw, Augustus Mcgraw, Erie Mcgraw, Gil Mcgraw, Raymon Mcgraw, Abelardo Mcgraw, Terance
Mcgraw, Tremaine Mcgraw, Cristian Mcgraw, Jedidiah Mcgraw, Christina Mcgraw, Homero Mcgraw, Rueben Mcgraw, Rey Mcgraw, Jace Mcgraw, Caesar Mcgraw, Hakim Mcgraw, Lydell Mcgraw, Marcello Mcgraw,
Parrish Mcgraw, Isaias Mcgraw, Johnie Mcgraw, Tavis Mcgraw, Eliot Mcgraw, Fermin Mcgraw, Nikolas Mcgraw, Orion Mcgraw, Alden Mcgraw, Clement Mcgraw, Lucio Mcgraw, Rondell Mcgraw, Ronnell Mcgraw,
Demetris Mcgraw, Juston Mcgraw, Len Mcgraw, Cade Mcgraw, Detrick Mcgraw, Sherwin Mcgraw, Truman Mcgraw, Reymundo Mcgraw, Danielle Mcgraw, Domenico Mcgraw, Octavius Mcgraw, Anwar Mcgraw, Christofer
Mcgraw, Dwaine Mcgraw, Leander Mcgraw, Reese Mcgraw, Houston Mcgraw, Conor Mcgraw, Dillon Mcgraw, Mitch Mcgraw, Demario Mcgraw, Artis Mcgraw, Boris Mcgraw, Samual Mcgraw, Dustan Mcgraw, Gorge Mcgraw,
Jory Mcgraw, Stanton Mcgraw, Amit Mcgraw, Julie Mcgraw, Kelby Mcgraw, Kody Mcgraw, Christos Mcgraw, Fritz Mcgraw, Lyndon Mcgraw, Constantine Mcgraw, Erasmo Mcgraw, Jodie Mcgraw, Brooke Mcgraw, Conan
Mcgraw, Kane Mcgraw, Merrill Mcgraw, Bryson Mcgraw, Isreal Mcgraw, Woody Mcgraw, Tadd Mcgraw, Emiliano Mcgraw, Cyril Mcgraw, Davey Mcgraw, Gabe Mcgraw, Pernell Mcgraw, Curtiss Mcgraw, Jacky Mcgraw,
Nicholaus Mcgraw, Hasan Mcgraw, Davy Mcgraw, Oswaldo Mcgraw, Tammy Mcgraw, Felton Mcgraw, Jarrad Mcgraw, Rosario Mcgraw, Yusef Mcgraw, Anthoney Mcgraw, Dimitri Mcgraw, Donn Mcgraw, Graig Mcgraw,
Glendon Mcgraw, Alain Mcgraw, Alonso Mcgraw, Dwane Mcgraw, Laura Mcgraw, Millard Mcgraw, Milan Mcgraw, Everardo Mcgraw, Taj Mcgraw, Kendal Mcgraw, Prentice Mcgraw, Hilario Mcgraw, Hayden Mcgraw,
Lukas Mcgraw, Lorin Mcgraw, Stevan Mcgraw, Benedict Mcgraw, Ferdinand Mcgraw, Renard Mcgraw, Geoff Mcgraw, Raleigh Mcgraw, Amanda Mcgraw, Jules Mcgraw, Marshal Mcgraw, Ephraim Mcgraw, Patricia
Mcgraw, Sarah Mcgraw, Kalvin Mcgraw, Sky Mcgraw, Vladimir Mcgraw, Kipp Mcgraw, Rashid Mcgraw, Burke Mcgraw, Derrek Mcgraw, Margarito Mcgraw, Rojelio Mcgraw, Von Mcgraw, Wayland Mcgraw, Mohammad
Mcgraw, Sandro Mcgraw, Renato Mcgraw, Shiloh Mcgraw, Torry Mcgraw, Kenji Mcgraw, Del Mcgraw, Elgin Mcgraw, Booker Mcgraw, Chistopher Mcgraw, Lavell Mcgraw, Mohammed Mcgraw, Refugio Mcgraw, Damen
Mcgraw, Nicola Mcgraw, Daryn Mcgraw, Elwood Mcgraw, Kenric Mcgraw, Andrae Mcgraw, Christine Mcgraw, Clemente Mcgraw, Corby Mcgraw, Edmundo Mcgraw, Channing Mcgraw, Maynard Mcgraw, Roderic Mcgraw,
Wilton Mcgraw, Kedrick Mcgraw, Kieran Mcgraw, Lucius Mcgraw, Cliffton Mcgraw, Geraldo Mcgraw, Adrien Mcgraw, Hershel Mcgraw, Reco Mcgraw, Dudley Mcgraw, Jame Mcgraw, Michal Mcgraw, Omari Mcgraw,
Christophe Mcgraw, Delton Mcgraw, Lindsay Mcgraw, Philippe Mcgraw, Faron Mcgraw, Brandan Mcgraw, Williams Mcgraw, Adonis Mcgraw, Jeanpaul Mcgraw, Mckinley Mcgraw, Bertram Mcgraw, Randel Mcgraw, Audie
Mcgraw, Fransisco Mcgraw, Gideon Mcgraw, Lafayette Mcgraw, Renaldo Mcgraw, Winfred Mcgraw, Lacy Mcgraw, Ravi Mcgraw, Denton Mcgraw, Bjorn Mcgraw, Demetrice Mcgraw, Duwayne Mcgraw, Lavon Mcgraw,
Porfirio Mcgraw, Eldridge Mcgraw, Hosea Mcgraw, Lupe Mcgraw, Corbett Mcgraw, Grayson Mcgraw, Sanjay Mcgraw, Emile Mcgraw, Emmitt Mcgraw, Olin Mcgraw, Ramone Mcgraw, Yusuf Mcgraw, Leandro Mcgraw,
Amado Mcgraw, Leighton Mcgraw, Malachi Mcgraw, Stephon Mcgraw, Wilford Mcgraw, Keon Mcgraw, Timmie Mcgraw, Errick Mcgraw, Jarad Mcgraw, Kaleb Mcgraw, Dayton Mcgraw, Jelani Mcgraw, Rance Mcgraw,
Corrie Mcgraw, Jerrad Mcgraw, Yancey Mcgraw, Jonpaul Mcgraw, Theo Mcgraw, Amador Mcgraw, Jamaine Mcgraw, Lorenza Mcgraw, Valentino Mcgraw, Dee Mcgraw, Eleazar Mcgraw, Jarett Mcgraw, Justen Mcgraw,
Lauren Mcgraw, Tyronne Mcgraw, Willy Mcgraw, Elroy Mcgraw, Esequiel Mcgraw, Ike Mcgraw, Renee Mcgraw, Karen Mcgraw, Herschel Mcgraw, Konstantinos Mcgraw, Arlo Mcgraw, Buford Mcgraw, Rasheen Mcgraw,
Reagan Mcgraw, Tobey Mcgraw, Haywood Mcgraw, Khristopher Mcgraw, Patricio Mcgraw, Zack Mcgraw, Jamon Mcgraw, Khary Mcgraw, Augustin Mcgraw, Benjiman Mcgraw, Ishmael Mcgraw, Wally Mcgraw, Faustino
Mcgraw, Hamilton Mcgraw, Jami Mcgraw, Vincente Mcgraw, Griffin Mcgraw, Jeramiah Mcgraw, Manny Mcgraw, Santino Mcgraw, Vern Mcgraw, Richardo Mcgraw, Franklyn Mcgraw, Wilmer Mcgraw, Christpher Mcgraw,
Derric Mcgraw, Drake Mcgraw, Lino Mcgraw, Nathen Mcgraw, Ryon Mcgraw, Vernell Mcgraw, Johann Mcgraw, Khalil Mcgraw, Cris Mcgraw, Foster Mcgraw, Gale Mcgraw, Teodoro Mcgraw, Creighton Mcgraw, Farrell
Mcgraw, Jai Mcgraw, Chirstopher Mcgraw, Lavelle Mcgraw, Mikal Mcgraw, Orin Mcgraw, Ceasar Mcgraw, Pietro Mcgraw, Sedric Mcgraw, Sydney Mcgraw, Val Mcgraw, Dawayne Mcgraw, Odis Mcgraw, Wardell Mcgraw,
Murphy Mcgraw, Nevin Mcgraw, Quenton Mcgraw, Santo Mcgraw, Slade Mcgraw, Dawn Mcgraw, Marko Mcgraw, Ricco Mcgraw, Rogers Mcgraw, Abe Mcgraw, Jerimy Mcgraw, Jessy Mcgraw, Montrell Mcgraw, Cassidy
Mcgraw, Hilton Mcgraw, Christen Mcgraw, Kamal Mcgraw, Leron Mcgraw, Miquel Mcgraw, Yosef Mcgraw, Judah Mcgraw, Malcom Mcgraw, Rayford Mcgraw, Arlen Mcgraw, Gaetano Mcgraw, Montez Mcgraw, Kareen
Mcgraw, Dajuan Mcgraw, Marlow Mcgraw, Uriel Mcgraw, Glynn Mcgraw, Bud Mcgraw, Eben Mcgraw, Geronimo Mcgraw, Deshaun Mcgraw, Hernan Mcgraw, Zebulon Mcgraw, Deshon Mcgraw, Gareth Mcgraw, Paolo Mcgraw,
Edison Mcgraw, Gaylon Mcgraw, Shamus Mcgraw, Carleton Mcgraw, Florentino Mcgraw, Osbaldo Mcgraw, Brodie Mcgraw, Jaret Mcgraw, Johathan Mcgraw, Tariq Mcgraw, Seamus Mcgraw, Devan Mcgraw, Richmond
Mcgraw, Cynthia Mcgraw, Cleo Mcgraw, Huey Mcgraw, Talmadge Mcgraw, Dain Mcgraw, Dennie Mcgraw, Paxton Mcgraw, Peyton Mcgraw, Darrius Mcgraw, Jodi Mcgraw, Ronell Mcgraw, Jeremi Mcgraw, Quinten Mcgraw,
Sandra Mcgraw, Waymon Mcgraw, Earle Mcgraw, Franz Mcgraw, Christoher Mcgraw, Micahel Mcgraw, Sal Mcgraw, Sharon Mcgraw, Keegan Mcgraw, Eusebio Mcgraw, Geoffery Mcgraw, Monica Mcgraw, Dontae Mcgraw,
Jimi Mcgraw, Kendric Mcgraw, Amin Mcgraw, Cleon Mcgraw, Kale Mcgraw, Tina Mcgraw, Ambrose Mcgraw, Armond Mcgraw, Ashton Mcgraw, Lucien Mcgraw, Maximo Mcgraw, Reno Mcgraw, Susan Mcgraw, Chico Mcgraw,
Jermel Mcgraw, Lennie Mcgraw, Rajesh Mcgraw, Artie Mcgraw, Kerwin Mcgraw, Sage Mcgraw, Dietrich Mcgraw, Eriberto Mcgraw, Niles Mcgraw, Rolland Mcgraw, Butch Mcgraw, Courtland Mcgraw, King Mcgraw, Kit
Mcgraw, Menachem Mcgraw, Rich Mcgraw, Florencio Mcgraw, Lovell Mcgraw, Jacque Mcgraw, Antwone Mcgraw, Darick Mcgraw, Kennth Mcgraw, Asher Mcgraw, Efrem Mcgraw, Lamarcus Mcgraw, Terron Mcgraw, Jacinto
Mcgraw, Karlos Mcgraw, Mohamed Mcgraw, Rahman Mcgraw, Ammon Mcgraw, Davie Mcgraw, Diallo Mcgraw, Kelton Mcgraw, Mathias Mcgraw, Sunny Mcgraw, Zeke Mcgraw, Jaison Mcgraw, Nate Mcgraw, Demetrious
Mcgraw, Maceo Mcgraw, Vasilios Mcgraw, Demetrio Mcgraw, Lyman Mcgraw, Angus Mcgraw, Arvin Mcgraw, Camron Mcgraw, Daryll Mcgraw, Ehren Mcgraw, Les Mcgraw, Nils Mcgraw, Akil Mcgraw, Markeith Mcgraw,
Santana Mcgraw, Warner Mcgraw, Alphonse Mcgraw, Arnaldo Mcgraw, Art Mcgraw, Domenick Mcgraw, Prentiss Mcgraw, Blas Mcgraw, Dwan Mcgraw, Geno Mcgraw, Kahlil Mcgraw, Feliciano Mcgraw, Hayward Mcgraw,
Ulises Mcgraw, Valentine Mcgraw, Darold Mcgraw, Deven Mcgraw, Mordechai Mcgraw, Rodriquez Mcgraw, Gentry Mcgraw, Jorje Mcgraw, Sunil Mcgraw, Wilburn Mcgraw, Anders Mcgraw, Damone Mcgraw, Sherrod
Mcgraw, Dashawn Mcgraw, Otha Mcgraw, Rahim Mcgraw, Javan Mcgraw, Wendy Mcgraw, Camilo Mcgraw, Jeromie Mcgraw, Jaysen Mcgraw, Modesto Mcgraw, Quincey Mcgraw, Tyran Mcgraw, Cleve Mcgraw, Raynard
Mcgraw, Tori Mcgraw, Antonino Mcgraw, Bartley Mcgraw, Kristofor Mcgraw, Sheridan Mcgraw, Alva Mcgraw, Correy Mcgraw, Dandre Mcgraw, Darek Mcgraw, Wylie Mcgraw, Anand Mcgraw, Decarlos Mcgraw, Domonic
Mcgraw, Rachel Mcgraw, Jamieson Mcgraw, Nikolaos Mcgraw, Rashaan Mcgraw, Roque Mcgraw, Shelly Mcgraw, Sven Mcgraw, Evans Mcgraw, Jakob Mcgraw, Lateef Mcgraw, Noble Mcgraw, Artemio Mcgraw, Kameron
Mcgraw, Kavin Mcgraw, Dick Mcgraw, Filiberto Mcgraw, Henri Mcgraw, Joby Mcgraw, Montgomery Mcgraw, Salomon Mcgraw, Vashon Mcgraw, Abdullah Mcgraw, Cal Mcgraw, Shain Mcgraw, Sherwood Mcgraw, Dewitt
Mcgraw, Newton Mcgraw, Nicklaus Mcgraw, Darvin Mcgraw, Jeron Mcgraw, Khari Mcgraw, Tyrome Mcgraw, Benjie Mcgraw, Kunta Mcgraw, Karsten Mcgraw, Patric Mcgraw, Reece Mcgraw, Olen Mcgraw, Rowdy Mcgraw,
Terrel Mcgraw, Addison Mcgraw, Blaise Mcgraw, Fidencio Mcgraw, Ibrahim Mcgraw, Idris Mcgraw, Lavern Mcgraw, Mickel Mcgraw, Shun Mcgraw, Athanasios Mcgraw, Danilo Mcgraw, Giles Mcgraw, Mateo Mcgraw,
Nino Mcgraw, Alonza Mcgraw, Cardell Mcgraw, Job Mcgraw, Mahlon Mcgraw, Anil Mcgraw, Destry Mcgraw, Lamon Mcgraw, Martez Mcgraw, Mikael Mcgraw, Nickey Mcgraw, Tywan Mcgraw, Abner Mcgraw, Jamell
Mcgraw, Nels Mcgraw, Raoul Mcgraw, Braulio Mcgraw, Randle Mcgraw, Roddy Mcgraw, Toni Mcgraw, Lynwood Mcgraw, Thurston Mcgraw, Chrisopher Mcgraw, Eron Mcgraw, Fausto Mcgraw, Keir Mcgraw, Kelcey
Mcgraw, Marquette Mcgraw, Mikeal Mcgraw, Anselmo Mcgraw, Hipolito Mcgraw, Neville Mcgraw, Rodriguez Mcgraw, Antron Mcgraw, Walton Mcgraw, Devlin Mcgraw, Lucian Mcgraw, Rudolfo Mcgraw, Avi Mcgraw,
Brantley Mcgraw, Dathan Mcgraw, Doran Mcgraw, Erica Mcgraw, Franky Mcgraw, Kary Mcgraw, Tiffany Mcgraw, Yuri Mcgraw, Bucky Mcgraw, Hardy Mcgraw, Justus Mcgraw, Cecilio Mcgraw, Hoyt Mcgraw, Jasson
Mcgraw, Jerardo Mcgraw, Johnson Mcgraw, Kenney Mcgraw, Panagiotis Mcgraw, Ajay Mcgraw, Alvis Mcgraw, Conrado Mcgraw, Mel Mcgraw, Burl Mcgraw, Daymon Mcgraw, Dorsey Mcgraw, Giancarlo Mcgraw, Riccardo
Mcgraw, Antuan Mcgraw, Braxton Mcgraw, Candelario Mcgraw, Colton Mcgraw, Rommel Mcgraw, Irwin Mcgraw, Reynold Mcgraw, Tino Mcgraw, Alessandro Mcgraw, Corie Mcgraw, Dameion Mcgraw, Dwyane Mcgraw,
Jabbar Mcgraw, Lajuan Mcgraw, Mose Mcgraw, Taiwan Mcgraw, Yisroel Mcgraw, Danniel Mcgraw, Darris Mcgraw, Jerred Mcgraw, Lori Mcgraw, Crispin Mcgraw, Maximilian Mcgraw, Skip Mcgraw, Yaakov Mcgraw,
Brodrick Mcgraw, Fabio Mcgraw, Gerrit Mcgraw, Iran Mcgraw, Trace Mcgraw, Amar Mcgraw, Aram Mcgraw, Buster Mcgraw, Byran Mcgraw, Jens Mcgraw, Joseluis Mcgraw, Karriem Mcgraw, Kenan Mcgraw, Tonya
Mcgraw, Trampas Mcgraw, Durand Mcgraw, Lou Mcgraw, Regis Mcgraw, Cain Mcgraw, Christ Mcgraw, Daivd Mcgraw, Marquise Mcgraw, Meredith Mcgraw, Dell Mcgraw, Matthias Mcgraw, Alphonzo Mcgraw, Latroy
Mcgraw, Allison Mcgraw, Darion Mcgraw, Melton Mcgraw, Tara Mcgraw, Boyce Mcgraw, Cletus Mcgraw, Derron Mcgraw, Madison Mcgraw, Tait Mcgraw, Yehuda Mcgraw, Arik Mcgraw, Donato Mcgraw, Duston Mcgraw,
Kimani Mcgraw, Lucious Mcgraw, Nader Mcgraw, Baldemar Mcgraw, Bertrand Mcgraw, Blane Mcgraw, Dujuan Mcgraw, Ellery Mcgraw, Kennedy Mcgraw, Misael Mcgraw, Tremain Mcgraw, Veronica Mcgraw, Christy
Mcgraw, Columbus Mcgraw, Little Mcgraw, Marius Mcgraw, Skyler Mcgraw, Alexandre Mcgraw, Celestino Mcgraw, Crystal Mcgraw, Jordon Mcgraw, Justine Mcgraw, Demon Mcgraw, Derreck Mcgraw, Flavio Mcgraw,
Kennard Mcgraw, Roby Mcgraw, Schuyler Mcgraw, Garrison Mcgraw, Cipriano Mcgraw, Cori Mcgraw, Denise Mcgraw, Keyon Mcgraw, Laverne Mcgraw, Obie Mcgraw, Tige Mcgraw, Gabino Mcgraw, Joselito Mcgraw,
Karlton Mcgraw, Lashon Mcgraw, Lucky Mcgraw, Neill Mcgraw, Shmuel Mcgraw, Ara Mcgraw, Blain Mcgraw, Pascual Mcgraw, Rashan Mcgraw, Serge Mcgraw, Thane Mcgraw, Bradlee Mcgraw, Connor Mcgraw, Cooper
Mcgraw, Jonny Mcgraw, Merrick Mcgraw, Porter Mcgraw, Zeb Mcgraw, April Mcgraw, Edrick Mcgraw, Flint Mcgraw, Heith Mcgraw, Jospeh Mcgraw, Luiz Mcgraw, Macario Mcgraw, Zach Mcgraw, Candido Mcgraw,
Dangelo Mcgraw, Dov Mcgraw, Dru Mcgraw, Eladio Mcgraw, Rasheem Mcgraw, Tarek Mcgraw, Willian Mcgraw, Ezell Mcgraw, Konrad Mcgraw, Mick Mcgraw, Tarrance Mcgraw, Yitzchok Mcgraw, Rahul Mcgraw, Shelley
Mcgraw, Theodis Mcgraw, Timonthy Mcgraw, Connie Mcgraw, Derrik Mcgraw, Ennis Mcgraw, Gail Mcgraw, Gaston Mcgraw, Gunnar Mcgraw, Justice Mcgraw, Romel Mcgraw, Bilal Mcgraw, Jamy Mcgraw, Jomo Mcgraw,
Keary Mcgraw, Nehemiah Mcgraw, Rashon Mcgraw, Rolf Mcgraw, Sameer Mcgraw, Jere Mcgraw, Jerone Mcgraw, Kennon Mcgraw, Rodd Mcgraw, Ronaldo Mcgraw, Teresa Mcgraw, Coley Mcgraw, Enos Mcgraw, Narciso
Mcgraw, Robinson Mcgraw, Tobby Mcgraw, Virgilio Mcgraw, Christiaan Mcgraw, Clarke Mcgraw, Demetruis Mcgraw, Jemal Mcgraw, Llewellyn Mcgraw, Miller Mcgraw, Shilo Mcgraw, Tarus Mcgraw, Young Mcgraw,
Arlie Mcgraw, Djuan Mcgraw, Marcial Mcgraw, Pamela Mcgraw, Rayshawn Mcgraw, Robyn Mcgraw, Tirrell Mcgraw, Dontay Mcgraw, Joshuah Mcgraw, Jovon Mcgraw, Lemar Mcgraw, Oran Mcgraw, Taron Mcgraw, Jaimie
Mcgraw, Kacey Mcgraw, Wendall Mcgraw, Armondo Mcgraw, Curry Mcgraw, Klint Mcgraw, Lauro Mcgraw, Maximillian Mcgraw, Rashard Mcgraw, Anastasios Mcgraw, Jermiah Mcgraw, Joao Mcgraw, Lonnell Mcgraw,
Matias Mcgraw, Rollin Mcgraw, Thimothy Mcgraw, Toma Mcgraw, Jahmal Mcgraw, Jerame Mcgraw, Skye Mcgraw, Alford Mcgraw, Baltazar Mcgraw, Gianni Mcgraw, Hayes Mcgraw, Lamond Mcgraw, Linda Mcgraw,
Neftali Mcgraw, Antwoine Mcgraw, Arden Mcgraw, Dionicio Mcgraw, Ely Mcgraw, Kacy Mcgraw, Syed Mcgraw, Tyrel Mcgraw, Windell Mcgraw, Allyn Mcgraw, Antwane Mcgraw, Dayne Mcgraw, Donal Mcgraw, Maxie
Mcgraw, Rashaun Mcgraw, Sampson Mcgraw, Torin Mcgraw, Axel Mcgraw, Delmer Mcgraw, Dickie Mcgraw, Evaristo Mcgraw, Francois Mcgraw, Radames Mcgraw, Sixto Mcgraw, Terri Mcgraw, Wyman Mcgraw, Ashanti
Mcgraw, Benjaman Mcgraw, Christophor Mcgraw, Delwin Mcgraw, Lemont Mcgraw, Lonzo Mcgraw, Mustafa Mcgraw, Sasha Mcgraw, Travon Mcgraw, Darryll Mcgraw, Garfield Mcgraw, Jarrell Mcgraw, Lawson Mcgraw,
Nikki Mcgraw, Terell Mcgraw, Ace Mcgraw, Donna Mcgraw, Gerome Mcgraw, Jermane Mcgraw, Joeseph Mcgraw, Raynaldo Mcgraw, Carrie Mcgraw, Demetri Mcgraw, Rupert Mcgraw, Shadrick Mcgraw, Sol Mcgraw,
Wendel Mcgraw, Zechariah Mcgraw, Delon Mcgraw, Demitrius Mcgraw, Mcarthur Mcgraw, Nickie Mcgraw, Tavaris Mcgraw, Baby Mcgraw, Cornelious Mcgraw, Deborah Mcgraw, Haven Mcgraw, Merritt Mcgraw, Vijay
Mcgraw, Granville Mcgraw, Rock Mcgraw, Romero Mcgraw, Tanya Mcgraw, Anastacio Mcgraw, Ciro Mcgraw, Gennaro Mcgraw, Jerrel Mcgraw, Justo Mcgraw, Maury Mcgraw, Stafford Mcgraw, Amon Mcgraw, Carol
Mcgraw, Cornelio Mcgraw, Dawson Mcgraw, Erskine Mcgraw, Tarrence Mcgraw, Torrie Mcgraw, Trevin Mcgraw, Breck Mcgraw, Delfino Mcgraw, Desean Mcgraw, Jerron Mcgraw, Milford Mcgraw, Nancy Mcgraw, Oswald
Mcgraw, Stefano Mcgraw, Demont Mcgraw, Lawerence Mcgraw, Parris Mcgraw, Godfrey Mcgraw, Hakeem Mcgraw, Leobardo Mcgraw, Loran Mcgraw, Garnett Mcgraw, Lawrance Mcgraw, Darryn Mcgraw, Eamon Mcgraw,
Geremy Mcgraw, Kobie Mcgraw, Leeroy Mcgraw, Maximino Mcgraw, Orrin Mcgraw, Wayman Mcgraw, Brenda Mcgraw, Celso Mcgraw, Daman Mcgraw, Denzil Mcgraw, Geary Mcgraw, Ladon Mcgraw, Marland Mcgraw,
Servando Mcgraw, Kamau Mcgraw, Migel Mcgraw, Ramond Mcgraw, Terrace Mcgraw, Ubaldo Mcgraw, Chaz Mcgraw, Demetrick Mcgraw, Freddrick Mcgraw, Gray Mcgraw, Ivy Mcgraw, Shlomo Mcgraw, Torre Mcgraw,
Damein Mcgraw, Deryl Mcgraw, Dusten Mcgraw, Katherine Mcgraw, Kreg Mcgraw, London Mcgraw, Patrice Mcgraw, Payton Mcgraw, Ramel Mcgraw, Rosalio Mcgraw, Serjio Mcgraw, Tramaine Mcgraw, Adolphus Mcgraw,
Ameer Mcgraw, Antwaun Mcgraw, Ashish Mcgraw, Benjamine Mcgraw, Charleston Mcgraw, Darry Mcgraw, Elmo Mcgraw, Nicklas Mcgraw, Nikolai Mcgraw, Ricki Mcgraw, Tai Mcgraw, Bernardino Mcgraw, Devaughn
Mcgraw, Epifanio Mcgraw, Gaylord Mcgraw, Mathieu Mcgraw, Romell Mcgraw, Arlin Mcgraw, Bradd Mcgraw, Clair Mcgraw, Dawud Mcgraw, Jeffory Mcgraw, Lyndell Mcgraw, Sara Mcgraw, Adriel Mcgraw, Algernon
Mcgraw, Benigno Mcgraw, Chancey Mcgraw, Elan Mcgraw, Jeromey Mcgraw, Lionell Mcgraw, Shone Mcgraw, Donyell Mcgraw, Herminio Mcgraw, Jerrett Mcgraw, Naim Mcgraw, Nikolaus Mcgraw, Soren Mcgraw, Agapito
Mcgraw, Carlin Mcgraw, Cass Mcgraw, Harmon Mcgraw, Johan Mcgraw, Junius Mcgraw, Kenrick Mcgraw, Kervin Mcgraw, Kori Mcgraw, Raj Mcgraw, Shimon Mcgraw, Basilio Mcgraw, Juventino Mcgraw, Kirkland
Mcgraw, Kwesi Mcgraw, Manish Mcgraw, Omer Mcgraw, Tedd Mcgraw, Vic Mcgraw, Antonie Mcgraw, Chauncy Mcgraw, Dagoberto Mcgraw, Darel Mcgraw, Jerrid Mcgraw, Julien Mcgraw, Layton Mcgraw, Quan Mcgraw,
Roddrick Mcgraw, Shaw Mcgraw, Silvestre Mcgraw, Tavon Mcgraw, Unknown Mcgraw, Daniell Mcgraw, Andrei Mcgraw, Hashim Mcgraw, Hobert Mcgraw, Terris Mcgraw, Aundre Mcgraw, Avrohom Mcgraw, Elden Mcgraw,
Guido Mcgraw, Holly Mcgraw, Marice Mcgraw, Martinez Mcgraw, Daneil Mcgraw, Mattew Mcgraw, Shamar Mcgraw, Shannan Mcgraw, Shomari Mcgraw, Daniele Mcgraw, Darrian Mcgraw, Demetrus Mcgraw, Erron Mcgraw,
Hasani Mcgraw, Robbin Mcgraw, Sabino Mcgraw, Sloan Mcgraw, Zev Mcgraw, Cassius Mcgraw, Dawan Mcgraw, Sekou Mcgraw, Tyjuan Mcgraw, Bentley Mcgraw, Ean Mcgraw, Edric Mcgraw, Gamaliel Mcgraw, Gerold
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Omarr Mcgraw, Tyre Mcgraw, Valente Mcgraw, Gavino Mcgraw, Kade Mcgraw, Kolby Mcgraw, Linton Mcgraw, Malvin Mcgraw, Moishe Mcgraw, Rhonda Mcgraw, Sherry Mcgraw, Verlin Mcgraw, Antoino Mcgraw, Benjy
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Antoni Mcgraw, Connell Mcgraw, Gustav Mcgraw, Meir Mcgraw, Reza Mcgraw, Rollie Mcgraw, Ronney Mcgraw, Sherron Mcgraw, Teron Mcgraw, Theotis Mcgraw, Vittorio Mcgraw, Athony Mcgraw, Burnell Mcgraw,
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Michiel Mcgraw, Pieter Mcgraw, Wilber Mcgraw, Christapher Mcgraw, Derk Mcgraw, Enrigue Mcgraw, Isidoro Mcgraw, Kriston Mcgraw, Ladell Mcgraw, Morton Mcgraw, Nash Mcgraw, Nickolaus Mcgraw, Paula
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Dione Mcgraw, Garren Mcgraw, Janson Mcgraw, Justyn Mcgraw, Megan Mcgraw, Niel Mcgraw, Niels Mcgraw, Rockie Mcgraw, Thornton Mcgraw, Turner Mcgraw, Viet Mcgraw, Albino Mcgraw, Alma Mcgraw, Artemus
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Charle Mcgraw, Erek Mcgraw, Evin Mcgraw, Graeme Mcgraw, Jerimie Mcgraw, Juanito Mcgraw, Keoni Mcgraw, Kwan Mcgraw, Lathan Mcgraw, Leonidas Mcgraw, Marrio Mcgraw, Mayer Mcgraw, Ozzie Mcgraw, Rajiv
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Dannon Mcgraw, Draper Mcgraw, Garet Mcgraw, Immanuel Mcgraw, Julia Mcgraw, Manfred Mcgraw, Monique Mcgraw, Monta Mcgraw, Nile Mcgraw, Rome Mcgraw, Tywon Mcgraw, Verne Mcgraw, Amando Mcgraw, Askia
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Dewight Mcgraw, Ferrell Mcgraw, Filippo Mcgraw, Garin Mcgraw, Herbie Mcgraw, Joachim Mcgraw, Khalif Mcgraw, Mikey Mcgraw, Severo Mcgraw, Tahir Mcgraw, Taras Mcgraw, Vinay Mcgraw, Aharon Mcgraw,
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Melvyn Mcgraw, Sabin Mcgraw, Tajuan Mcgraw, Terrick Mcgraw, Thadius Mcgraw, Timoteo Mcgraw, Treavor Mcgraw, Tyrese Mcgraw, Victoria Mcgraw, Vishal Mcgraw, Aran Mcgraw, Christoffer Mcgraw, Devron
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Claudia Mcgraw, Dartagnan Mcgraw, Errin Mcgraw, Hung Mcgraw, Juvenal Mcgraw, Kaseem Mcgraw, Keri Mcgraw, Marwan Mcgraw, Matteo Mcgraw, Raymone Mcgraw, Rennie Mcgraw, Rondale Mcgraw, Shadd Mcgraw, Yul
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Delane Mcgraw, Erric Mcgraw, Gerhard Mcgraw, Hillel Mcgraw, Jansen Mcgraw, Jenny Mcgraw, Keelan Mcgraw, Michell Mcgraw, Petros Mcgraw, Saeed Mcgraw, Shamon Mcgraw, Starsky Mcgraw, Tamir Mcgraw, Umar
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Sinclair Mcgraw, Stephane Mcgraw, Torian Mcgraw, Tylor Mcgraw, Tzvi Mcgraw, Umberto Mcgraw, Baruch Mcgraw, Benard Mcgraw, Glenwood Mcgraw, Jaun Mcgraw, Jermie Mcgraw, Johnney Mcgraw, Kenard Mcgraw,
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Gregroy Mcgraw, Heyward Mcgraw, Jebediah Mcgraw, Kye Mcgraw, Larue Mcgraw, Lonell Mcgraw, Nevada Mcgraw, Norma Mcgraw, Renny Mcgraw, Rosa Mcgraw, Ana Mcgraw, Blue Mcgraw, Cresencio Mcgraw, Danon
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Marvel Mcgraw, Ocie Mcgraw, Raudel Mcgraw, Romaine Mcgraw, Rush Mcgraw, Ruston Mcgraw, Talib Mcgraw, Therman Mcgraw, Vann Mcgraw, Vaughan Mcgraw, Wilberto Mcgraw, Albin Mcgraw, Antwuan Mcgraw, Aramis
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Remy Mcgraw, Rion Mcgraw, Sanchez Mcgraw, Shahid Mcgraw, Valdez Mcgraw, Acie Mcgraw, Aundrey Mcgraw, Buckley Mcgraw, Eliyahu Mcgraw, Erroll Mcgraw, Leotis Mcgraw, Levern Mcgraw, Mace Mcgraw, Perrin
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Luz Mcgraw, Montel Mcgraw, Sylvan Mcgraw, Tobi Mcgraw, Twan Mcgraw, Ashraf Mcgraw, Avram Mcgraw, Benzion Mcgraw, Casimiro Mcgraw, Chane Mcgraw, Christie Mcgraw, Chritopher Mcgraw, Cirilo Mcgraw,
Constantino Mcgraw, Dupree Mcgraw, Gregor Mcgraw, Jathan Mcgraw, Keaton Mcgraw, Kostas Mcgraw, Larenzo Mcgraw, Parag Mcgraw, Rudi Mcgraw, Sameul Mcgraw, Sanjeev Mcgraw, Soloman Mcgraw, Tarrell
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Felicia Mcgraw, Juwan Mcgraw, Kemp Mcgraw, Kenn Mcgraw, Kenyatte Mcgraw, Michail Mcgraw, Orval Mcgraw, Salvadore Mcgraw, Shirley Mcgraw, Boone Mcgraw, Dionte Mcgraw, Durwood Mcgraw, Ellison Mcgraw,
Garron Mcgraw, Gerson Mcgraw, Jerrald Mcgraw, Lavaughn Mcgraw, Musa Mcgraw, Rayvon Mcgraw, Richrd Mcgraw, Sharrod Mcgraw, Stony Mcgraw, Vikas Mcgraw, Ector Mcgraw, Elia Mcgraw, Jerret Mcgraw, Josua
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Kregg Mcgraw, Larkin Mcgraw, Lenell Mcgraw, Natalie Mcgraw, Ruel Mcgraw, Sabrina Mcgraw, Thom Mcgraw, Yale Mcgraw, Adnan Mcgraw, Ankur Mcgraw, Arnett Mcgraw, Bishop Mcgraw, Brently Mcgraw, Carla
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Mcgraw, Olivier Mcgraw, Ramin Mcgraw, Rikki Mcgraw, Aleksandar Mcgraw, Burley Mcgraw, Kawan Mcgraw, Kourtney Mcgraw, Lary Mcgraw, Laszlo Mcgraw, Loyal Mcgraw, Marin Mcgraw, Richey Mcgraw, Rio Mcgraw,
Rogerio Mcgraw, Salem Mcgraw, Shalom Mcgraw, Sharron Mcgraw, Tamer Mcgraw, Torie Mcgraw, Antar Mcgraw, Barrie Mcgraw, Cletis Mcgraw, Colter Mcgraw, Duaine Mcgraw, Elpidio Mcgraw, Erice Mcgraw, Farid
Mcgraw, Fortunato Mcgraw, Gerrad Mcgraw, Jad Mcgraw, Jenaro Mcgraw, Jones Mcgraw, Kamran Mcgraw, Kelli Mcgraw, Keneth Mcgraw, Kortney Mcgraw, Levell Mcgraw, Minh Mcgraw, Monti Mcgraw, Rodell Mcgraw,
Ronal Mcgraw, Sheddrick Mcgraw, Sonia Mcgraw, Terryl Mcgraw, York Mcgraw, Andrian Mcgraw, Anne Mcgraw, Attila Mcgraw, Avis Mcgraw, Blaze Mcgraw, Chi Mcgraw, Corin Mcgraw, Dann Mcgraw, Darik Mcgraw,
Deleon Mcgraw, Demarko Mcgraw, Dennison Mcgraw, Dovid Mcgraw, Dutch Mcgraw, Emmit Mcgraw, Evert Mcgraw, Iman Mcgraw, Joshue Mcgraw, Lindy Mcgraw, Mohamad Mcgraw, Purnell Mcgraw, Rayshon Mcgraw,
Tyreese Mcgraw, Alistair Mcgraw, Arman Mcgraw, Blakely Mcgraw, Brit Mcgraw, Carlis Mcgraw, Doniel Mcgraw, Earnie Mcgraw, Hershell Mcgraw, Jasin Mcgraw, Kendale Mcgraw, Laine Mcgraw, Leandrew Mcgraw,
Lenin Mcgraw, Macon Mcgraw, Merrell Mcgraw, Mychal Mcgraw, Nat Mcgraw, Nicanor Mcgraw, Nirav Mcgraw, Waldemar Mcgraw, Yakov Mcgraw, Adrion Mcgraw, Bowen Mcgraw, Carolyn Mcgraw, Creg Mcgraw, Farley
Mcgraw, Favian Mcgraw, Ferman Mcgraw, Hurley Mcgraw, Jairus Mcgraw, Kevyn Mcgraw, Korry Mcgraw, Mandell Mcgraw, Melchor Mcgraw, Osualdo Mcgraw, Shanti Mcgraw, Sheila Mcgraw, Sotirios Mcgraw, Taft
Mcgraw, Tijuan Mcgraw, Antonyo Mcgraw, Antwoin Mcgraw, Arley Mcgraw, Carlisle Mcgraw, Christerpher Mcgraw, Derell Mcgraw, Dimitrius Mcgraw, Fabrizio Mcgraw, Gibran Mcgraw, Jasn Mcgraw, June Mcgraw,
Kenta Mcgraw, Kenyetta Mcgraw, Lantz Mcgraw, Mart Mcgraw, Naaman Mcgraw, Nasser Mcgraw, Norvell Mcgraw, Ole Mcgraw, Perfecto Mcgraw, Philemon Mcgraw, Primitivo Mcgraw, Quint Mcgraw, Rony Mcgraw,
Shanta Mcgraw, Tighe Mcgraw, Trini Mcgraw, Antionne Mcgraw, Antwaine Mcgraw, Bradon Mcgraw, Campbell Mcgraw, Chao Mcgraw, Donel Mcgraw, Isa Mcgraw, Kendra Mcgraw, Kiel Mcgraw, Lejon Mcgraw, Lonn
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Elson Mcgraw, Fulton Mcgraw, Hillard Mcgraw, Jedadiah Mcgraw, Jihad Mcgraw, Keevin Mcgraw, Kile Mcgraw, Kito Mcgraw, Kwane Mcgraw, Lamount Mcgraw, Lanell Mcgraw, Leith Mcgraw, Marshawn Mcgraw, Marwin
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Salome Mcgraw, Savalas Mcgraw, Shamel Mcgraw, Sigfredo Mcgraw, Tylon Mcgraw, Yehoshua Mcgraw, Damean Mcgraw, Gualberto Mcgraw, Guiseppe Mcgraw, Huy Mcgraw, Jaymes Mcgraw, Jaymie Mcgraw, Jeshua
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Cederick Mcgraw, Colleen Mcgraw, Cortland Mcgraw, Deanna Mcgraw, Farron Mcgraw, Gayland Mcgraw, Jaeson Mcgraw, Jamerson Mcgraw, Kenon Mcgraw, Keoki Mcgraw, Kevon Mcgraw, Kristy Mcgraw, Ladale Mcgraw,
Mare Mcgraw, Maurilio Mcgraw, Niko Mcgraw, Nikos Mcgraw, Orland Mcgraw, Remigio Mcgraw, Rhys Mcgraw, Ruby Mcgraw, Simcha Mcgraw, Thatcher Mcgraw, Travas Mcgraw, Zachry Mcgraw, Zenon Mcgraw, Amish
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Shante Mcgraw, Sullivan Mcgraw, Waylan Mcgraw, Westly Mcgraw, Woodie Mcgraw, Algie Mcgraw, Arvel Mcgraw, Benedetto Mcgraw, Bradrick Mcgraw, Chancellor Mcgraw, Chawn Mcgraw, Daylon Mcgraw, Devonne
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Barnard Mcgraw, Darriel Mcgraw, Darvis Mcgraw, Derryl Mcgraw, Dewain Mcgraw, Dietrick Mcgraw, Glyn Mcgraw, Jasmine Mcgraw, Kenyan Mcgraw, Michae Mcgraw, Montoya Mcgraw, Oracio Mcgraw, Roney Mcgraw,
Sascha Mcgraw, Shermaine Mcgraw, Sigifredo Mcgraw, Theodor Mcgraw, Virginia Mcgraw, Waco Mcgraw, Yaphet Mcgraw, Zebadiah Mcgraw, Alika Mcgraw, Aundrea Mcgraw, Brendt Mcgraw, Carvin Mcgraw,
Christophere Mcgraw, Clevon Mcgraw, Drayton Mcgraw, Evelio Mcgraw, Gerasimos Mcgraw, Jemaine Mcgraw, Kian Mcgraw, Lucan Mcgraw, Moise Mcgraw, Richy Mcgraw, Tyshon Mcgraw, Zebedee Mcgraw, Alcides
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Vashawn Mcgraw, Victorio Mcgraw, Aubry Mcgraw, Audwin Mcgraw, Babatunde Mcgraw, Bejamin Mcgraw, Cantrell Mcgraw, Carman Mcgraw, Chaddrick Mcgraw, Claiborne Mcgraw, Creston Mcgraw, Deshone Mcgraw,
Dewane Mcgraw, Gabriele Mcgraw, Hason Mcgraw, Jessi Mcgraw, Keenon Mcgraw, Keithan Mcgraw, Kostantinos Mcgraw, Leah Mcgraw, Ludwig Mcgraw, Maverick Mcgraw, Neilson Mcgraw, Octaviano Mcgraw, Orestes
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Dayle Mcgraw, Dermot Mcgraw, Encarnacion Mcgraw, Hendrick Mcgraw, Imani Mcgraw, Jackey Mcgraw, Keion Mcgraw, Kennis Mcgraw, Langdon Mcgraw, Lawayne Mcgraw, Leticia Mcgraw, Levan Mcgraw, Liberty
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Apostolos Mcgraw, Clayborn Mcgraw, Devone Mcgraw, Genesis Mcgraw, Giorgio Mcgraw, Jaleel Mcgraw, Jarmar Mcgraw, Jasan Mcgraw, Jermar Mcgraw, Jojo Mcgraw, Jondavid Mcgraw, Martyn Mcgraw, Melecio
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Aldon Mcgraw, Alejo Mcgraw, Atif Mcgraw, Cason Mcgraw, Daiel Mcgraw, Davian Mcgraw, Donaciano Mcgraw, Dywane Mcgraw, Eliberto Mcgraw, Ephriam Mcgraw, Galo Mcgraw, Giulio Mcgraw, Hanif Mcgraw, Harlon
Mcgraw, Hermon Mcgraw, Homar Mcgraw, Jhonny Mcgraw, Jonnathan Mcgraw, Kara Mcgraw, Keath Mcgraw, Kennieth Mcgraw, Kirsten Mcgraw, Kriss Mcgraw, Lennis Mcgraw, Leonides Mcgraw, Levelle Mcgraw, Marlyn
Mcgraw, Merced Mcgraw, Montrel Mcgraw, Naveen Mcgraw, Newman Mcgraw, Osiris Mcgraw, Raymondo Mcgraw, Renzo Mcgraw, Royden Mcgraw, Shawnn Mcgraw, Shuan Mcgraw, Thurmond Mcgraw, Treg Mcgraw, Tremell
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Mcgraw, Doroteo Mcgraw, Fontaine Mcgraw, Hadley Mcgraw, Jeanclaude Mcgraw, Josha Mcgraw, Kenyada Mcgraw, Laban Mcgraw, Neeraj Mcgraw, Nilesh Mcgraw, Oral Mcgraw, Quran Mcgraw, Raheim Mcgraw, Rockford
Mcgraw, Rondy Mcgraw, Sesar Mcgraw, Thayer Mcgraw, Vincient Mcgraw, Amilcar Mcgraw, Antion Mcgraw, Antrone Mcgraw, Ardis Mcgraw, Arvil Mcgraw, Avid Mcgraw, Branton Mcgraw, Carmon Mcgraw, Claudell
Mcgraw, Deion Mcgraw, Dejon Mcgraw, Delonte Mcgraw, Delvecchio Mcgraw, Dickson Mcgraw, Elwyn Mcgraw, Fredie Mcgraw, Garnet Mcgraw, Gunther Mcgraw, Jawanza Mcgraw, Jermone Mcgraw, Katrina Mcgraw,
Krista Mcgraw, Landy Mcgraw, Levester Mcgraw, Levin Mcgraw, Rahiem Mcgraw, Rashun Mcgraw, Rinaldo Mcgraw, Ronson Mcgraw, Terril Mcgraw, Ambrosio Mcgraw, Arvid Mcgraw, Atlee Mcgraw, Calogero Mcgraw,
Cobey Mcgraw, Donathan Mcgraw, Donelle Mcgraw, Dorman Mcgraw, Dung Mcgraw, Garcia Mcgraw, Jase Mcgraw, Jeanette Mcgraw, Kolin Mcgraw, Kurtiss Mcgraw, Lorenz Mcgraw, Mathis Mcgraw, Natale Mcgraw,
Patton Mcgraw, Petar Mcgraw, Richar Mcgraw, Thadd Mcgraw, Timithy Mcgraw, Tomislav Mcgraw, Tommaso Mcgraw, Tyris Mcgraw, Uvaldo Mcgraw, Anothy Mcgraw, Aristides Mcgraw, Arno Mcgraw, Deshannon Mcgraw,
Eamonn Mcgraw, Hiawatha Mcgraw, Jamol Mcgraw, Jasun Mcgraw, Joedy Mcgraw, Lenn Mcgraw, Lord Mcgraw, Manoj Mcgraw, Merl Mcgraw, Michaelangelo Mcgraw, Navin Mcgraw, Per Mcgraw, Peterson Mcgraw, Sherri
Mcgraw, Simmie Mcgraw, Socrates Mcgraw, Tramell Mcgraw, Trevino Mcgraw, Winslow Mcgraw, Andrell Mcgraw, Arlington Mcgraw, Austen Mcgraw, Aziz Mcgraw, Benuel Mcgraw, Cristofer Mcgraw, Delrico Mcgraw,
Dereke Mcgraw, Edmon Mcgraw, Finis Mcgraw, Frederich Mcgraw, Hussein Mcgraw, Jerman Mcgraw, Johnathen Mcgraw, Joni Mcgraw, Kariem Mcgraw, Kasim Mcgraw, Klayton Mcgraw, Lafe Mcgraw, Lamario Mcgraw,
Manning Mcgraw, Markos Mcgraw, Natasha Mcgraw, Nicholis Mcgraw, Nickolaos Mcgraw, Nima Mcgraw, Nitin Mcgraw, Piero Mcgraw, Robertson Mcgraw, Sabastian Mcgraw, Sandon Mcgraw, Shandy Mcgraw, Sylvia
Mcgraw, Tacuma Mcgraw, Tarance Mcgraw, Tarl Mcgraw, Traver Mcgraw, Ander Mcgraw, Bartolo Mcgraw, Brentt Mcgraw, Chace Mcgraw, Chay Mcgraw, Clete Mcgraw, Colbert Mcgraw, Domonick Mcgraw, Dondre
Mcgraw, Emanuele Mcgraw, Enoc Mcgraw, Hercules Mcgraw, Inocencio Mcgraw, Ivery Mcgraw, Jaja Mcgraw, Jalal Mcgraw, Jamelle Mcgraw, Jin Mcgraw, Jumaane Mcgraw, Kendel Mcgraw, Kurk Mcgraw, Lendell
Mcgraw, Linc Mcgraw, Louise Mcgraw, Luc Mcgraw, Mackie Mcgraw, Mehul Mcgraw, Myrick Mcgraw, Nasir Mcgraw, Omero Mcgraw, Osiel Mcgraw, Rhyan Mcgraw, Roni Mcgraw, Roshan Mcgraw, Sander Mcgraw, Thai
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Vinnie Mcgraw, Arland Mcgraw, Atul Mcgraw, Baudelio Mcgraw, Bernhard Mcgraw, Chadric Mcgraw, Charon Mcgraw, Damaris Mcgraw, Derin Mcgraw, Donavin Mcgraw, Feliz Mcgraw, Ferdinando Mcgraw, Fortino
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Arion Mcgraw, Burgess Mcgraw, Chadwin Mcgraw, Collier Mcgraw, Derico Mcgraw, Egan Mcgraw, Foy Mcgraw, Jacobi Mcgraw, Jemar Mcgraw, Kay Mcgraw, Lake Mcgraw, Markey Mcgraw, Micaiah Mcgraw, Nasario
Mcgraw, Oakley Mcgraw, Remo Mcgraw, Richad Mcgraw, Ringo Mcgraw, Romano Mcgraw, Thierry Mcgraw, Vergil Mcgraw, Vinton Mcgraw, Ajamu Mcgraw, Andrzej Mcgraw, Annette Mcgraw, Asif Mcgraw, Aundray
Mcgraw, Benn Mcgraw, Bernice Mcgraw, Byrant Mcgraw, Clemon Mcgraw, Cleotha Mcgraw, Cully Mcgraw, Darious Mcgraw, Diondre Mcgraw, Donivan Mcgraw, Fard Mcgraw, Gianfranco Mcgraw, Gyasi Mcgraw, Hosie
Mcgraw, Inez Mcgraw, Jaden Mcgraw, Julious Mcgraw, Junious Mcgraw, Ketan Mcgraw, Kieron Mcgraw, Loring Mcgraw, Noal Mcgraw, Pepe Mcgraw, Pharoah Mcgraw, Pilar Mcgraw, Raquel Mcgraw, Remon Mcgraw,
Sederick Mcgraw, Severiano Mcgraw, Shawna Mcgraw, Shawnee Mcgraw, Sim Mcgraw, Spurgeon Mcgraw, Stace Mcgraw, Taber Mcgraw, Tarron Mcgraw, Thorin Mcgraw, Wil Mcgraw, Zaire Mcgraw, Andray Mcgraw,
Berlin Mcgraw, Betty Mcgraw, Biran Mcgraw, Chaney Mcgraw, Chon Mcgraw, Dearl Mcgraw, Demetria Mcgraw, Dewaine Mcgraw, Edan Mcgraw, Ediberto Mcgraw, Froilan Mcgraw, Henrique Mcgraw, Jamille Mcgraw,
Jessey Mcgraw, Kaine Mcgraw, Kendricks Mcgraw, Kynan Mcgraw, Laray Mcgraw, Laren Mcgraw, Mandy Mcgraw, Marilyn Mcgraw, Neale Mcgraw, Nyle Mcgraw, Okey Mcgraw, Ramesh Mcgraw, Ricard Mcgraw, Saladin
Mcgraw, Sherrill Mcgraw, Shonta Mcgraw, Sione Mcgraw, Steaven Mcgraw, Stefon Mcgraw, Taji Mcgraw, Toris Mcgraw, Travell Mcgraw, Urbano Mcgraw, Wolf Mcgraw, Alvah Mcgraw, Ante Mcgraw, Arlon Mcgraw,
Auther Mcgraw, Bran Mcgraw, Chevy Mcgraw, Cormac Mcgraw, Damione Mcgraw, Delando Mcgraw, Deno Mcgraw, Dow Mcgraw, Earvin Mcgraw, Erinn Mcgraw, Florian Mcgraw, Georg Mcgraw, Gerren Mcgraw, Henery
Mcgraw, Isac Mcgraw, Jamond Mcgraw, Jaramie Mcgraw, Kinta Mcgraw, Kylan Mcgraw, Lashan Mcgraw, Mckay Mcgraw, Molly Mcgraw, Ngai Mcgraw, Nolberto Mcgraw, Quang Mcgraw, Solon Mcgraw, Tobiah Mcgraw,
Tricia Mcgraw, Wai Mcgraw, Willem Mcgraw, Yaron Mcgraw, Agostino Mcgraw, Alfonse Mcgraw, Antonious Mcgraw, Antwion Mcgraw, Bryen Mcgraw, Camille Mcgraw, Carsten Mcgraw, Cheo Mcgraw, Chevis Mcgraw,
Colen Mcgraw, Corneilus Mcgraw, Davidson Mcgraw, Dawon Mcgraw, Deldrick Mcgraw, Dia Mcgraw, Dimitris Mcgraw, Donnelle Mcgraw, Dory Mcgraw, Ewell Mcgraw, Hai Mcgraw, Heron Mcgraw, Hope Mcgraw, Imari
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Mcgraw, Toribio Mcgraw, Trebor Mcgraw, Vick Mcgraw, Worth Mcgraw, Zaid Mcgraw, Adin Mcgraw, Alim Mcgraw, Artez Mcgraw, Bashir Mcgraw, Dameian Mcgraw, Demorris Mcgraw, Deryk Mcgraw, Estill Mcgraw,
Gautam Mcgraw, Hyman Mcgraw, Isom Mcgraw, Jeral Mcgraw, Juanita Mcgraw, Kalman Mcgraw, Kee Mcgraw, Keithen Mcgraw, Kipling Mcgraw, Lesean Mcgraw, Love Mcgraw, Nadir Mcgraw, Norton Mcgraw, Philander
Mcgraw, Sebastien Mcgraw, Sherif Mcgraw, Tabitha Mcgraw, Webb Mcgraw, Akin Mcgraw, Blu Mcgraw, Calbert Mcgraw, Cassandra Mcgraw, Cephus Mcgraw, Dalvin Mcgraw, Daxton Mcgraw, Delfin Mcgraw, Drexel
Mcgraw, Elijio Mcgraw, Fareed Mcgraw, Geffrey Mcgraw, Jabin Mcgraw, Jodey Mcgraw, Jomar Mcgraw, Judy Mcgraw, Jujuan Mcgraw, Klay Mcgraw, Kyriakos Mcgraw, Laderrick Mcgraw, Landen Mcgraw, Latoya
Mcgraw, Lebron Mcgraw, Mat Mcgraw, Rashi Mcgraw, Roberta Mcgraw, Rodderick Mcgraw, Shant Mcgraw, Summer Mcgraw, Viktor Mcgraw, Abayomi Mcgraw, Adrienne Mcgraw, Akeem Mcgraw, Amaury Mcgraw, Andrez
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Mcgraw, Janmichael Mcgraw, Jeston Mcgraw, Jobie Mcgraw, Kalon Mcgraw, Kumar Mcgraw, Lemond Mcgraw, Raimundo Mcgraw, Rainier Mcgraw, Robbert Mcgraw, Rulon Mcgraw, Skeeter Mcgraw, Starling Mcgraw, Sun
Mcgraw, Tammie Mcgraw, Terrelle Mcgraw, Tiwan Mcgraw, Toddrick Mcgraw, Varick Mcgraw, Zavier Mcgraw, Zion Mcgraw, Aniceto Mcgraw, Asmar Mcgraw, Boaz Mcgraw, Cayetano Mcgraw, Dakarai Mcgraw, Deadrick
Mcgraw, Debbie Mcgraw, Dominik Mcgraw, Dugan Mcgraw, Emad Mcgraw, Glennon Mcgraw, Haralambos Mcgraw, Harlin Mcgraw, Hilliard Mcgraw, Hristopher Mcgraw, Humphrey Mcgraw, Javar Mcgraw, Javin Mcgraw,
Jeffre Mcgraw, Josuha Mcgraw, Kashif Mcgraw, Keino Mcgraw, Kemper Mcgraw, Kendrix Mcgraw, Keshawn Mcgraw, Lamone Mcgraw, Lashone Mcgraw, Lekeith Mcgraw, Markham Mcgraw, Maruice Mcgraw, Miguelangel
Mcgraw, Nichalos Mcgraw, Olanda Mcgraw, Rodrigues Mcgraw, Tarig Mcgraw, Tarius Mcgraw, Tonnie Mcgraw, Velton Mcgraw, Walden Mcgraw, Yasin Mcgraw, Alexie Mcgraw, Autry Mcgraw, Burnett Mcgraw, Burnis
Mcgraw, Cephas Mcgraw, Crandall Mcgraw, Curvin Mcgraw, Dashaun Mcgraw, Donyel Mcgraw, Emir Mcgraw, Eston Mcgraw, Farrel Mcgraw, Giovani Mcgraw, Jamaul Mcgraw, Jarmel Mcgraw, Jarold Mcgraw, Jeorge
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Mcgraw, Rasaan Mcgraw, Ravon Mcgraw, Romone Mcgraw, Stefen Mcgraw, Veron Mcgraw, Vondell Mcgraw, Airrion Mcgraw, Aki Mcgraw, Alegandro Mcgraw, Author Mcgraw, Autumn Mcgraw, Brannan Mcgraw, Caliph
Mcgraw, Carles Mcgraw, Cartez Mcgraw, Codey Mcgraw, Dammon Mcgraw, Daunte Mcgraw, Demonte Mcgraw, Devell Mcgraw, Domonique Mcgraw, Donyale Mcgraw, Dornell Mcgraw, Duwan Mcgraw, Edvardo Mcgraw,
Emeterio Mcgraw, Gerado Mcgraw, Gordan Mcgraw, Jahi Mcgraw, Jair Mcgraw, Jasmin Mcgraw, Jaydee Mcgraw, Judas Mcgraw, Kennan Mcgraw, Lehman Mcgraw, Macy Mcgraw, Mayo Mcgraw, Mazen Mcgraw, Mikhail
Mcgraw, Osama Mcgraw, Rainer Mcgraw, Raman Mcgraw, Rocio Mcgraw, Ryen Mcgraw, Shayn Mcgraw, Tee Mcgraw, Temple Mcgraw, Timathy Mcgraw, Vinh Mcgraw, Adarryl Mcgraw, Adrin Mcgraw, Alastair Mcgraw, Anh
Mcgraw, Braun Mcgraw, Burrell Mcgraw, Dalon Mcgraw, Demetrias Mcgraw, Derrill Mcgraw, Donnis Mcgraw, Dwon Mcgraw, Fredick Mcgraw, Gasper Mcgraw, Glade Mcgraw, Hulon Mcgraw, Jonatha Mcgraw, Jorma
Mcgraw, Kaipo Mcgraw, Karel Mcgraw, Kejuan Mcgraw, Kristine Mcgraw, Larrie Mcgraw, Majid Mcgraw, Mar Mcgraw, Mareo Mcgraw, Nahshon Mcgraw, Ondre Mcgraw, Rasool Mcgraw, Rayn Mcgraw, Reade Mcgraw,
Roshon Mcgraw, Rozell Mcgraw, Ryann Mcgraw, Salbador Mcgraw, Seon Mcgraw, Sharone Mcgraw, Shown Mcgraw, Taggart Mcgraw, Tarell Mcgraw, Tymon Mcgraw, Akira Mcgraw, Alice Mcgraw, Arend Mcgraw, Baretta
Mcgraw, Billyjack Mcgraw, Breton Mcgraw, Cleophas Mcgraw, Dejan Mcgraw, Delante Mcgraw, Delmus Mcgraw, Derren Mcgraw, Doni Mcgraw, Ezekial Mcgraw, Fernand Mcgraw, Gable Mcgraw, Gilford Mcgraw, Irfan
Mcgraw, Jamarcus Mcgraw, Jamason Mcgraw, Jermond Mcgraw, Jozef Mcgraw, Kal Mcgraw, Kalin Mcgraw, Kayne Mcgraw, Kosta Mcgraw, Lancelot Mcgraw, Lois Mcgraw, Marlos Mcgraw, Penny Mcgraw, Rodriques
Mcgraw, Rogerick Mcgraw, Roldan Mcgraw, Ronel Mcgraw, Sha Mcgraw, Shana Mcgraw, Taris Mcgraw, Torres Mcgraw, Trapper Mcgraw, Venancio Mcgraw, Vinod Mcgraw, Whit Mcgraw, Allah Mcgraw, Angelos Mcgraw,
Antawan Mcgraw, Bobbi Mcgraw, Declan Mcgraw, Eliasar Mcgraw, Emanual Mcgraw, Ericson Mcgraw, Erol Mcgraw, Faris Mcgraw, Filemon Mcgraw, Guthrie Mcgraw, Helder Mcgraw, Jahn Mcgraw, Jamine Mcgraw,
Jeoffrey Mcgraw, Jervis Mcgraw, Ketih Mcgraw, Kimble Mcgraw, Kole Mcgraw, Krzysztof Mcgraw, Lorena Mcgraw, Martice Mcgraw, Orie Mcgraw, Oziel Mcgraw, Ralphael Mcgraw, Randie Mcgraw, Seldon Mcgraw,
Shanna Mcgraw, Shawndale Mcgraw, Shawnte Mcgraw, Smokey Mcgraw, Teri Mcgraw, Tyrrell Mcgraw, Wilburt Mcgraw, Winton Mcgraw, Yonatan Mcgraw, Brand Mcgraw, Briton Mcgraw, Budd Mcgraw, Charlene Mcgraw,
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Donold Mcgraw, Eddrick Mcgraw, Elder Mcgraw, Gaines Mcgraw, Giovanny Mcgraw, Hermilo Mcgraw, Ivo Mcgraw, Jermale Mcgraw, Jibri Mcgraw, Joshawa Mcgraw, Kendrell Mcgraw, Kerin Mcgraw, Kert Mcgraw, Kojo
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Scotti Mcgraw, Shaunn Mcgraw, Shedric Mcgraw, Sundance Mcgraw, Tally Mcgraw, Tiant Mcgraw, Tysen Mcgraw, Ulyses Mcgraw, Vivian Mcgraw, Ajani Mcgraw, Anthany Mcgraw, Antowan Mcgraw, Brack Mcgraw,
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Jamale Mcgraw, Jamone Mcgraw, Jane Mcgraw, Jerri Mcgraw, Karla Mcgraw, Kelan Mcgraw, Kerrie Mcgraw, Kobi Mcgraw, Krist Mcgraw, Lalo Mcgraw, Latif Mcgraw, Leman Mcgraw, Levert Mcgraw, Lorenso Mcgraw,
Marcy Mcgraw, Markese Mcgraw, Merlyn Mcgraw, Milburn Mcgraw, Nicolai Mcgraw, Norval Mcgraw, Oris Mcgraw, Ranjit Mcgraw, Rashidi Mcgraw, Richardson Mcgraw, Rohn Mcgraw, Roshaun Mcgraw, Shepherd
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Tylan Mcgraw, Wheeler Mcgraw, Zalman Mcgraw, Aleksander Mcgraw, Alok Mcgraw, Alvan Mcgraw, Angelica Mcgraw, Ato Mcgraw, Aureliano Mcgraw, Ayman Mcgraw, Berkeley Mcgraw, Bracken Mcgraw, Bren Mcgraw,
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Enrrique Mcgraw, Frederik Mcgraw, French Mcgraw, Hani Mcgraw, Herby Mcgraw, Howie Mcgraw, Joab Mcgraw, Kylie Mcgraw, Leshon Mcgraw, Maribel Mcgraw, Marquese Mcgraw, Martine Mcgraw, Maurico Mcgraw,
Melford Mcgraw, Mikie Mcgraw, Myreon Mcgraw, Nabeel Mcgraw, Nabor Mcgraw, Orlandus Mcgraw, Rafi Mcgraw, Rahn Mcgraw, Ramont Mcgraw, Romain Mcgraw, Ronold Mcgraw, Rosco Mcgraw, Shabazz Mcgraw,
Shandell Mcgraw, Shaughn Mcgraw, Sumner Mcgraw, Tamika Mcgraw, Tarry Mcgraw, Tedric Mcgraw, Tellas Mcgraw, Theordore Mcgraw, Tolbert Mcgraw, Wanda Mcgraw, Winthrop Mcgraw, Wyndell Mcgraw, Abrahm
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Trung Mcgraw, Zeferino Mcgraw, Abu Mcgraw, Adaryll Mcgraw, Adolf Mcgraw, Allon Mcgraw, Almon Mcgraw, Alphonza Mcgraw, Andras Mcgraw, Angelito Mcgraw, Aurelius Mcgraw, Banjamin Mcgraw, Bartlett
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Holden Mcgraw, Iram Mcgraw, Jalon Mcgraw, Jams Mcgraw, Jaquan Mcgraw, Jerimey Mcgraw, Johnthomas Mcgraw, Kain Mcgraw, Lonney Mcgraw, Lukus Mcgraw, Marquet Mcgraw, Melody Mcgraw, Michall Mcgraw,
Nicodemus Mcgraw, Octavia Mcgraw, Petro Mcgraw, Prashant Mcgraw, Primo Mcgraw, Rahmel Mcgraw, Rita Mcgraw, Shepard Mcgraw, Siddhartha Mcgraw, Talbot Mcgraw, Tomy Mcgraw, Tyon Mcgraw, Tyrie Mcgraw,
Victory Mcgraw, Aaren Mcgraw, Ade Mcgraw, Adel Mcgraw, Akida Mcgraw, Anjel Mcgraw, Arrick Mcgraw, Becky Mcgraw, Bekim Mcgraw, Camillo Mcgraw, Cerrone Mcgraw, Chett Mcgraw, Cicero Mcgraw, Collie
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Gregrey Mcgraw, Greig Mcgraw, Gust Mcgraw, Jaycee Mcgraw, Jes Mcgraw, Jocob Mcgraw, Judith Mcgraw, Karlin Mcgraw, Kaveh Mcgraw, Keil Mcgraw, Kino Mcgraw, Kwaku Mcgraw, Lamel Mcgraw, Lea Mcgraw,
Leodis Mcgraw, Loni Mcgraw, Lonie Mcgraw, Miriam Mcgraw, Nnamdi Mcgraw, Olufemi Mcgraw, Patrik Mcgraw, Raun Mcgraw, Refujio Mcgraw, Rossie Mcgraw, Rydell Mcgraw, Severin Mcgraw, Shadeed Mcgraw,
Shannen Mcgraw, Shellie Mcgraw, Sheri Mcgraw, Sumit Mcgraw, Tamiko Mcgraw, Tien Mcgraw, Tirso Mcgraw, Tonie Mcgraw, Trino Mcgraw, Tyra Mcgraw, Wright Mcgraw, Zalmen Mcgraw, Abdon Mcgraw, Abrahan
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Elihu Mcgraw, Eustacio Mcgraw, Fernado Mcgraw, Filbert Mcgraw, Free Mcgraw, Goran Mcgraw, Habib Mcgraw, Haile Mcgraw, Han Mcgraw, Heraclio Mcgraw, Jarin Mcgraw, Javaris Mcgraw, Jeramine Mcgraw,
Joanna Mcgraw, Johnel Mcgraw, Joie Mcgraw, Juddson Mcgraw, Kason Mcgraw, Leverne Mcgraw, Liston Mcgraw, Marck Mcgraw, Marcoantonio Mcgraw, Margarita Mcgraw, Messiah Mcgraw, Nicholaos Mcgraw, Onofre
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Akram Mcgraw, Antino Mcgraw, Bengy Mcgraw, Benjimen Mcgraw, Bohdan Mcgraw, Canyon Mcgraw, Chandra Mcgraw, Cheron Mcgraw, Christhoper Mcgraw, Cordney Mcgraw, Covey Mcgraw, Danyale Mcgraw, Daquan
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Herny Mcgraw, Hervey Mcgraw, Italo Mcgraw, Jeno Mcgraw, Joell Mcgraw, Kaiser Mcgraw, Kalem Mcgraw, Kijana Mcgraw, Knute Mcgraw, Korby Mcgraw, Leaf Mcgraw, Lejuan Mcgraw, Lin Mcgraw, Mahdi Mcgraw,
Mindy Mcgraw, Mithcell Mcgraw, Mylon Mcgraw, Nichlas Mcgraw, Nickolus Mcgraw, Quanah Mcgraw, Rachael Mcgraw, Rafiq Mcgraw, Ranier Mcgraw, Rashaud Mcgraw, Rebel Mcgraw, Reuel Mcgraw, Rodgerick Mcgraw,
Ryun Mcgraw, Samad Mcgraw, Steffon Mcgraw, Tan Mcgraw, Tonio Mcgraw, Trevell Mcgraw, Truett Mcgraw, Woodley Mcgraw, Ziad Mcgraw, Alisha Mcgraw, Amory Mcgraw, Ashford Mcgraw, Bard Mcgraw, Brenan
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Dorion Mcgraw, Eberardo Mcgraw, Ethen Mcgraw, Eyal Mcgraw, Fedrick Mcgraw, Gonsalo Mcgraw, Gorje Mcgraw, Hart Mcgraw, Irby Mcgraw, Jarmal Mcgraw, Jaworski Mcgraw, Johnston Mcgraw, Joshuwa Mcgraw,
Json Mcgraw, Kenzie Mcgraw, Kristie Mcgraw, Labaron Mcgraw, Lan Mcgraw, Leroi Mcgraw, Lysander Mcgraw, Marten Mcgraw, Matthieu Mcgraw, Meghan Mcgraw, Misha Mcgraw, Norvel Mcgraw, Ohn Mcgraw, Olivia
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Vencent Mcgraw, Vida Mcgraw, Virgle Mcgraw, Weslee Mcgraw, Wilfrido Mcgraw, Ahron Mcgraw, Alanzo Mcgraw, Amol Mcgraw, Asad Mcgraw, Avel Mcgraw, Bao Mcgraw, Benancio Mcgraw, Britten Mcgraw, Cassie
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Emeka Mcgraw, Eyad Mcgraw, Fabricio Mcgraw, Graviel Mcgraw, Gunner Mcgraw, Harun Mcgraw, Hiroshi Mcgraw, Ihsan Mcgraw, Irineo Mcgraw, Jacon Mcgraw, Jamile Mcgraw, Jawad Mcgraw, Jeramia Mcgraw, Joann
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Munir Mcgraw, Murrell Mcgraw, Nachman Mcgraw, Natan Mcgraw, Nguyen Mcgraw, Nickalus Mcgraw, Nikitas Mcgraw, Orlin Mcgraw, Phillippe Mcgraw, Rasean Mcgraw, Rino Mcgraw, Robi Mcgraw, Rodgers Mcgraw,
Rolan Mcgraw, Roswell Mcgraw, Sagar Mcgraw, Schaun Mcgraw, Segundo Mcgraw, Senaca Mcgraw, Severino Mcgraw, Sloane Mcgraw, Spenser Mcgraw, Talvin Mcgraw, Teddrick Mcgraw, Theran Mcgraw, Tiron Mcgraw,
Tshombe Mcgraw, Zephaniah Mcgraw, Abbas Mcgraw, Adil Mcgraw, Alireza Mcgraw, Ancil Mcgraw, Antoan Mcgraw, Atanacio Mcgraw, Benedicto Mcgraw, Benjimin Mcgraw, Blandon Mcgraw, Bradey Mcgraw, Brannen
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Farhad Mcgraw, Gabor Mcgraw, Garreth Mcgraw, Heshimu Mcgraw, Ines Mcgraw, Ivar Mcgraw, Jahan Mcgraw, Jamael Mcgraw, Janes Mcgraw, Jaycen Mcgraw, Jett Mcgraw, Johnhenry Mcgraw, Jona Mcgraw, Josey
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Nicolaos Mcgraw, Nkosi Mcgraw, Ozie Mcgraw, Ponciano Mcgraw, Praveen Mcgraw, Ramar Mcgraw, Rizwan Mcgraw, Ronen Mcgraw, Saad Mcgraw, Shalin Mcgraw, Shalon Mcgraw, Shaunte Mcgraw, Steele Mcgraw, Sten
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Nicklos Mcgraw, Raiford Mcgraw, Rashed Mcgraw, Rayshaun Mcgraw, Reubin Mcgraw, Senica Mcgraw, Shaan Mcgraw, Shermon Mcgraw, Sherrell Mcgraw, Sierra Mcgraw, Tamarcus Mcgraw, Tavius Mcgraw, Thurmon
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Eutimio Mcgraw, Gustavus Mcgraw, Hebert Mcgraw, Hyun Mcgraw, Janssen Mcgraw, Jaxon Mcgraw, Jerre Mcgraw, Joanne Mcgraw, Jorden Mcgraw, Joseantonio Mcgraw, Kainoa Mcgraw, Kalif Mcgraw, Karin Mcgraw,
Katina Mcgraw, Keisha Mcgraw, Khalfani Mcgraw, Leanthony Mcgraw, Michaelpaul Mcgraw, Murl Mcgraw, Nazareth Mcgraw, Pearl Mcgraw, Purcell Mcgraw, Quay Mcgraw, Raylon Mcgraw, Ren Mcgraw, Renell Mcgraw,
Reshard Mcgraw, Rodel Mcgraw, Shaen Mcgraw, Talon Mcgraw, Torsten Mcgraw, Victorino Mcgraw, Vonzell Mcgraw, Yong Mcgraw, Zaki Mcgraw, Abdiel Mcgraw, Adon Mcgraw, Alferd Mcgraw, Amjad Mcgraw, Arlester
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Dayna Mcgraw, Domanic Mcgraw, Dougles Mcgraw, Elex Mcgraw, Elrico Mcgraw, Estaban Mcgraw, Ferlando Mcgraw, Filipe Mcgraw, Geovanni Mcgraw, Glenford Mcgraw, Gumaro Mcgraw, Gustabo Mcgraw, Haley
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Rivers Mcgraw, Rody Mcgraw, Saint Mcgraw, Santee Mcgraw, Shai Mcgraw, Shelvin Mcgraw, Shyam Mcgraw, Silvino Mcgraw, Squire Mcgraw, Star Mcgraw, Steen Mcgraw, Stein Mcgraw, Tae Mcgraw, Taryn Mcgraw,
Tavare Mcgraw, Thedore Mcgraw, Yehudah Mcgraw, Zacchaeus Mcgraw, Zachory Mcgraw, Zan Mcgraw, Adewale Mcgraw, Aleksandr Mcgraw, Alf Mcgraw, Alvester Mcgraw, Amiri Mcgraw, Araceli Mcgraw, Arcenio
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Delmont Mcgraw, Dena Mcgraw, Devlon Mcgraw, Dimitry Mcgraw, Duwane Mcgraw, Esteven Mcgraw, Fergus Mcgraw, Harles Mcgraw, Hutch Mcgraw, Ion Mcgraw, Isidore Mcgraw, Jamus Mcgraw, Jerett Mcgraw, Jerrit
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Mcgraw, Wei Mcgraw, Zackariah Mcgraw, Aleem Mcgraw, Antwione Mcgraw, Ayodele Mcgraw, Barkley Mcgraw, Bartt Mcgraw, Bethany Mcgraw, Bing Mcgraw, Braheem Mcgraw, Brek Mcgraw, Caron Mcgraw, Cathy
Mcgraw, Cecilia Mcgraw, Cristina Mcgraw, Dainel Mcgraw, Dal Mcgraw, Darrek Mcgraw, Darrelle Mcgraw, Dashun Mcgraw, Delrick Mcgraw, Demitrus Mcgraw, Dishon Mcgraw, Donnald Mcgraw, Ehab Mcgraw, Erico
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Mcgraw, Khaalis Mcgraw, Kion Mcgraw, Lam Mcgraw, Larence Mcgraw, Lavonne Mcgraw, Marti Mcgraw, Mclean Mcgraw, Mercury Mcgraw, Miki Mcgraw, Norvin Mcgraw, Onnie Mcgraw, Paulmichael Mcgraw, Pavan
Mcgraw, Paz Mcgraw, Phuong Mcgraw, Race Mcgraw, Ralston Mcgraw, Ramadan Mcgraw, Rashee Mcgraw, Roan Mcgraw, Sally Mcgraw, Shanga Mcgraw, Slater Mcgraw, Tarvaris Mcgraw, Tevin Mcgraw, Travus Mcgraw,
Viviano Mcgraw, Zerick Mcgraw, Abdula Mcgraw, Achilles Mcgraw, Adarryll Mcgraw, Ananias Mcgraw, Antinio Mcgraw, Antrell Mcgraw, Anzio Mcgraw, Aristidis Mcgraw, Ashante Mcgraw, Atom Mcgraw, Audley
Mcgraw, Azim Mcgraw, Bandy Mcgraw, Bary Mcgraw, Britain Mcgraw, Bryne Mcgraw, Calen Mcgraw, Ceddrick Mcgraw, Chalmers Mcgraw, Chanc Mcgraw, Chap Mcgraw, Charly Mcgraw, Charron Mcgraw, Cornellius
Mcgraw, Darth Mcgraw, Dayan Mcgraw, Deran Mcgraw, Dewand Mcgraw, Fran Mcgraw, Girolamo Mcgraw, Harvie Mcgraw, Haskel Mcgraw, Hoy Mcgraw, Islam Mcgraw, Ivon Mcgraw, Jafari Mcgraw, Jamario Mcgraw, Jehu
Mcgraw, Josemanuel Mcgraw, Jung Mcgraw, Kabir Mcgraw, Kael Mcgraw, Khan Mcgraw, Khayree Mcgraw, Kimber Mcgraw, Koran Mcgraw, Lamonta Mcgraw, Lavalle Mcgraw, Lavel Mcgraw, Lenardo Mcgraw, Lydia
Mcgraw, Mandrill Mcgraw, Mannie Mcgraw, Marguis Mcgraw, Marisa Mcgraw, Marissa Mcgraw, Marshaun Mcgraw, Medardo Mcgraw, Naomi Mcgraw, Naser Mcgraw, Natanael Mcgraw, Oji Mcgraw, Orvil Mcgraw, Pasqual
Mcgraw, Previn Mcgraw, Reo Mcgraw, Ruddy Mcgraw, Shahram Mcgraw, Sharad Mcgraw, Shaunta Mcgraw, Shuron Mcgraw, Teo Mcgraw, Thalmus Mcgraw, Theodoric Mcgraw, Tibor Mcgraw, Tres Mcgraw, Ulrich Mcgraw,
Winter Mcgraw, Yasser Mcgraw, Ygnacio Mcgraw, Abasi Mcgraw, Alwin Mcgraw, Alwyn Mcgraw, Amalio Mcgraw, Arty Mcgraw, Baker Mcgraw, Bishara Mcgraw, Briane Mcgraw, Catalino Mcgraw, Darrall Mcgraw,
Dartanyon Mcgraw, Deano Mcgraw, Demico Mcgraw, Deontae Mcgraw, Deveron Mcgraw, Devonn Mcgraw, Donnovan Mcgraw, Durant Mcgraw, Duy Mcgraw, Evander Mcgraw, Ever Mcgraw, Ewing Mcgraw, Fouad Mcgraw,
Froylan Mcgraw, Gabriela Mcgraw, Garnell Mcgraw, Gerlad Mcgraw, Ginger Mcgraw, Glendale Mcgraw, Grafton Mcgraw, Haig Mcgraw, Hale Mcgraw, Iver Mcgraw, Ivin Mcgraw, Jarel Mcgraw, Javen Mcgraw, Jeran
Mcgraw, Jereld Mcgraw, Jermichael Mcgraw, Jimel Mcgraw, Karreem Mcgraw, Keiron Mcgraw, Kester Mcgraw, Laronn Mcgraw, Lavor Mcgraw, Lehi Mcgraw, Lemon Mcgraw, Lourdes Mcgraw, Marqui Mcgraw, Marsh
Mcgraw, Marx Mcgraw, Montego Mcgraw, Murice Mcgraw, Mychael Mcgraw, Nashawn Mcgraw, Nichalas Mcgraw, Nickalas Mcgraw, Nuri Mcgraw, Obinna Mcgraw, Onaje Mcgraw, Patrich Mcgraw, Pavel Mcgraw, Priscilla
Mcgraw, Rakeem Mcgraw, Rande Mcgraw, Rashann Mcgraw, Rayan Mcgraw, Raynold Mcgraw, Ricko Mcgraw, Riki Mcgraw, Rooney Mcgraw, Saulo Mcgraw, Schad Mcgraw, Seiji Mcgraw, Shahn Mcgraw, Shantel Mcgraw,
Shanton Mcgraw, Sisto Mcgraw, Taiwo Mcgraw, Travion Mcgraw, Tristen Mcgraw, Truong Mcgraw, Tywann Mcgraw, Tywayne Mcgraw, Uhuru Mcgraw, Welby Mcgraw, Wesam Mcgraw, Alexius Mcgraw, Ammar Mcgraw,
Andrews Mcgraw, Atticus Mcgraw, Audy Mcgraw, Belinda Mcgraw, Bennet Mcgraw, Bomani Mcgraw, Bulmaro Mcgraw, Charlies Mcgraw, Chato Mcgraw, Chelsea Mcgraw, Chipper Mcgraw, Clearence Mcgraw, Cleotis
Mcgraw, Cleven Mcgraw, Corydon Mcgraw, Dadrian Mcgraw, Damiano Mcgraw, Darrold Mcgraw, Davell Mcgraw, Derone Mcgraw, Derral Mcgraw, Derwood Mcgraw, Django Mcgraw, Doc Mcgraw, Dontell Mcgraw, Doren
Mcgraw, Elza Mcgraw, Esmeralda Mcgraw, Estil Mcgraw, Finn Mcgraw, Fredrik Mcgraw, Garrie Mcgraw, Greogory Mcgraw, Harlow Mcgraw, Iris Mcgraw, Jascha Mcgraw, Jediah Mcgraw, Jef Mcgraw, Jefferie
Mcgraw, Jerimah Mcgraw, Jerrie Mcgraw, Joal Mcgraw, Johsua Mcgraw, Joon Mcgraw, Joson Mcgraw, Kamar Mcgraw, Kellen Mcgraw, Kelvis Mcgraw, Ketrick Mcgraw, Kimberley Mcgraw, Kingston Mcgraw, Meldon
Mcgraw, Mendell Mcgraw, Michaeljohn Mcgraw, Moss Mcgraw, Nuno Mcgraw, Onofrio Mcgraw, Phineas Mcgraw, Ramell Mcgraw, Raydell Mcgraw, Regie Mcgraw, Ronnel Mcgraw, Sadat Mcgraw, Sajid Mcgraw, Shanard
Mcgraw, Shaul Mcgraw, Shondale Mcgraw, Shyrone Mcgraw, Simuel Mcgraw, Sjon Mcgraw, Stephenson Mcgraw, Tadeusz Mcgraw, Takeshi Mcgraw, Tilman Mcgraw, Tomasz Mcgraw, Travares Mcgraw, Truitt Mcgraw,
Ulrick Mcgraw, Vandy Mcgraw, Yvon Mcgraw, Zebulan Mcgraw, Adolpho Mcgraw, Alek Mcgraw, Amery Mcgraw, Andrej Mcgraw, Aniello Mcgraw, Anurag Mcgraw, Audra Mcgraw, Boruch Mcgraw, Burr Mcgraw, Cable
Mcgraw, Cheney Mcgraw, Cimarron Mcgraw, Clell Mcgraw, Clinten Mcgraw, Daimen Mcgraw, Dameyon Mcgraw, Dason Mcgraw, Deanglo Mcgraw, Delaine Mcgraw, Demeco Mcgraw, Denim Mcgraw, Derryck Mcgraw, Domnick
Mcgraw, Doy Mcgraw, Dwyne Mcgraw, Edwards Mcgraw, Elisa Mcgraw, Esmond Mcgraw, Gawain Mcgraw, Green Mcgraw, Hallie Mcgraw, Hameed Mcgraw, Heinrich Mcgraw, Jeri Mcgraw, Kaj Mcgraw, Kalan Mcgraw, Kash
Mcgraw, Kente Mcgraw, Koji Mcgraw, Kunte Mcgraw, Laurin Mcgraw, Livingston Mcgraw, Meliton Mcgraw, Michaeal Mcgraw, Myran Mcgraw, Nephtali Mcgraw, Nicasio Mcgraw, Nyles Mcgraw, Obediah Mcgraw, Pride
Mcgraw, Primus Mcgraw, Rae Mcgraw, Rahmaan Mcgraw, Rhoderick Mcgraw, Rip Mcgraw, River Mcgraw, Roberts Mcgraw, Rodricus Mcgraw, Rommie Mcgraw, Romney Mcgraw, Roverto Mcgraw, Samar Mcgraw, Sergei
Mcgraw, Shadrack Mcgraw, Shavon Mcgraw, Stevin Mcgraw, Tarone Mcgraw, Tashawn Mcgraw, Tora Mcgraw, Tri Mcgraw, Vinny Mcgraw, Yesenia Mcgraw, Yvan Mcgraw, Zubin Mcgraw, Absalom Mcgraw, Aldrick Mcgraw,
Alif Mcgraw, Amad Mcgraw, Anacleto Mcgraw, Andru Mcgraw, Argelio Mcgraw, Arrington Mcgraw, Ashely Mcgraw, Auston Mcgraw, Baird Mcgraw, Beryl Mcgraw, Celester Mcgraw, Chadron Mcgraw, Chante Mcgraw,
Dack Mcgraw, Darain Mcgraw, Darvell Mcgraw, Daune Mcgraw, Daylan Mcgraw, Dedan Mcgraw, Deitrick Mcgraw, Dejaun Mcgraw, Denzel Mcgraw, Deverick Mcgraw, Eris Mcgraw, Flor Mcgraw, Frankey Mcgraw,
Gaberial Mcgraw, Gianluca Mcgraw, Hesham Mcgraw, Iban Mcgraw, Ilya Mcgraw, Isham Mcgraw, Issiah Mcgraw, Jacey Mcgraw, Jahmel Mcgraw, Jaimey Mcgraw, Jamane Mcgraw, Jammey Mcgraw, Jareb Mcgraw, Jarek
Mcgraw, Jemery Mcgraw, Jeremaih Mcgraw, Jernard Mcgraw, Jerramy Mcgraw, Jese Mcgraw, Johnaton Mcgraw, Jontae Mcgraw, Kayode Mcgraw, Kaz Mcgraw, Kedar Mcgraw, Kell Mcgraw, Kewan Mcgraw, Kiyoshi
Mcgraw, Ladarryl Mcgraw, Lamart Mcgraw, Lataurus Mcgraw, Lavarr Mcgraw, Lavert Mcgraw, Lawan Mcgraw, Loney Mcgraw, Lonnel Mcgraw, Luan Mcgraw, Macio Mcgraw, Mansa Mcgraw, Miachel Mcgraw, Neftaly
Mcgraw, Nichlos Mcgraw, Nicklous Mcgraw, Nik Mcgraw, Orlander Mcgraw, Quadir Mcgraw, Raeford Mcgraw, Rajohn Mcgraw, Ranson Mcgraw, Rasul Mcgraw, Renald Mcgraw, Reynald Mcgraw, Roben Mcgraw,
Rutherford Mcgraw, Salathiel Mcgraw, Savoy Mcgraw, Sharief Mcgraw, Sherrick Mcgraw, Siddharth Mcgraw, Staci Mcgraw, Steed Mcgraw, Tauheed Mcgraw, Thaddus Mcgraw, Toland Mcgraw, Toran Mcgraw, Torriano
Mcgraw, Turon Mcgraw, Vashaun Mcgraw, Zebediah Mcgraw, Aasim Mcgraw, Abron Mcgraw, Ahman Mcgraw, Aiden Mcgraw, Alexsander Mcgraw, Alias Mcgraw, Aly Mcgraw, Ameet Mcgraw, Anothny Mcgraw, Ansley
Mcgraw, Ballard Mcgraw, Bedford Mcgraw, Beecher Mcgraw, Benaiah Mcgraw, Clent Mcgraw, Colvin Mcgraw, Dakari Mcgraw, Darric Mcgraw, Daved Mcgraw, Demichael Mcgraw, Deray Mcgraw, Deren Mcgraw,
Dionisios Mcgraw, Dionta Mcgraw, Dorothy Mcgraw, Dorrian Mcgraw, Dorris Mcgraw, Dyke Mcgraw, Elmon Mcgraw, Erle Mcgraw, Eunice Mcgraw, Eustace Mcgraw, Evon Mcgraw, Fay Mcgraw, Federick Mcgraw, Feras
Mcgraw, Gabrielle Mcgraw, Gaven Mcgraw, Gay Mcgraw, Gjon Mcgraw, Grier Mcgraw, Guillaume Mcgraw, Hansen Mcgraw, Harrel Mcgraw, Hazel Mcgraw, Heathe Mcgraw, Herschell Mcgraw, Ingram Mcgraw, Isauro
Mcgraw, Jaimy Mcgraw, Jaymar Mcgraw, Jerin Mcgraw, Jermall Mcgraw, Jevin Mcgraw, Jillian Mcgraw, Johnmark Mcgraw, Jovaughn Mcgraw, Kamuela Mcgraw, Kendon Mcgraw, Khoa Mcgraw, Lamberto Mcgraw, Latham
Mcgraw, Luster Mcgraw, Malon Mcgraw, Micholas Mcgraw, Mitul Mcgraw, Montrice Mcgraw, Moroni Mcgraw, Mylan Mcgraw, Nason Mcgraw, Nic Mcgraw, Nowell Mcgraw, Oryan Mcgraw, Owens Mcgraw, Phill Mcgraw,
Raed Mcgraw, Raji Mcgraw, Ramzy Mcgraw, Rebekah Mcgraw, Reeve Mcgraw, Regenald Mcgraw, Roddie Mcgraw, Roshun Mcgraw, Salvator Mcgraw, Seanpaul Mcgraw, Shanne Mcgraw, Shawntay Mcgraw, Sheldrick
Mcgraw, Sheppard Mcgraw, Snapper Mcgraw, Tarick Mcgraw, Terelle Mcgraw, Tery Mcgraw, Thelonious Mcgraw, Trampis Mcgraw, Tushar Mcgraw, Vernice Mcgraw, Whalen Mcgraw, Willim Mcgraw, Alban Mcgraw,
Alexy Mcgraw, Almalik Mcgraw, Amedeo Mcgraw, Antoney Mcgraw, Asia Mcgraw, Aurthur Mcgraw, Avelardo Mcgraw, Barth Mcgraw, Cartrell Mcgraw, Chaderick Mcgraw, Champ Mcgraw, Chano Mcgraw, Charity Mcgraw,
Christhopher Mcgraw, Christino Mcgraw, Clavin Mcgraw, Cuyler Mcgraw, Dalbert Mcgraw, Darcel Mcgraw, Darcell Mcgraw, Darrie Mcgraw, Denney Mcgraw, Denson Mcgraw, Dezmond Mcgraw, Donyea Mcgraw, Dorien
Mcgraw, Emidio Mcgraw, Etan Mcgraw, Evagelos Mcgraw, Everado Mcgraw, Farren Mcgraw, Fate Mcgraw, Filomeno Mcgraw, Fonzie Mcgraw, Franko Mcgraw, Garrod Mcgraw, Graciano Mcgraw, Gustin Mcgraw, Isrrael
Mcgraw, Jac Mcgraw, Jakie Mcgraw, Jamas Mcgraw, Jamiyl Mcgraw, Jaylon Mcgraw, Jeanmarc Mcgraw, Jebadiah Mcgraw, Jeffey Mcgraw, Jiles Mcgraw, Johnnell Mcgraw, Josip Mcgraw, Juanmanuel Mcgraw, Kemo
Mcgraw, Kemuel Mcgraw, Kindrick Mcgraw, Kirtis Mcgraw, Kyley Mcgraw, Lacorey Mcgraw, Lansing Mcgraw, Lawyer Mcgraw, Lenzie Mcgraw, Lessie Mcgraw, Lindley Mcgraw, Lynette Mcgraw, Lyonel Mcgraw, Magnus
Mcgraw, Marki Mcgraw, Marrell Mcgraw, Mccoy Mcgraw, Mcdonald Mcgraw, Meade Mcgraw, Mickell Mcgraw, Mikah Mcgraw, Miranda Mcgraw, Mitchael Mcgraw, Mitcheal Mcgraw, Montee Mcgraw, Morio Mcgraw, Murat
Mcgraw, Nashon Mcgraw, Nova Mcgraw, Olusegun Mcgraw, Ondra Mcgraw, Orian Mcgraw, Orren Mcgraw, Reggy Mcgraw, Rodrico Mcgraw, Ronzell Mcgraw, Runako Mcgraw, Ryszard Mcgraw, Shady Mcgraw, Shango
Mcgraw, Shawan Mcgraw, Terez Mcgraw, Thang Mcgraw, Tomothy Mcgraw, Toron Mcgraw, Torrin Mcgraw, Toya Mcgraw, Trayon Mcgraw, Trellis Mcgraw, Tremel Mcgraw, Tryon Mcgraw, Tyrek Mcgraw, Wessley Mcgraw,
Yan Mcgraw, Zed Mcgraw, Ako Mcgraw, Aldrich Mcgraw, Alesandro Mcgraw, Alrick Mcgraw, Ayo Mcgraw, Bobbyjoe Mcgraw, Broadus Mcgraw, Brockton Mcgraw, Callie Mcgraw, Ched Mcgraw, Conn Mcgraw, Cristo
Mcgraw, Dallen Mcgraw, Dannel Mcgraw, Dasan Mcgraw, Datron Mcgraw, Dawaun Mcgraw, Delance Mcgraw, Delawrence Mcgraw, Demarrio Mcgraw, Denell Mcgraw, Derak Mcgraw, Derrie Mcgraw, Deundra Mcgraw, Dorin
Mcgraw, Dyon Mcgraw, Elizardo Mcgraw, Filimon Mcgraw, Fitzroy Mcgraw, Gari Mcgraw, Glover Mcgraw, Gretchen Mcgraw, Hartley Mcgraw, Holt Mcgraw, Jabe Mcgraw, Jaclyn Mcgraw, Jadrien Mcgraw, Jamah
Mcgraw, Jarman Mcgraw, Jarreau Mcgraw, Jearld Mcgraw, Jerimi Mcgraw, Jerris Mcgraw, Joeph Mcgraw, Jorel Mcgraw, Joshus Mcgraw, Jovany Mcgraw, Kaven Mcgraw, Kawaski Mcgraw, Kiron Mcgraw, Korrey
Mcgraw, Kort Mcgraw, Lavale Mcgraw, Lenorris Mcgraw, Lizandro Mcgraw, Lynard Mcgraw, Manson Mcgraw, Marky Mcgraw, Marlando Mcgraw, Maxime Mcgraw, Mcgarrett Mcgraw, Merwin Mcgraw, Miko Mcgraw, Niklas
Mcgraw, Nkrumah Mcgraw, Orman Mcgraw, Osmar Mcgraw, Paresh Mcgraw, Ponce Mcgraw, Prem Mcgraw, Rachelle Mcgraw, Rajendra Mcgraw, Rashone Mcgraw, Rawn Mcgraw, Raynor Mcgraw, Rechard Mcgraw, Rogan
Mcgraw, Rollo Mcgraw, Romando Mcgraw, Royd Mcgraw, Salah Mcgraw, Scoey Mcgraw, Selvin Mcgraw, Swain Mcgraw, Tarun Mcgraw, Tawn Mcgraw, Teal Mcgraw, Telesforo Mcgraw, Thaddeaus Mcgraw, Timoth Mcgraw,
Toren Mcgraw, Vicki Mcgraw, Wael Mcgraw, Whitfield Mcgraw, Winifred Mcgraw, Wyeth Mcgraw, Abdu Mcgraw, Abdulla Mcgraw, Akio Mcgraw, Aleck Mcgraw, Antwian Mcgraw, Aquiles Mcgraw, Armad Mcgraw, Averill
Mcgraw, Bijan Mcgraw, Billyjo Mcgraw, Bladimir Mcgraw, Bray Mcgraw, Brigido Mcgraw, Candice Mcgraw, Changa Mcgraw, Chanon Mcgraw, Charlotte Mcgraw, Colie Mcgraw, Colonel Mcgraw, Crespin Mcgraw,
Dabney Mcgraw, Dace Mcgraw, Damaine Mcgraw, Danna Mcgraw, Danya Mcgraw, Darelle Mcgraw, Darrnell Mcgraw, Deaundre Mcgraw, Dijon Mcgraw, Dina Mcgraw, Dontel Mcgraw, Dracy Mcgraw, Duayne Mcgraw, Easton
Mcgraw, Eban Mcgraw, Elric Mcgraw, Ervey Mcgraw, Festus Mcgraw, Filip Mcgraw, Furnell Mcgraw, Gardiner Mcgraw, Gleen Mcgraw, Herve Mcgraw, Hogan Mcgraw, Hossein Mcgraw, Ikaika Mcgraw, Imad Mcgraw,
Jacek Mcgraw, Jamaica Mcgraw, Jamichael Mcgraw, Jammal Mcgraw, Jarmon Mcgraw, Jerren Mcgraw, Joshu Mcgraw, Juanjose Mcgraw, Kanoa Mcgraw, Kia Mcgraw, Kiernan Mcgraw, Kirtus Mcgraw, Lacedric Mcgraw,
Legrand Mcgraw, Lekendrick Mcgraw, Lemoyne Mcgraw, Lige Mcgraw, Loranzo Mcgraw, Loyce Mcgraw, Masai Mcgraw, Master Mcgraw, Michah Mcgraw, Min Mcgraw, Mir Mcgraw, Mohan Mcgraw, Montrail Mcgraw,
Montray Mcgraw, Nacoma Mcgraw, Nakoa Mcgraw, Navid Mcgraw, Nole Mcgraw, Opie Mcgraw, Orpheus Mcgraw, Payam Mcgraw, Pearce Mcgraw, Ramal Mcgraw, Recco Mcgraw, Renn Mcgraw, Rigel Mcgraw, Roe Mcgraw,
Rolondo Mcgraw, Romaldo Mcgraw, Severn Mcgraw, Shakim Mcgraw, Shulem Mcgraw, Silviano Mcgraw, Strider Mcgraw, Tayari Mcgraw, Tiburcio Mcgraw, Tilmon Mcgraw, Tomar Mcgraw, Tran Mcgraw, Travaris
Mcgraw, Trayvon Mcgraw, Trev Mcgraw, Trevar Mcgraw, Tyreek Mcgraw, Windle Mcgraw, Yaw Mcgraw, Yechezkel Mcgraw, Zeth Mcgraw, Zollie Mcgraw, Acey Mcgraw, Adisa Mcgraw, Arlanda Mcgraw, Artavius Mcgraw,
Avin Mcgraw, Beauford Mcgraw, Bernadette Mcgraw, Birch Mcgraw, Brittan Mcgraw, Brown Mcgraw, Cabe Mcgraw, Calton Mcgraw, Caribe Mcgraw, Carrington Mcgraw, Chadwich Mcgraw, Chrstopher Mcgraw, Chun
Mcgraw, Crescencio Mcgraw, Damyon Mcgraw, Dantrell Mcgraw, Dariusz Mcgraw, Dartanion Mcgraw, Delio Mcgraw, Demarius Mcgraw, Deniz Mcgraw, Dennard Mcgraw, Domingos Mcgraw, Dominie Mcgraw, Doyal
Mcgraw, Dwanye Mcgraw, Earon Mcgraw, Efstathios Mcgraw, Elkin Mcgraw, Ellen Mcgraw, Fabain Mcgraw, Frans Mcgraw, Gopal Mcgraw, Haleem Mcgraw, Iverson Mcgraw, Japeth Mcgraw, Johanna Mcgraw, Jorgen
Mcgraw, Jourdan Mcgraw, Kassim Mcgraw, Keidrick Mcgraw, Keif Mcgraw, Knox Mcgraw, Lacharles Mcgraw, Ladarius Mcgraw, Ladarrell Mcgraw, Leonce Mcgraw, Lipa Mcgraw, Lorren Mcgraw, Luqman Mcgraw,
Mcihael Mcgraw, Mercer Mcgraw, Micharl Mcgraw, Mikle Mcgraw, Montre Mcgraw, Morad Mcgraw, Nichael Mcgraw, Phi Mcgraw, Quantez Mcgraw, Quency Mcgraw, Quinlan Mcgraw, Quintrell Mcgraw, Redell Mcgraw,
Regnald Mcgraw, Renauld Mcgraw, Ronelle Mcgraw, Rudell Mcgraw, Ruffin Mcgraw, Salil Mcgraw, Seanmichael Mcgraw, Shaheem Mcgraw, Sharieff Mcgraw, Shawndel Mcgraw, Stamatis Mcgraw, Tadashi Mcgraw,
Tiras Mcgraw, Toan Mcgraw, Toben Mcgraw, Tyrik Mcgraw, Tyshaun Mcgraw, Ulisses Mcgraw, Visente Mcgraw, Wadell Mcgraw, Wilfrid Mcgraw, Won Mcgraw, Aashish Mcgraw, Adama Mcgraw, Aja Mcgraw, Alcindor
Mcgraw, Alejandra Mcgraw, Alyn Mcgraw, Ami Mcgraw, Andrel Mcgraw, Antonine Mcgraw, Artemas Mcgraw, Artimus Mcgraw, Ary Mcgraw, Atiim Mcgraw, Avinash Mcgraw, Brittain Mcgraw, Callan Mcgraw, Cari
Mcgraw, Claire Mcgraw, Conard Mcgraw, Danan Mcgraw, Darol Mcgraw, Day Mcgraw, Delma Mcgraw, Demetricus Mcgraw, Dewone Mcgraw, Dougals Mcgraw, Elester Mcgraw, Espiridion Mcgraw, Estanislado Mcgraw,
Esther Mcgraw, Fabien Mcgraw, Famous Mcgraw, Fredderick Mcgraw, Friedrich Mcgraw, Garrette Mcgraw, Greogry Mcgraw, Gwendolyn Mcgraw, Hardin Mcgraw, Hendrik Mcgraw, Hendrix Mcgraw, Jabulani Mcgraw,
Jacin Mcgraw, Jacquelyn Mcgraw, Jantzen Mcgraw, Jarren Mcgraw, Jasyn Mcgraw, Jayde Mcgraw, Jeris Mcgraw, Jery Mcgraw, Jolyon Mcgraw, Karem Mcgraw, Key Mcgraw, Kibwe Mcgraw, Kyon Mcgraw, Lamin Mcgraw,
Lasaro Mcgraw, Ledon Mcgraw, Malcomb Mcgraw, Mando Mcgraw, Marcelus Mcgraw, Margus Mcgraw, Marzell Mcgraw, Mearl Mcgraw, Miron Mcgraw, Mitesh Mcgraw, Montay Mcgraw, Montrey Mcgraw, Morey Mcgraw,
Morrie Mcgraw, Mozell Mcgraw, Narada Mcgraw, Naveed Mcgraw, Nenad Mcgraw, Nicolaas Mcgraw, Nilton Mcgraw, Ojay Mcgraw, Olga Mcgraw, Oman Mcgraw, Pancho Mcgraw, Pepper Mcgraw, Poul Mcgraw, Rajon
Mcgraw, Rawley Mcgraw, Rayon Mcgraw, Read Mcgraw, Rees Mcgraw, Reeves Mcgraw, Romond Mcgraw, Rondrick Mcgraw, Sancho Mcgraw, Santi Mcgraw, Selso Mcgraw, Sevan Mcgraw, Shondel Mcgraw, Tavarius Mcgraw,
Thelbert Mcgraw, Theodoro Mcgraw, Theus Mcgraw, Thien Mcgraw, Timoty Mcgraw, Timur Mcgraw, Travius Mcgraw, Trisha Mcgraw, Tyquan Mcgraw, Usbaldo Mcgraw, Varnell Mcgraw, Videl Mcgraw, Willi Mcgraw,
Zacariah Mcgraw, Zain Mcgraw, Aamir Mcgraw, Abdallah Mcgraw, Almando Mcgraw, Almond Mcgraw, Alpheus Mcgraw, Alvarez Mcgraw, Alverto Mcgraw, Andrick Mcgraw, Anthonie Mcgraw, Anthoy Mcgraw, Antiona
Mcgraw, Antwand Mcgraw, Arrie Mcgraw, Artur Mcgraw, Baldwin Mcgraw, Bardo Mcgraw, Carols Mcgraw, Chau Mcgraw, Clayborne Mcgraw, Conell Mcgraw, Confesor Mcgraw, Cottrell Mcgraw, Craven Mcgraw, Curits
Mcgraw, Damiel Mcgraw, Deandrae Mcgraw, Derrius Mcgraw, Dirrick Mcgraw, Dora Mcgraw, Dorain Mcgraw, Dushaun Mcgraw, Eliodoro Mcgraw, Elvie Mcgraw, Faraz Mcgraw, Fernanda Mcgraw, Ferron Mcgraw,
Frankin Mcgraw, Fraser Mcgraw, Giuliano Mcgraw, Godwin Mcgraw, Hakan Mcgraw, Hershal Mcgraw, Jacobie Mcgraw, Jada Mcgraw, Jaimeson Mcgraw, Jamill Mcgraw, Jarrard Mcgraw, Jeromi Mcgraw, Jerral Mcgraw,
Jilberto Mcgraw, Jonel Mcgraw, Joseangel Mcgraw, Josedejesus Mcgraw, Joshva Mcgraw, Jovani Mcgraw, Juana Mcgraw, Juvencio Mcgraw, Karol Mcgraw, Keanon Mcgraw, Keli Mcgraw, Kelven Mcgraw, Kemal
Mcgraw, Khanh Mcgraw, Kiah Mcgraw, Kimmy Mcgraw, Kitt Mcgraw, Laith Mcgraw, Latravis Mcgraw, Lebarron Mcgraw, Lenon Mcgraw, Leondre Mcgraw, Lue Mcgraw, Luka Mcgraw, Lyndal Mcgraw, Macedonio Mcgraw,
Mahdee Mcgraw, Marcellas Mcgraw, Mardell Mcgraw, Monnie Mcgraw, Naron Mcgraw, Nehal Mcgraw, Nicholai Mcgraw, Nicol Mcgraw, Nimesh Mcgraw, Orlan Mcgraw, Pal Mcgraw, Paras Mcgraw, Pasha Mcgraw, Pio
Mcgraw, Piotr Mcgraw, Rahmon Mcgraw, Reilly Mcgraw, Rommell Mcgraw, Romondo Mcgraw, Ronie Mcgraw, Rontae Mcgraw, Roth Mcgraw, Ruven Mcgraw, Senecca Mcgraw, Sequoia Mcgraw, Shadwick Mcgraw, Shaquan
Mcgraw, Shaundell Mcgraw, Shaya Mcgraw, Shloma Mcgraw, Shonnon Mcgraw, Shunta Mcgraw, Silver Mcgraw, Stanislaw Mcgraw, Sumeet Mcgraw, Sylvain Mcgraw, Tajiddin Mcgraw, Tejas Mcgraw, Tereso Mcgraw,
Tilton Mcgraw, Tjay Mcgraw, Tramon Mcgraw, Troi Mcgraw, Troye Mcgraw, Tyrelle Mcgraw, Usher Mcgraw, Valeriano Mcgraw, Verlyn Mcgraw, Vickie Mcgraw, Wallis Mcgraw, Yahya Mcgraw, Yair Mcgraw, Zacharie
Mcgraw, Zakery Mcgraw, Zoe Mcgraw, Abdual Mcgraw, Adeyemi Mcgraw, Alisa Mcgraw, Alois Mcgraw, Alondo Mcgraw, Alprentice Mcgraw, Altariq Mcgraw, Amaro Mcgraw, Amen Mcgraw, Andria Mcgraw, Anthoni
Mcgraw, Apurva Mcgraw, Arrow Mcgraw, Aryn Mcgraw, Atilano Mcgraw, Auburn Mcgraw, Auturo Mcgraw, Aven Mcgraw, Averil Mcgraw, Bernerd Mcgraw, Bethel Mcgraw, Biju Mcgraw, Bogdan Mcgraw, Boston Mcgraw,
Bree Mcgraw, Camaron Mcgraw, Candace Mcgraw, Cara Mcgraw, Cato Mcgraw, Ceferino Mcgraw, Chantry Mcgraw, Chappell Mcgraw, Charels Mcgraw, Ciaran Mcgraw, Codi Mcgraw, Colan Mcgraw, Crayton Mcgraw,
Dakar Mcgraw, Dallis Mcgraw, Danen Mcgraw, Daoud Mcgraw, Davien Mcgraw, Deny Mcgraw, Derius Mcgraw, Derrico Mcgraw, Devere Mcgraw, Dmarcus Mcgraw, Dwright Mcgraw, Dyami Mcgraw, Eileen Mcgraw, Eitan
Mcgraw, Elgie Mcgraw, Elimelech Mcgraw, Ellie Mcgraw, Ellwood Mcgraw, Emmanual Mcgraw, Eryk Mcgraw, Esdras Mcgraw, Estanislao Mcgraw, Euell Mcgraw, Everard Mcgraw, Hagen Mcgraw, Happy Mcgraw, Harald
Mcgraw, Harrington Mcgraw, Hays Mcgraw, Henrik Mcgraw, Hillery Mcgraw, Honorio Mcgraw, Hristos Mcgraw, Ilario Mcgraw, Indalecio Mcgraw, Izaac Mcgraw, Jahmar Mcgraw, Jarome Mcgraw, Jarone Mcgraw,
Javaughn Mcgraw, Jeanne Mcgraw, Jeannie Mcgraw, Jearl Mcgraw, Jenner Mcgraw, Jennie Mcgraw, Jerit Mcgraw, Jet Mcgraw, Jonath Mcgraw, Jumal Mcgraw, Kahil Mcgraw, Kealii Mcgraw, Keeley Mcgraw, Kehinde
Mcgraw, Kene Mcgraw, Kermitt Mcgraw, Keshon Mcgraw, Kore Mcgraw, Lapatrick Mcgraw, Larmar Mcgraw, Laroyce Mcgraw, Larren Mcgraw, Latisha Mcgraw, Latonio Mcgraw, Lewayne Mcgraw, Malachy Mcgraw,
Manford Mcgraw, Marcanthony Mcgraw, Marks Mcgraw, Marley Mcgraw, Marlone Mcgraw, Marquies Mcgraw, Mihir Mcgraw, Minas Mcgraw, Montae Mcgraw, Mortimer Mcgraw, Mykel Mcgraw, Nadia Mcgraw, Nadim Mcgraw,
Naftoli Mcgraw, Nakai Mcgraw, Nashid Mcgraw, Nello Mcgraw, Nichola Mcgraw, Nilo Mcgraw, Nthony Mcgraw, Orenthial Mcgraw, Palani Mcgraw, Pericles Mcgraw, Quentine Mcgraw, Rafiel Mcgraw, Rainey Mcgraw,
Remberto Mcgraw, Renan Mcgraw, Riyad Mcgraw, Robey Mcgraw, Rochell Mcgraw, Romaro Mcgraw, Roosvelt Mcgraw, Satish Mcgraw, Scotte Mcgraw, Sentell Mcgraw, Serapio Mcgraw, Shabaka Mcgraw, Srinivas
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Mcgraw, Xzavier Mcgraw, Yaser Mcgraw, Zahir Mcgraw, Zeus Mcgraw, Abdurrahim Mcgraw, Ahmet Mcgraw, Albion Mcgraw, Andi Mcgraw, Andri Mcgraw, Angello Mcgraw, Angle Mcgraw, Aquilla Mcgraw, Arlee Mcgraw,
Arliss Mcgraw, Asante Mcgraw, Ashwin Mcgraw, Benford Mcgraw, Bernd Mcgraw, Bevin Mcgraw, Blanton Mcgraw, Brandie Mcgraw, Brint Mcgraw, Carle Mcgraw, Carvel Mcgraw, Carvell Mcgraw, Chang Mcgraw,
Charls Mcgraw, Christa Mcgraw, Deidrick Mcgraw, Derrion Mcgraw, Dervin Mcgraw, Devine Mcgraw, Dixie Mcgraw, Dom Mcgraw, Dwone Mcgraw, Edman Mcgraw, Efthimios Mcgraw, Eian Mcgraw, Elazar Mcgraw,
Elridge Mcgraw, Emma Mcgraw, Eoin Mcgraw, Ericka Mcgraw, Everick Mcgraw, Everitt Mcgraw, Franck Mcgraw, Fredi Mcgraw, Ganesh Mcgraw, Gaurav Mcgraw, Geza Mcgraw, Gianpaolo Mcgraw, Grahm Mcgraw, Haden
Mcgraw, Haitham Mcgraw, Hansford Mcgraw, Heberto Mcgraw, Hieu Mcgraw, Jabali Mcgraw, Jabir Mcgraw, Jamilah Mcgraw, Jarriel Mcgraw, Javares Mcgraw, Jawaun Mcgraw, Jessen Mcgraw, Jonanthan Mcgraw,
Joshoa Mcgraw, Joslyn Mcgraw, Jovian Mcgraw, Jozsef Mcgraw, Juanpablo Mcgraw, Justun Mcgraw, Kalief Mcgraw, Kaseen Mcgraw, Kato Mcgraw, Keagan Mcgraw, Keefer Mcgraw, Kishan Mcgraw, Kitwana Mcgraw,
Konata Mcgraw, Lajohn Mcgraw, Lakeisha Mcgraw, Lamichael Mcgraw, Latonia Mcgraw, Lelan Mcgraw, Lendon Mcgraw, Leovardo Mcgraw, Leray Mcgraw, Linh Mcgraw, Lovie Mcgraw, Manasseh Mcgraw, Marcella
Mcgraw, Marguette Mcgraw, Marke Mcgraw, Marlene Mcgraw, Martie Mcgraw, Martrell Mcgraw, Marus Mcgraw, Masaki Mcgraw, Mathhew Mcgraw, Maurisio Mcgraw, Mayur Mcgraw, Meco Mcgraw, Meguel Mcgraw, Meko
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Mcgraw, Nelvin Mcgraw, Nichol Mcgraw, Nicholous Mcgraw, Norm Mcgraw, Olander Mcgraw, Pace Mcgraw, Paschal Mcgraw, Patick Mcgraw, Pearson Mcgraw, Phuc Mcgraw, Ramil Mcgraw, Ravinder Mcgraw, Rayman
Mcgraw, Remond Mcgraw, Rendall Mcgraw, Reshawn Mcgraw, Rhian Mcgraw, Romy Mcgraw, Roozbeh Mcgraw, Rutledge Mcgraw, Ryker Mcgraw, Satoshi Mcgraw, Shafton Mcgraw, Shahab Mcgraw, Shantell Mcgraw,
Shontez Mcgraw, Standley Mcgraw, Taha Mcgraw, Tarren Mcgraw, Tavio Mcgraw, Terah Mcgraw, Texas Mcgraw, Thabiti Mcgraw, Theodric Mcgraw, Timtohy Mcgraw, Torence Mcgraw, Torrick Mcgraw, Towan Mcgraw,
Travor Mcgraw, Tyge Mcgraw, Vandell Mcgraw, Virginio Mcgraw, Wilkie Mcgraw, Windsor Mcgraw, Yarrow Mcgraw, Yuji Mcgraw, Zedric Mcgraw, Ziyad Mcgraw, Abie Mcgraw, Abundio Mcgraw, Adi Mcgraw, Aisha
Mcgraw, Akintunde Mcgraw, Alfio Mcgraw, Anival Mcgraw, Antonnio Mcgraw, Asencion Mcgraw, Ashly Mcgraw, Beaux Mcgraw, Bodhi Mcgraw, Bowie Mcgraw, Brandal Mcgraw, Bristol Mcgraw, Carrick Mcgraw, Chales
Mcgraw, Chantz Mcgraw, Chenier Mcgraw, Chima Mcgraw, Chrystopher Mcgraw, Cian Mcgraw, Clearance Mcgraw, Cohen Mcgraw, Colman Mcgraw, Corneil Mcgraw, Dannis Mcgraw, Dantonio Mcgraw, Danyal Mcgraw,
Darus Mcgraw, Daudi Mcgraw, Dejohn Mcgraw, Demeko Mcgraw, Dewaun Mcgraw, Dondrick Mcgraw, Donley Mcgraw, Donney Mcgraw, Donyelle Mcgraw, Dorjan Mcgraw, Drummond Mcgraw, Dwayn Mcgraw, Dywayne Mcgraw,
Euclides Mcgraw, Farhan Mcgraw, Fritzgerald Mcgraw, Gabreil Mcgraw, Gerrald Mcgraw, Greer Mcgraw, Halim Mcgraw, Haroon Mcgraw, Hatim Mcgraw, Herald Mcgraw, Ismeal Mcgraw, Jamesmichael Mcgraw, Jamir
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Mcgraw, Keifer Mcgraw, Keldric Mcgraw, Kenichi Mcgraw, Kery Mcgraw, Khris Mcgraw, Kiko Mcgraw, Kimothy Mcgraw, Kippy Mcgraw, Kivin Mcgraw, Kojak Mcgraw, Kortez Mcgraw, Lambros Mcgraw, Lealand Mcgraw,
Lillian Mcgraw, Llewelyn Mcgraw, Loron Mcgraw, Maneesh Mcgraw, Mansoor Mcgraw, Mariah Mcgraw, Maricus Mcgraw, Marta Mcgraw, Marte Mcgraw, Martese Mcgraw, Maston Mcgraw, Maylon Mcgraw, Mcclain Mcgraw,
Mikko Mcgraw, Montrez Mcgraw, Najee Mcgraw, Najeeb Mcgraw, Nakoma Mcgraw, Nakuma Mcgraw, Oliverio Mcgraw, Orlondo Mcgraw, Osceola Mcgraw, Osric Mcgraw, Othoniel Mcgraw, Ovid Mcgraw, Peterjohn Mcgraw,
Rad Mcgraw, Rogue Mcgraw, Rolly Mcgraw, Roshad Mcgraw, Sahwn Mcgraw, Santosh Mcgraw, Shakeem Mcgraw, Shanen Mcgraw, Shey Mcgraw, Sina Mcgraw, Sonja Mcgraw, Stone Mcgraw, Stphen Mcgraw, Sue Mcgraw,
Sylvanus Mcgraw, Tameka Mcgraw, Tauris Mcgraw, Tawayne Mcgraw, Tennille Mcgraw, Terrice Mcgraw, Thalamus Mcgraw, Tore Mcgraw, Tranell Mcgraw, Travone Mcgraw, Trevan Mcgraw, Tristram Mcgraw, Tywaun
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Mcgraw, Audre Mcgraw, Bertrum Mcgraw, Bond Mcgraw, Canon Mcgraw, Careem Mcgraw, Carly Mcgraw, Charmaine Mcgraw, Corian Mcgraw, Coye Mcgraw, Daisuke Mcgraw, Damacio Mcgraw, Damarius Mcgraw, Damiane
Mcgraw, Darrik Mcgraw, Dartanian Mcgraw, Dary Mcgraw, Davan Mcgraw, Deana Mcgraw, Deaundra Mcgraw, Delos Mcgraw, Develle Mcgraw, Dontray Mcgraw, Dorsett Mcgraw, Dyrell Mcgraw, Gered Mcgraw, Gilad
Mcgraw, Hadrian Mcgraw, Haji Mcgraw, Harl Mcgraw, Hideki Mcgraw, Husani Mcgraw, Jacson Mcgraw, Jaimes Mcgraw, Jama Mcgraw, Janathan Mcgraw, Jaran Mcgraw, Jauron Mcgraw, Jeffie Mcgraw, Jejuan Mcgraw,
Jerick Mcgraw, Jermery Mcgraw, Jerramie Mcgraw, Kalei Mcgraw, Karam Mcgraw, Keevan Mcgraw, Kei Mcgraw, Kel Mcgraw, Keo Mcgraw, Khaleel Mcgraw, Klye Mcgraw, Korrie Mcgraw, Kunal Mcgraw, Kyrone Mcgraw,
Ladaryl Mcgraw, Laman Mcgraw, Larome Mcgraw, Latwan Mcgraw, Lavaris Mcgraw, Lavoy Mcgraw, Lenoard Mcgraw, Levarr Mcgraw, Matilde Mcgraw, Mazin Mcgraw, Nakie Mcgraw, Nikolos Mcgraw, Oak Mcgraw, Orrie
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Torance Mcgraw, Travelle Mcgraw, Tulio Mcgraw, Tyrance Mcgraw, Ustin Mcgraw, Verle Mcgraw, Vicky Mcgraw, Wlliam Mcgraw, Xerxes Mcgraw, Yero Mcgraw, Adetokunbo Mcgraw, Adika Mcgraw, Adriane Mcgraw,
Afshin Mcgraw, Alto Mcgraw, Alvon Mcgraw, Anant Mcgraw, Anatole Mcgraw, Aston Mcgraw, Aul Mcgraw, Aza Mcgraw, Azriel Mcgraw, Bay Mcgraw, Bee Mcgraw, Benoit Mcgraw, Bradden Mcgraw, Brayton Mcgraw,
Buren Mcgraw, Caton Mcgraw, Cesareo Mcgraw, Chandar Mcgraw, Charistopher Mcgraw, Chike Mcgraw, Chong Mcgraw, Cline Mcgraw, Dabid Mcgraw, Damascus Mcgraw, Delron Mcgraw, Delvis Mcgraw, Demeterius
Mcgraw, Demetries Mcgraw, Demorio Mcgraw, Demtrius Mcgraw, Derrian Mcgraw, Dilanjan Mcgraw, Donti Mcgraw, Doulgas Mcgraw, Doyce Mcgraw, Eber Mcgraw, Elena Mcgraw, Ellington Mcgraw, Eros Mcgraw,
Esiquio Mcgraw, Esley Mcgraw, Ewan Mcgraw, Firas Mcgraw, Freeland Mcgraw, Frenchie Mcgraw, Garlan Mcgraw, Gaylin Mcgraw, Hamin Mcgraw, Hoyle Mcgraw, Hubbard Mcgraw, Ikechukwu Mcgraw, Issam Mcgraw,
Jameil Mcgraw, Janard Mcgraw, Jann Mcgraw, Jarmain Mcgraw, Jayjay Mcgraw, Jemell Mcgraw, Jerem Mcgraw, Jeremian Mcgraw, Jerritt Mcgraw, Juda Mcgraw, Kahn Mcgraw, Kartik Mcgraw, Katrell Mcgraw, Kavon
Mcgraw, Kelsie Mcgraw, Kelsy Mcgraw, Kenna Mcgraw, Kenner Mcgraw, Kentaro Mcgraw, Kenyotta Mcgraw, Keonte Mcgraw, Kindell Mcgraw, Kodie Mcgraw, Kurtus Mcgraw, Lacory Mcgraw, Latanya Mcgraw, Lathaniel
Mcgraw, Lauriano Mcgraw, Lavares Mcgraw, Lemarr Mcgraw, Lomont Mcgraw, Lyndel Mcgraw, Mandeep Mcgraw, Mansur Mcgraw, Marrion Mcgraw, Mathan Mcgraw, Maurie Mcgraw, Melburn Mcgraw, Mia Mcgraw, Micheil
Mcgraw, Milledge Mcgraw, Mourice Mcgraw, Mousa Mcgraw, Muneer Mcgraw, Naphtali Mcgraw, Navarro Mcgraw, Nehemias Mcgraw, Nickalous Mcgraw, Nickolous Mcgraw, Ola Mcgraw, Oshua Mcgraw, Osmond Mcgraw,
Quinta Mcgraw, Quintel Mcgraw, Rahshawn Mcgraw, Raney Mcgraw, Raye Mcgraw, Rayne Mcgraw, Rockne Mcgraw, Romar Mcgraw, Sebastain Mcgraw, Shang Mcgraw, Sherrard Mcgraw, Smauel Mcgraw, Stamatios Mcgraw,
Talal Mcgraw, Taurin Mcgraw, Terdell Mcgraw, Thong Mcgraw, Tishawn Mcgraw, Tizoc Mcgraw, Trad Mcgraw, Trafton Mcgraw, Trenity Mcgraw, Tyray Mcgraw, Ulices Mcgraw, Vareck Mcgraw, Videll Mcgraw,
Voltaire Mcgraw, Wilmot Mcgraw, Zacary Mcgraw, Aaran Mcgraw, Abdalla Mcgraw, Abdias Mcgraw, Abiodun Mcgraw, Adlai Mcgraw, Ahamad Mcgraw, Alexandar Mcgraw, Alie Mcgraw, Alissa Mcgraw, Alterick Mcgraw,
Amitabh Mcgraw, Andros Mcgraw, Angelia Mcgraw, Antwonne Mcgraw, Arnez Mcgraw, Arrion Mcgraw, Asaf Mcgraw, Auden Mcgraw, Bacilio Mcgraw, Brynn Mcgraw, Byan Mcgraw, Can Mcgraw, Carlen Mcgraw, Carliss
Mcgraw, Carlose Mcgraw, Cazzie Mcgraw, Chadney Mcgraw, Chapman Mcgraw, Chedrick Mcgraw, Cherokee Mcgraw, Christophen Mcgraw, Clearnce Mcgraw, Cleto Mcgraw, Cliford Mcgraw, Colburn Mcgraw, Cordy
Mcgraw, Craigory Mcgraw, Dalan Mcgraw, Dam Mcgraw, Damonn Mcgraw, Dandrea Mcgraw, Danford Mcgraw, Danieal Mcgraw, Dano Mcgraw, Daril Mcgraw, Darrence Mcgraw, Dartanyan Mcgraw, Dawone Mcgraw, Deatrick
Mcgraw, Delayne Mcgraw, Delfred Mcgraw, Delshon Mcgraw, Demark Mcgraw, Demea Mcgraw, Demetra Mcgraw, Demetress Mcgraw, Demeturis Mcgraw, Dene Mcgraw, Derran Mcgraw, Dewann Mcgraw, Dezi Mcgraw,
Dondrell Mcgraw, Donsha Mcgraw, Dontai Mcgraw, Durelle Mcgraw, Ebay Mcgraw, Edin Mcgraw, Efran Mcgraw, Emelio Mcgraw, Erasto Mcgraw, Erby Mcgraw, Eriks Mcgraw, Erine Mcgraw, Ester Mcgraw, Everet
Mcgraw, Excell Mcgraw, Falando Mcgraw, Gaius Mcgraw, Garald Mcgraw, Garlon Mcgraw, Germane Mcgraw, Gernard Mcgraw, Glenden Mcgraw, Graciela Mcgraw, Gram Mcgraw, Grayling Mcgraw, Gunter Mcgraw, Gustaf
Mcgraw, Gwyn Mcgraw, Hamp Mcgraw, Hanley Mcgraw, Harrold Mcgraw, Harvest Mcgraw, Hermenegildo Mcgraw, Hooman Mcgraw, Husam Mcgraw, Idrees Mcgraw, Jaaron Mcgraw, Jaben Mcgraw, Jacent Mcgraw, Jacyn
Mcgraw, Jadrian Mcgraw, Jaisen Mcgraw, Jamard Mcgraw, Jamare Mcgraw, Jarl Mcgraw, Jarvin Mcgraw, Jary Mcgraw, Jashon Mcgraw, Jaso Mcgraw, Jaton Mcgraw, Jawann Mcgraw, Jebidiah Mcgraw, Jedadia Mcgraw,
Jenard Mcgraw, Jeramaine Mcgraw, Jerrimy Mcgraw, Jerrin Mcgraw, Jibreel Mcgraw, Jigar Mcgraw, Joah Mcgraw, Jolene Mcgraw, Jonell Mcgraw, Jordy Mcgraw, Josuah Mcgraw, Jwan Mcgraw, Kantrell Mcgraw,
Keithrick Mcgraw, Kennet Mcgraw, Keshaun Mcgraw, Kief Mcgraw, Kisha Mcgraw, Kiwan Mcgraw, Kizzy Mcgraw, Koury Mcgraw, Kylon Mcgraw, Lakisha Mcgraw, Lamarco Mcgraw, Lavonte Mcgraw, Lemanuel Mcgraw,
Lennell Mcgraw, Letcher Mcgraw, Lethaniel Mcgraw, Levone Mcgraw, Lilton Mcgraw, Lindel Mcgraw, Linzy Mcgraw, Lora Mcgraw, Lorie Mcgraw, Lovelle Mcgraw, Lowry Mcgraw, Lucia Mcgraw, Luisa Mcgraw,
Macheal Mcgraw, Magdiel Mcgraw, Mahmood Mcgraw, Manrique Mcgraw, Manus Mcgraw, Maricela Mcgraw, Marjoe Mcgraw, Marquett Mcgraw, Mars Mcgraw, Mathews Mcgraw, Mattthew Mcgraw, Mehmet Mcgraw, Mehran
Mcgraw, Menashe Mcgraw, Merril Mcgraw, Moe Mcgraw, Morgen Mcgraw, Natalio Mcgraw, Nero Mcgraw, Nickolos Mcgraw, Nidal Mcgraw, Nishant Mcgraw, Nixon Mcgraw, Oreste Mcgraw, Osie Mcgraw, Parks Mcgraw,
Pax Mcgraw, Phillipp Mcgraw, Qaadir Mcgraw, Quantas Mcgraw, Rameen Mcgraw, Ranell Mcgraw, Rani Mcgraw, Rasheim Mcgraw, Rawlin Mcgraw, Rena Mcgraw, Rett Mcgraw, Riko Mcgraw, Rimas Mcgraw, Roark
Mcgraw, Roma Mcgraw, Romale Mcgraw, Rosemary Mcgraw, Roshell Mcgraw, Sadiki Mcgraw, Saquan Mcgraw, Sargon Mcgraw, Shantez Mcgraw, Shawntel Mcgraw, Shelia Mcgraw, Sheryl Mcgraw, Sholem Mcgraw, Sly
Mcgraw, Sohail Mcgraw, Spero Mcgraw, Stepehn Mcgraw, Tabor Mcgraw, Taki Mcgraw, Talley Mcgraw, Tamboura Mcgraw, Tarrus Mcgraw, Tau Mcgraw, Taz Mcgraw, Tejuan Mcgraw, Terik Mcgraw, Terone Mcgraw,
Terreance Mcgraw, Terriel Mcgraw, Terriss Mcgraw, Theodus Mcgraw, Tobian Mcgraw, Tolan Mcgraw, Tollie Mcgraw, Townsend Mcgraw, Tramain Mcgraw, Trennis Mcgraw, Tshaka Mcgraw, Tynell Mcgraw, Tyone
Mcgraw, Tywain Mcgraw, Uganda Mcgraw, Unique Mcgraw, Vahe Mcgraw, Vashone Mcgraw, Verna Mcgraw, Welsey Mcgraw, Woodson Mcgraw, Wydell Mcgraw, Youssef Mcgraw, Zebulin Mcgraw, Zeyad Mcgraw, Aaryn
Mcgraw, Ademola Mcgraw, Adom Mcgraw, Alandus Mcgraw, Alcide Mcgraw, Aldridge Mcgraw, Alfonsa Mcgraw, Alfonzia Mcgraw, Allyson Mcgraw, Alter Mcgraw, Amari Mcgraw, Amr Mcgraw, Ancel Mcgraw, Andie
Mcgraw, Andrus Mcgraw, Antiwan Mcgraw, Antoinette Mcgraw, Antonis Mcgraw, Antwyne Mcgraw, Anupam Mcgraw, Aquila Mcgraw, Arville Mcgraw, Asael Mcgraw, Assad Mcgraw, Aurohom Mcgraw, Autrey Mcgraw, Ayal
Mcgraw, Azikiwe Mcgraw, Bashar Mcgraw, Basim Mcgraw, Beauregard Mcgraw, Belal Mcgraw, Benyamin Mcgraw, Bertis Mcgraw, Bertran Mcgraw, Boban Mcgraw, Boby Mcgraw, Bracy Mcgraw, Branndon Mcgraw, Burtis
Mcgraw, Cameo Mcgraw, Carley Mcgraw, Carnelius Mcgraw, Caroll Mcgraw, Carolos Mcgraw, Carvis Mcgraw, Cashawn Mcgraw, Cean Mcgraw, Charod Mcgraw, Chevelle Mcgraw, Chrisotpher Mcgraw, Christerfer
Mcgraw, Christi Mcgraw, Clois Mcgraw, Cloyce Mcgraw, Constancio Mcgraw, Corkey Mcgraw, Corvin Mcgraw, Corye Mcgraw, Costello Mcgraw, Coury Mcgraw, Crecencio Mcgraw, Cuahutemoc Mcgraw, Curby Mcgraw,
Curly Mcgraw, Dade Mcgraw, Dag Mcgraw, Daimeon Mcgraw, Dakin Mcgraw, Damiean Mcgraw, Dandy Mcgraw, Danian Mcgraw, Davinder Mcgraw, Daylin Mcgraw, Deljuan Mcgraw, Delone Mcgraw, Delta Mcgraw, Demaris
Mcgraw, Demel Mcgraw, Demerick Mcgraw, Demetrous Mcgraw, Demiko Mcgraw, Demorrio Mcgraw, Deonne Mcgraw, Derian Mcgraw, Detron Mcgraw, Deva Mcgraw, Dillion Mcgraw, Divine Mcgraw, Dniel Mcgraw, Doanld
Mcgraw, Donaldo Mcgraw, Donshay Mcgraw, Dontee Mcgraw, Doyne Mcgraw, Dubois Mcgraw, Dulani Mcgraw, Duong Mcgraw, Dupre Mcgraw, Durante Mcgraw, Edith Mcgraw, Edris Mcgraw, Edwyn Mcgraw, Efrin Mcgraw,
Ehrin Mcgraw, Ehsan Mcgraw, Eldwin Mcgraw, Elic Mcgraw, Elis Mcgraw, Elven Mcgraw, Eren Mcgraw, Errik Mcgraw, Erubey Mcgraw, Evertt Mcgraw, Exavier Mcgraw, Fahad Mcgraw, Fares Mcgraw, Fatmir Mcgraw,
Ferando Mcgraw, Fonta Mcgraw, Fotis Mcgraw, Gaither Mcgraw, Gar Mcgraw, Garvey Mcgraw, Gilmer Mcgraw, Godofredo Mcgraw, Greggery Mcgraw, Gregoire Mcgraw, Gregorey Mcgraw, Hagan Mcgraw, Haim Mcgraw,
Hanz Mcgraw, Hartford Mcgraw, Hassen Mcgraw, Haynes Mcgraw, Herrick Mcgraw, Herry Mcgraw, Hoa Mcgraw, Hud Mcgraw, Hurbert Mcgraw, Ilir Mcgraw, Ilyas Mcgraw, Ivica Mcgraw, Jabarr Mcgraw, Jamara
Mcgraw, Jamarco Mcgraw, Jana Mcgraw, Janis Mcgraw, Jarry Mcgraw, Javad Mcgraw, Javas Mcgraw, Javed Mcgraw, Jayden Mcgraw, Jebb Mcgraw, Jeptha Mcgraw, Jerimia Mcgraw, Joffrey Mcgraw, Johanthan Mcgraw,
Johari Mcgraw, Johnanthony Mcgraw, Johnnathan Mcgraw, Johnta Mcgraw, Jolly Mcgraw, Jonahtan Mcgraw, Jong Mcgraw, Jonh Mcgraw, Josep Mcgraw, Joss Mcgraw, Juergen Mcgraw, Juliocesar Mcgraw, Junichi
Mcgraw, Jusitn Mcgraw, Kalid Mcgraw, Kamali Mcgraw, Kannon Mcgraw, Karlon Mcgraw, Kashawn Mcgraw, Keilan Mcgraw, Kelcy Mcgraw, Kellis Mcgraw, Kelon Mcgraw, Kendrich Mcgraw, Kentrel Mcgraw, Kerwyn
Mcgraw, Kesley Mcgraw, Kimon Mcgraw, Kindu Mcgraw, Kinneth Mcgraw, Kondwani Mcgraw, Krystal Mcgraw, Kyran Mcgraw, Lacarlos Mcgraw, Laney Mcgraw, Lelon Mcgraw, Lem Mcgraw, Lena Mcgraw, Leno Mcgraw,
Leshan Mcgraw, Lexie Mcgraw, Lisle Mcgraw, Lonne Mcgraw, Lorence Mcgraw, Lowery Mcgraw, Lundy Mcgraw, Lynford Mcgraw, Maleek Mcgraw, Maliek Mcgraw, Mano Mcgraw, Marcellis Mcgraw, Marcellius Mcgraw,
Marci Mcgraw, Maris Mcgraw, Marl Mcgraw, Marston Mcgraw, Matthe Mcgraw, Matther Mcgraw, Maximiano Mcgraw, Mecca Mcgraw, Melesio Mcgraw, Mic Mcgraw, Mican Mcgraw, Mickal Mcgraw, Mitchum Mcgraw, Mohit
Mcgraw, Myrl Mcgraw, Nael Mcgraw, Najja Mcgraw, Nedal Mcgraw, Nektarios Mcgraw, Nemiah Mcgraw, Niclas Mcgraw, Nilay Mcgraw, Nina Mcgraw, Noberto Mcgraw, Noell Mcgraw, Obrian Mcgraw, Orbie Mcgraw,
Ormond Mcgraw, Orvin Mcgraw, Osborn Mcgraw, Pasco Mcgraw, Pele Mcgraw, Penn Mcgraw, Perryn Mcgraw, Philipe Mcgraw, Prakash Mcgraw, Quarterrio Mcgraw, Quiency Mcgraw, Rahaman Mcgraw, Rahshan Mcgraw,
Rahson Mcgraw, Raimond Mcgraw, Ramsay Mcgraw, Ranard Mcgraw, Ranulfo Mcgraw, Rasha Mcgraw, Rashine Mcgraw, Rason Mcgraw, Ravis Mcgraw, Raylan Mcgraw, Rayme Mcgraw, Raymel Mcgraw, Redrick Mcgraw, Regi
Mcgraw, Renier Mcgraw, Rhea Mcgraw, Riad Mcgraw, Richerd Mcgraw, Rishard Mcgraw, Rockland Mcgraw, Rolfe Mcgraw, Roudy Mcgraw, Saed Mcgraw, Salaam Mcgraw, Samie Mcgraw, Savino Mcgraw, Selim Mcgraw,
Senneca Mcgraw, Seung Mcgraw, Shadric Mcgraw, Shaman Mcgraw, Sharn Mcgraw, Shawen Mcgraw, Shiron Mcgraw, Shohn Mcgraw, Shunn Mcgraw, Sims Mcgraw, Skeet Mcgraw, Soctt Mcgraw, Sohn Mcgraw, Standly
Mcgraw, Starbuck Mcgraw, Stratton Mcgraw, Sulton Mcgraw, Sunday Mcgraw, Sylas Mcgraw, Taff Mcgraw, Tallis Mcgraw, Tamal Mcgraw, Tarrant Mcgraw, Taurice Mcgraw, Teko Mcgraw, Telvis Mcgraw, Terek
Mcgraw, Terrius Mcgraw, Thermon Mcgraw, Thorsten Mcgraw, Timothee Mcgraw, Tirus Mcgraw, Tomi Mcgraw, Torren Mcgraw, Travolta Mcgraw, Travoris Mcgraw, Tyan Mcgraw, Vasco Mcgraw, Vashion Mcgraw, Verron
Mcgraw, Vinicio Mcgraw, Vyron Mcgraw, Wilkin Mcgraw, Winson Mcgraw, Worthy Mcgraw, Wynne Mcgraw, Yance Mcgraw, Yates Mcgraw, Yecheskel Mcgraw, Yonah Mcgraw, Yuseff Mcgraw, Abby Mcgraw, Adaryl Mcgraw,
Addis Mcgraw, Afrim Mcgraw, Ahmand Mcgraw, Ahmon Mcgraw, Alejos Mcgraw, Alphonsa Mcgraw, Alphonsus Mcgraw, Altie Mcgraw, Anastasio Mcgraw, Andris Mcgraw, Ankit Mcgraw, Antario Mcgraw, Antero Mcgraw,
Arafat Mcgraw, Armard Mcgraw, Armel Mcgraw, Arrin Mcgraw, Artavious Mcgraw, Ashlee Mcgraw, Atari Mcgraw, Aundrae Mcgraw, Avan Mcgraw, Avion Mcgraw, Azad Mcgraw, Azariah Mcgraw, Babe Mcgraw, Banks
Mcgraw, Bentura Mcgraw, Bily Mcgraw, Blakeley Mcgraw, Bonner Mcgraw, Bradshaw Mcgraw, Braeden Mcgraw, Brainard Mcgraw, Brianne Mcgraw, Brigg Mcgraw, Broch Mcgraw, Bryian Mcgraw, Buell Mcgraw,
Burchell Mcgraw, Butler Mcgraw, Callen Mcgraw, Cardale Mcgraw, Carlester Mcgraw, Carwin Mcgraw, Cassell Mcgraw, Casy Mcgraw, Chancelor Mcgraw, Chapin Mcgraw, Chee Mcgraw, Chey Mcgraw, Chianti Mcgraw,
Clarenc Mcgraw, Clayburn Mcgraw, Cleofas Mcgraw, Clevester Mcgraw, Clydell Mcgraw, Constance Mcgraw, Coran Mcgraw, Correll Mcgraw, Cotton Mcgraw, Currie Mcgraw, Dae Mcgraw, Dai Mcgraw, Daisy Mcgraw,
Dametrius Mcgraw, Damico Mcgraw, Danal Mcgraw, Danien Mcgraw, Danis Mcgraw, Dar Mcgraw, Darivs Mcgraw, Darmon Mcgraw, Deejay Mcgraw, Delia Mcgraw, Demarquis Mcgraw, Demetrik Mcgraw, Demonta Mcgraw,
Denys Mcgraw, Derec Mcgraw, Derl Mcgraw, Deshay Mcgraw, Destiny Mcgraw, Detroy Mcgraw, Devar Mcgraw, Dewyane Mcgraw, Dierre Mcgraw, Dillan Mcgraw, Din Mcgraw, Dionysios Mcgraw, Dishawn Mcgraw, Doak
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Mcgraw, Effrem Mcgraw, Eliel Mcgraw, Elizah Mcgraw, Elray Mcgraw, Elsa Mcgraw, Emerick Mcgraw, Epigmenio Mcgraw, Epimenio Mcgraw, Erasmus Mcgraw, Erez Mcgraw, Erman Mcgraw, Esgar Mcgraw, Everrett
Mcgraw, Fateen Mcgraw, Felimon Mcgraw, Felis Mcgraw, Ferguson Mcgraw, Firman Mcgraw, Fitz Mcgraw, France Mcgraw, Freddick Mcgraw, Frederi Mcgraw, Gadiel Mcgraw, Galdino Mcgraw, Gant Mcgraw, Gared
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Mcgraw, Griselda Mcgraw, Gussie Mcgraw, Hall Mcgraw, Hammad Mcgraw, Hamza Mcgraw, Handy Mcgraw, Hardie Mcgraw, Hassel Mcgraw, Heiko Mcgraw, Henley Mcgraw, Hersh Mcgraw, Hewitt Mcgraw, Hiran Mcgraw,
Hoke Mcgraw, Holmes Mcgraw, Hunt Mcgraw, Ikenna Mcgraw, Imre Mcgraw, Irma Mcgraw, Isaul Mcgraw, Ishmail Mcgraw, Ishmel Mcgraw, Iyad Mcgraw, Jaamal Mcgraw, Jabaar Mcgraw, Jacquin Mcgraw, Jameison
Mcgraw, Jasom Mcgraw, Jefery Mcgraw, Jeffy Mcgraw, Jermanine Mcgraw, Jerol Mcgraw, Jesses Mcgraw, Jibril Mcgraw, Jimy Mcgraw, Joffre Mcgraw, Johm Mcgraw, Johne Mcgraw, Johnothan Mcgraw, Josias
Mcgraw, Joslin Mcgraw, Jovanny Mcgraw, Juel Mcgraw, Kaare Mcgraw, Kalib Mcgraw, Kalim Mcgraw, Kalum Mcgraw, Kedron Mcgraw, Kerim Mcgraw, Kerman Mcgraw, Keyan Mcgraw, Khai Mcgraw, Khareem Mcgraw,
Kierre Mcgraw, Kimmie Mcgraw, Kipton Mcgraw, Kord Mcgraw, Lajaune Mcgraw, Lamor Mcgraw, Lancy Mcgraw, Lando Mcgraw, Landrum Mcgraw, Lani Mcgraw, Larmont Mcgraw, Laurens Mcgraw, Lawrenc Mcgraw, Lealon
Mcgraw, Leeman Mcgraw, Legrande Mcgraw, Leone Mcgraw, Lewie Mcgraw, Loretta Mcgraw, Lorrin Mcgraw, Loukas Mcgraw, Luisito Mcgraw, Lutalo Mcgraw, Machel Mcgraw, Macklin Mcgraw, Mahir Mcgraw, Manpreet
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Montique Mcgraw, Morice Mcgraw, Nataniel Mcgraw, Neiman Mcgraw, Noa Mcgraw, Obdulio Mcgraw, Odilon Mcgraw, Oma Mcgraw, Onathan Mcgraw, Orlandis Mcgraw, Osamah Mcgraw, Pastor Mcgraw, Patterson Mcgraw,
Pattrick Mcgraw, Pavlos Mcgraw, Phoenix Mcgraw, Pierson Mcgraw, Puneet Mcgraw, Quartez Mcgraw, Quetin Mcgraw, Rahsheen Mcgraw, Randee Mcgraw, Ras Mcgraw, Raymie Mcgraw, Raymound Mcgraw, Raysean
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Sedgwick Mcgraw, Shaddrick Mcgraw, Shaffer Mcgraw, Shamal Mcgraw, Shamone Mcgraw, Sharman Mcgraw, Sharrief Mcgraw, Shawnell Mcgraw, Shawntae Mcgraw, Shimshon Mcgraw, Shontay Mcgraw, Shuaib Mcgraw,
Steveland Mcgraw, Stonewall Mcgraw, Sujal Mcgraw, Tage Mcgraw, Tarin Mcgraw, Tarrick Mcgraw, Tarryl Mcgraw, Tauras Mcgraw, Tayvon Mcgraw, Terrol Mcgraw, Thearon Mcgraw, Thinh Mcgraw, Tirell Mcgraw,
Tobius Mcgraw, Tobyn Mcgraw, Toderick Mcgraw, Torben Mcgraw, Torran Mcgraw, Torray Mcgraw, Traven Mcgraw, Trek Mcgraw, Trina Mcgraw, Twayne Mcgraw, Tyee Mcgraw, Tyrick Mcgraw, Tyvon Mcgraw, Ubong
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Choice Mcgraw, Cosmos Mcgraw, Cymande Mcgraw, Cyprian Mcgraw, Dang Mcgraw, Derrus Mcgraw, Dorwin Mcgraw, Everest Mcgraw, Farzad Mcgraw, Gralin Mcgraw, Idi Mcgraw, Jakari Mcgraw, Jezreel Mcgraw,
Jorell Mcgraw, Kong Mcgraw, Liza Mcgraw, Mendy Mcgraw, Onix Mcgraw, Rajaee Mcgraw, Random Mcgraw, Rashean Mcgraw, Revis Mcgraw, Saman Mcgraw, Santini Mcgraw, Shahin Mcgraw, Talton Mcgraw, Teco
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Jermont Mcgraw, Jerud Mcgraw, Johncharles Mcgraw, Jolan Mcgraw, Kairaba Mcgraw, Kawon Mcgraw, Kemon Mcgraw, Kennen Mcgraw, Kenyun Mcgraw, Khevin Mcgraw, Kiowa Mcgraw, Kwabene Mcgraw, Ladrick Mcgraw,
Lamarc Mcgraw, Laramy Mcgraw, Larance Mcgraw, Lebrone Mcgraw, Loc Mcgraw, Lynda Mcgraw, Marcius Mcgraw, Mariel Mcgraw, Marisela Mcgraw, Meshach Mcgraw, Milon Mcgraw, Monterio Mcgraw, Najib Mcgraw,
Natha Mcgraw, Nery Mcgraw, Niklaus Mcgraw, Pinchus Mcgraw, Quaid Mcgraw, Rael Mcgraw, Rain Mcgraw, Ralpheal Mcgraw, Ramiah Mcgraw, Redmond Mcgraw, Regniald Mcgraw, Rennard Mcgraw, Rhet Mcgraw, Ritesh
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Terren Mcgraw, Traun Mcgraw, Trong Mcgraw, Trygve Mcgraw, Unborn Mcgraw, Vaden Mcgraw, Valery Mcgraw, Vuong Mcgraw, Waseem Mcgraw, Zebulen Mcgraw, Abed Mcgraw, Abhay Mcgraw, Adebayo Mcgraw, Ahmid
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Bianca Mcgraw, Bonny Mcgraw, Boy Mcgraw, Brach Mcgraw, Brishen Mcgraw, Britney Mcgraw, Burlin Mcgraw, Buzz Mcgraw, Carolina Mcgraw, Casimer Mcgraw, Cedeno Mcgraw, Cemal Mcgraw, Chadly Mcgraw, Chen
Mcgraw, Chirs Mcgraw, Christion Mcgraw, Christohpher Mcgraw, Christpoher Mcgraw, Clate Mcgraw, Cleaven Mcgraw, Clif Mcgraw, Coburn Mcgraw, Conroy Mcgraw, Corbitt Mcgraw, Crispus Mcgraw, Daiman
Mcgraw, Daimion Mcgraw, Damarco Mcgraw, Dammian Mcgraw, Darr Mcgraw, Darreyl Mcgraw, Davar Mcgraw, Dawain Mcgraw, Dayron Mcgraw, Dectrick Mcgraw, Dederick Mcgraw, Demarkis Mcgraw, Denman Mcgraw,
Derril Mcgraw, Devanand Mcgraw, Dharma Mcgraw, Dionysus Mcgraw, Domanick Mcgraw, Dorey Mcgraw, Duel Mcgraw, Ebenezer Mcgraw, Edon Mcgraw, Effie Mcgraw, Ehron Mcgraw, Eliceo Mcgraw, Elija Mcgraw,
Elijha Mcgraw, Elizandro Mcgraw, Enriquez Mcgraw, Eryn Mcgraw, Essie Mcgraw, Etoyi Mcgraw, Fabrice Mcgraw, Farouk Mcgraw, Garick Mcgraw, Gean Mcgraw, Geovani Mcgraw, Germar Mcgraw, Gillian Mcgraw,
Gilmore Mcgraw, Giuseppi Mcgraw, Governor Mcgraw, Graden Mcgraw, Gresham Mcgraw, Haashim Mcgraw, Hagop Mcgraw, Hamzah Mcgraw, Harlie Mcgraw, Hatem Mcgraw, Hien Mcgraw, Hose Mcgraw, Ibe Mcgraw, Ingo
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Mcgraw, Jemiah Mcgraw, Jerek Mcgraw, Jeren Mcgraw, Jeriah Mcgraw, Jerode Mcgraw, Jimmel Mcgraw, Jitu Mcgraw, Jmichael Mcgraw, Johnas Mcgraw, Johnwilliam Mcgraw, Joran Mcgraw, Jori Mcgraw, Josehua
Mcgraw, Joshlin Mcgraw, Joushua Mcgraw, Juane Mcgraw, Juba Mcgraw, Juvon Mcgraw, Kamaal Mcgraw, Kasib Mcgraw, Kate Mcgraw, Kayin Mcgraw, Keffer Mcgraw, Kentay Mcgraw, Kentrail Mcgraw, Kenyell Mcgraw,
Kerr Mcgraw, Khalik Mcgraw, Khurram Mcgraw, Kipchoge Mcgraw, Koa Mcgraw, Kowan Mcgraw, Kreig Mcgraw, Kristien Mcgraw, Krystopher Mcgraw, Kurry Mcgraw, Kylee Mcgraw, Lamaine Mcgraw, Lamell Mcgraw,
Lamour Mcgraw, Lanoris Mcgraw, Laquann Mcgraw, Laquinn Mcgraw, Lavone Mcgraw, Leanord Mcgraw, Leeandrew Mcgraw, Lenzy Mcgraw, Lequient Mcgraw, Leshun Mcgraw, Lopaka Mcgraw, Lopez Mcgraw, Lyall
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Mcgraw, Quante Mcgraw, Quindell Mcgraw, Raeshawn Mcgraw, Rahshon Mcgraw, Ran Mcgraw, Real Mcgraw, Rhodes Mcgraw, Rhodney Mcgraw, Rodeny Mcgraw, Rondi Mcgraw, Rozelle Mcgraw, Rutilio Mcgraw, Saif
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Tag Mcgraw, Tamon Mcgraw, Tanisha Mcgraw, Tearle Mcgraw, Tellys Mcgraw, Terriance Mcgraw, Timber Mcgraw, Timthoy Mcgraw, Tomiko Mcgraw, Torr Mcgraw, Torrez Mcgraw, Toshiro Mcgraw, Trammell Mcgraw,
Trance Mcgraw, Tregg Mcgraw, Treston Mcgraw, Tristian Mcgraw, Tymaine Mcgraw, Tyne Mcgraw, Tyrane Mcgraw, Vada Mcgraw, Valerio Mcgraw, Varion Mcgraw, Vasean Mcgraw, Vaugh Mcgraw, Viviana Mcgraw,
Welford Mcgraw, Windel Mcgraw, Yaacov Mcgraw, Yaniv Mcgraw, Yusuke Mcgraw, Zebulah Mcgraw, Zuberi Mcgraw, Aarin Mcgraw, Abbott Mcgraw, Abdule Mcgraw, Abdulrahman Mcgraw, Abrahim Mcgraw, Abrian
Mcgraw, Adham Mcgraw, Aditya Mcgraw, Adre Mcgraw, Akito Mcgraw, Ala Mcgraw, Alberico Mcgraw, Alexsandro Mcgraw, Aljandro Mcgraw, Allin Mcgraw, Alyssa Mcgraw, Amond Mcgraw, Angelique Mcgraw, Ankoma
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Barnie Mcgraw, Barrick Mcgraw, Baruti Mcgraw, Bascom Mcgraw, Basheer Mcgraw, Bear Mcgraw, Beatriz Mcgraw, Bengt Mcgraw, Bennette Mcgraw, Berl Mcgraw, Berley Mcgraw, Berwin Mcgraw, Biren Mcgraw, Blase
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Cali Mcgraw, Carlan Mcgraw, Carmello Mcgraw, Carmichael Mcgraw, Carney Mcgraw, Carr Mcgraw, Carrell Mcgraw, Carzell Mcgraw, Castro Mcgraw, Caswell Mcgraw, Catrell Mcgraw, Cedrice Mcgraw, Celeste
Mcgraw, Celina Mcgraw, Cesear Mcgraw, Chade Mcgraw, Chadwell Mcgraw, Chalres Mcgraw, Chamar Mcgraw, Chanan Mcgraw, Chanda Mcgraw, Chanin Mcgraw, Charo Mcgraw, Chayim Mcgraw, Chelton Mcgraw, Chevez
Mcgraw, Chih Mcgraw, Chin Mcgraw, Chino Mcgraw, Christaphor Mcgraw, Ciprian Mcgraw, Conny Mcgraw, Corbit Mcgraw, Corinthians Mcgraw, Cosby Mcgraw, Curlee Mcgraw, Cyrano Mcgraw, Daimian Mcgraw, Daks
Mcgraw, Dalin Mcgraw, Damir Mcgraw, Damisi Mcgraw, Damiso Mcgraw, Dantae Mcgraw, Danthony Mcgraw, Dantoni Mcgraw, Darcus Mcgraw, Darence Mcgraw, Darrious Mcgraw, Darylle Mcgraw, Datril Mcgraw,
Davidlee Mcgraw, Dawane Mcgraw, Daylen Mcgraw, Deandrew Mcgraw, Deloy Mcgraw, Delson Mcgraw, Demaine Mcgraw, Dementrius Mcgraw, Demetrise Mcgraw, Demien Mcgraw, Demingo Mcgraw, Demitris Mcgraw, Deni
Mcgraw, Dennys Mcgraw, Denon Mcgraw, Dent Mcgraw, Deondrick Mcgraw, Derelle Mcgraw, Derike Mcgraw, Derrall Mcgraw, Des Mcgraw, Deshea Mcgraw, Deston Mcgraw, Deval Mcgraw, Devery Mcgraw, Deville
Mcgraw, Devrin Mcgraw, Dewell Mcgraw, Diondray Mcgraw, Dishan Mcgraw, Dkwon Mcgraw, Dmario Mcgraw, Dolph Mcgraw, Dominador Mcgraw, Donae Mcgraw, Donni Mcgraw, Dontea Mcgraw, Donye Mcgraw, Dossie
Mcgraw, Dray Mcgraw, Dwann Mcgraw, Earic Mcgraw, Earley Mcgraw, Eaton Mcgraw, Edwar Mcgraw, Eiad Mcgraw, Eirc Mcgraw, Eith Mcgraw, Elba Mcgraw, Elefterios Mcgraw, Eleodoro Mcgraw, Elisabeth Mcgraw,
Elrod Mcgraw, Elzy Mcgraw, Emilo Mcgraw, Emmanouel Mcgraw, Enis Mcgraw, Ephrain Mcgraw, Erec Mcgraw, Erique Mcgraw, Eriverto Mcgraw, Erven Mcgraw, Esco Mcgraw, Esquiel Mcgraw, Essa Mcgraw, Ettore
Mcgraw, Eulis Mcgraw, Evay Mcgraw, Fabius Mcgraw, Faiz Mcgraw, Felando Mcgraw, Fenwick Mcgraw, Fernell Mcgraw, Francesca Mcgraw, Francisca Mcgraw, Franke Mcgraw, Gabiel Mcgraw, Garrit Mcgraw, Garyn
Mcgraw, Geffery Mcgraw, Generoso Mcgraw, Genghis Mcgraw, Georgia Mcgraw, Geovanny Mcgraw, Gerell Mcgraw, Gerred Mcgraw, Gerrett Mcgraw, Ghassan Mcgraw, Gilliam Mcgraw, Gilson Mcgraw, Givon Mcgraw,
Goeffrey Mcgraw, Gorgonio Mcgraw, Grahame Mcgraw, Grayland Mcgraw, Grzegorz Mcgraw, Guan Mcgraw, Gumercindo Mcgraw, Gwen Mcgraw, Hannah Mcgraw, Harden Mcgraw, Harsha Mcgraw, Harvy Mcgraw, Hassain
Mcgraw, Helio Mcgraw, Hemant Mcgraw, Herchel Mcgraw, Herold Mcgraw, Hershy Mcgraw, Hillis Mcgraw, Hinton Mcgraw, Hommy Mcgraw, Hong Mcgraw, Huberto Mcgraw, Hughes Mcgraw, Husain Mcgraw, Hussien
Mcgraw, Ibis Mcgraw, Ilias Mcgraw, Irl Mcgraw, Iron Mcgraw, Isak Mcgraw, Ishaq Mcgraw, Issaac Mcgraw, Iva Mcgraw, Jabon Mcgraw, Jachin Mcgraw, Jacobus Mcgraw, Janeiro Mcgraw, Japhet Mcgraw, Jarion
Mcgraw, Jaris Mcgraw, Jarmell Mcgraw, Jarnell Mcgraw, Jarvon Mcgraw, Jaryl Mcgraw, Jasonn Mcgraw, Jauan Mcgraw, Javian Mcgraw, Javoris Mcgraw, Jawhar Mcgraw, Jawwaad Mcgraw, Jayesh Mcgraw, Jeffree
Mcgraw, Jeffri Mcgraw, Jerauld Mcgraw, Jereny Mcgraw, Jereomy Mcgraw, Jerom Mcgraw, Jerrall Mcgraw, Jerran Mcgraw, Jerrico Mcgraw, Jessiah Mcgraw, Jesten Mcgraw, Jevan Mcgraw, Jimbob Mcgraw, Jiro
Mcgraw, Johnpatrick Mcgraw, Jolon Mcgraw, Jonothon Mcgraw, Jonthomas Mcgraw, Josephine Mcgraw, Jovonne Mcgraw, Juanantonio Mcgraw, Julis Mcgraw, Jumah Mcgraw, Jwyanza Mcgraw, Kail Mcgraw, Kaleem
Mcgraw, Kalil Mcgraw, Kamari Mcgraw, Kamien Mcgraw, Kanon Mcgraw, Kanta Mcgraw, Karac Mcgraw, Karega Mcgraw, Karras Mcgraw, Kealoha Mcgraw, Kearston Mcgraw, Kedryn Mcgraw, Keiven Mcgraw, Keland
Mcgraw, Kelechi Mcgraw, Kelson Mcgraw, Kendle Mcgraw, Kenjuan Mcgraw, Kennell Mcgraw, Kenson Mcgraw, Kerick Mcgraw, Ketric Mcgraw, Kevine Mcgraw, Keylon Mcgraw, Keynan Mcgraw, Khaaliq Mcgraw, Khali
Mcgraw, Khang Mcgraw, Khoi Mcgraw, Kinshasa Mcgraw, Kono Mcgraw, Konstandinos Mcgraw, Kontar Mcgraw, Kou Mcgraw, Kreston Mcgraw, Kristapher Mcgraw, Kristophor Mcgraw, Kumasi Mcgraw, Kuntakinte
Mcgraw, Kwami Mcgraw, Kwon Mcgraw, Laderick Mcgraw, Lajon Mcgraw, Lakesha Mcgraw, Lakieth Mcgraw, Lakota Mcgraw, Lamarkus Mcgraw, Lameen Mcgraw, Lamondo Mcgraw, Lamonica Mcgraw, Landers Mcgraw,
Landrick Mcgraw, Lanis Mcgraw, Lapriest Mcgraw, Laquane Mcgraw, Larvell Mcgraw, Larwence Mcgraw, Latavius Mcgraw, Latimer Mcgraw, Laton Mcgraw, Latrelle Mcgraw, Latwon Mcgraw, Laurencio Mcgraw,
Lavelton Mcgraw, Laver Mcgraw, Lavester Mcgraw, Lavonn Mcgraw, Lawanda Mcgraw, Lawarren Mcgraw, Lazerick Mcgraw, Lecedric Mcgraw, Leconte Mcgraw, Leib Mcgraw, Lemel Mcgraw, Lenis Mcgraw, Lenox
Mcgraw, Lenward Mcgraw, Leonid Mcgraw, Leory Mcgraw, Lerin Mcgraw, Leroyal Mcgraw, Linnell Mcgraw, Linville Mcgraw, Lional Mcgraw, Loel Mcgraw, Lofton Mcgraw, Londale Mcgraw, Lonza Mcgraw, Lovis
Mcgraw, Lucy Mcgraw, Luie Mcgraw, Lyndall Mcgraw, Macker Mcgraw, Makarios Mcgraw, Malcon Mcgraw, Manly Mcgraw, Marell Mcgraw, Margo Mcgraw, Markco Mcgraw, Markeis Mcgraw, Markian Mcgraw, Marsden
Mcgraw, Martina Mcgraw, Martinus Mcgraw, Mashon Mcgraw, Masud Mcgraw, Mathaniel Mcgraw, Mazi Mcgraw, Merel Mcgraw, Merwyn Mcgraw, Michaelanthony Mcgraw, Michaeljames Mcgraw, Mico Mcgraw, Mihran
Mcgraw, Mikki Mcgraw, Milik Mcgraw, Milos Mcgraw, Miran Mcgraw, Mirza Mcgraw, Mishael Mcgraw, Mondale Mcgraw, Mondre Mcgraw, Montsho Mcgraw, Morse Mcgraw, Mostafa Mcgraw, Mumin Mcgraw, Mynor Mcgraw,
Nantambu Mcgraw, Narayana Mcgraw, Naseem Mcgraw, Nathane Mcgraw, Navada Mcgraw, Nazar Mcgraw, Nectarios Mcgraw, Nell Mcgraw, Nevelle Mcgraw, Nichalaus Mcgraw, Nichlous Mcgraw, Nicholson Mcgraw,
Nicklus Mcgraw, Nickolai Mcgraw, Nicolino Mcgraw, Nihar Mcgraw, Nishan Mcgraw, Nizar Mcgraw, Nocholas Mcgraw, Noelle Mcgraw, Novel Mcgraw, Nyerere Mcgraw, Obbie Mcgraw, Ocean Mcgraw, Oded Mcgraw,
Oden Mcgraw, Olaniyan Mcgraw, Ondray Mcgraw, Orentha Mcgraw, Osby Mcgraw, Othell Mcgraw, Othniel Mcgraw, Ottie Mcgraw, Panayiotis Mcgraw, Parren Mcgraw, Pedram Mcgraw, Petey Mcgraw, Petter Mcgraw,
Pharaoh Mcgraw, Precious Mcgraw, Pressley Mcgraw, Procopio Mcgraw, Pryor Mcgraw, Quaashie Mcgraw, Quinte Mcgraw, Quintez Mcgraw, Quinzell Mcgraw, Quirino Mcgraw, Rabon Mcgraw, Rachard Mcgraw, Rade
Mcgraw, Raju Mcgraw, Ramee Mcgraw, Ranjan Mcgraw, Ransford Mcgraw, Rashawd Mcgraw, Raynald Mcgraw, Regenal Mcgraw, Reiko Mcgraw, Reinhold Mcgraw, Remijio Mcgraw, Renaud Mcgraw, Ritchard Mcgraw,
Roanld Mcgraw, Roarke Mcgraw, Robley Mcgraw, Rochester Mcgraw, Rodolph Mcgraw, Roi Mcgraw, Romelle Mcgraw, Romey Mcgraw, Romolo Mcgraw, Rondo Mcgraw, Ronni Mcgraw, Ronta Mcgraw, Roselio Mcgraw,
Rosheen Mcgraw, Ruban Mcgraw, Rustan Mcgraw, Sajan Mcgraw, Salik Mcgraw, Samel Mcgraw, Santonia Mcgraw, Sargent Mcgraw, Saud Mcgraw, Saxton Mcgraw, Schane Mcgraw, Sebron Mcgraw, Shadley Mcgraw,
Shadron Mcgraw, Shafi Mcgraw, Shalako Mcgraw, Shanell Mcgraw, Shani Mcgraw, Shantae Mcgraw, Shari Mcgraw, Shauntay Mcgraw, Shauwn Mcgraw, Shawkat Mcgraw, Shell Mcgraw, Shephen Mcgraw, Sherief Mcgraw,
Shin Mcgraw, Shneur Mcgraw, Shontell Mcgraw, Shoun Mcgraw, Shundell Mcgraw, Sigmond Mcgraw, Spanky Mcgraw, Stpehen Mcgraw, Suhail Mcgraw, Sulieman Mcgraw, Sundown Mcgraw, Taboris Mcgraw, Taijuan
Mcgraw, Talion Mcgraw, Tamario Mcgraw, Tamel Mcgraw, Tana Mcgraw, Tari Mcgraw, Tarnell Mcgraw, Tashon Mcgraw, Tasos Mcgraw, Tavar Mcgraw, Tawon Mcgraw, Tay Mcgraw, Tederick Mcgraw, Tedman Mcgraw,
Teejay Mcgraw, Tephen Mcgraw, Terranc Mcgraw, Terric Mcgraw, Tevita Mcgraw, Thao Mcgraw, Theoplis Mcgraw, Thuan Mcgraw, Tiger Mcgraw, Tigh Mcgraw, Timmon Mcgraw, Tin Mcgraw, Tionne Mcgraw, Tipton
Mcgraw, Todo Mcgraw, Tomatra Mcgraw, Tomeka Mcgraw, Tomie Mcgraw, Toriono Mcgraw, Torivio Mcgraw, Toye Mcgraw, Tramine Mcgraw, Tre Mcgraw, Treaver Mcgraw, Tredell Mcgraw, Treva Mcgraw, Trinton
Mcgraw, TRUE Mcgraw, Tydus Mcgraw, Tyeson Mcgraw, Tyle Mcgraw, Tyrae Mcgraw, Tyrand Mcgraw, Tyresse Mcgraw, Tyri Mcgraw, Tysean Mcgraw, Tyus Mcgraw, Uland Mcgraw, Usman Mcgraw, Vahid Mcgraw, Vang
Mcgraw, Vassilios Mcgraw, Vernis Mcgraw, Vidale Mcgraw, Vikash Mcgraw, Vishnu Mcgraw, Volney Mcgraw, Vondale Mcgraw, Warees Mcgraw, Webber Mcgraw, Whitman Mcgraw, Willilam Mcgraw, Windy Mcgraw, Winn
Mcgraw, Wynell Mcgraw, Xan Mcgraw, Yeshaya Mcgraw, Yuma Mcgraw, Zackory Mcgraw, Zahid Mcgraw, Zarak Mcgraw, Zayd Mcgraw, Zebula Mcgraw, Zef Mcgraw, Zeljko Mcgraw, Zephyr Mcgraw, Zeshan Mcgraw, Aakash
Mcgraw, Aalon Mcgraw, Aaraon Mcgraw, Aaric Mcgraw, Aarn Mcgraw, Abduel Mcgraw, Abid Mcgraw, Abigail Mcgraw, Abijah Mcgraw, Achille Mcgraw, Ada Mcgraw, Adali Mcgraw, Adante Mcgraw, Adarsh Mcgraw,
Adedayo Mcgraw, Adeyinka Mcgraw, Adib Mcgraw, Adran Mcgraw, Adrell Mcgraw, Adrew Mcgraw, Adriaan Mcgraw, Adriene Mcgraw, Afton Mcgraw, Agim Mcgraw, Ahmond Mcgraw, Aiman Mcgraw, Akiba Mcgraw, Alamin
Mcgraw, Aland Mcgraw, Alanson Mcgraw, Albertico Mcgraw, Albertus Mcgraw, Alcario Mcgraw, Aldrin Mcgraw, Alejando Mcgraw, Alexey Mcgraw, Alfreda Mcgraw, Alfreddie Mcgraw, Allateef Mcgraw, Almer
Mcgraw, Almondo Mcgraw, Alpesh Mcgraw, Alrahman Mcgraw, Alson Mcgraw, Altarik Mcgraw, Alven Mcgraw, Amadi Mcgraw, Amancio Mcgraw, Amandeep Mcgraw, Amauris Mcgraw, Amdrew Mcgraw, Amel Mcgraw, Amelia
Mcgraw, Amhad Mcgraw, Amrit Mcgraw, Ande Mcgraw, Anderw Mcgraw, Andon Mcgraw, Andren Mcgraw, Andretti Mcgraw, Andrius Mcgraw, Andrue Mcgraw, Andrw Mcgraw, Aneesh Mcgraw, Aneudy Mcgraw, Anglo Mcgraw,
Anissa Mcgraw, Annie Mcgraw, Anquan Mcgraw, Ansar Mcgraw, Ansen Mcgraw, Anslem Mcgraw, Anthone Mcgraw, Anthory Mcgraw, Antoinio Mcgraw, Antojuan Mcgraw, Antonial Mcgraw, Antowain Mcgraw, Antowine
Mcgraw, Antrione Mcgraw, Antroy Mcgraw, Anttwan Mcgraw, Antwin Mcgraw, Antwonn Mcgraw, Antyon Mcgraw, Anwon Mcgraw, Aquan Mcgraw, Aquarius Mcgraw, Aracely Mcgraw, Aragorn Mcgraw, Arbie Mcgraw, Arhtur
Mcgraw, Aristede Mcgraw, Aristotelis Mcgraw, Arkee Mcgraw, Arlandus Mcgraw, Arlondo Mcgraw, Armstead Mcgraw, Arnab Mcgraw, Aroldo Mcgraw, Arric Mcgraw, Arshad Mcgraw, Artice Mcgraw, Arvelle Mcgraw,
Arvie Mcgraw, Aryan Mcgraw, Arzell Mcgraw, Asbury Mcgraw, Aseem Mcgraw, Ash Mcgraw, Ashan Mcgraw, Ashkan Mcgraw, Ashlin Mcgraw, Ashur Mcgraw, Astor Mcgraw, Asuncion Mcgraw, Ata Mcgraw, Atsushi
Mcgraw, Aubery Mcgraw, Audi Mcgraw, Augustino Mcgraw, Aurther Mcgraw, Aviv Mcgraw, Ayan Mcgraw, Azael Mcgraw, Azar Mcgraw, Azhar Mcgraw, Azizi Mcgraw, Bai Mcgraw, Bain Mcgraw, Baldo Mcgraw, Balentin
Mcgraw, Banyon Mcgraw, Barbaro Mcgraw, Barett Mcgraw, Barnell Mcgraw, Barren Mcgraw, Bartolomeo Mcgraw, Basem Mcgraw, Basilios Mcgraw, Bassey Mcgraw, Bayete Mcgraw, Baylen Mcgraw, Bayne Mcgraw,
Bayron Mcgraw, Beaufort Mcgraw, Bejan Mcgraw, Belvin Mcgraw, Benajmin Mcgraw, Bengamin Mcgraw, Bently Mcgraw, Berel Mcgraw, Berman Mcgraw, Bernadino Mcgraw, Bernis Mcgraw, Bernon Mcgraw, Bertie
Mcgraw, Bertin Mcgraw, Bibiano Mcgraw, Biko Mcgraw, Binyamin Mcgraw, Birche Mcgraw, Blakley Mcgraw, Bleu Mcgraw, Blythe Mcgraw, Bobak Mcgraw, Bowden Mcgraw, Boz Mcgraw, Brahim Mcgraw, Brahin Mcgraw,
Brandell Mcgraw, Brandley Mcgraw, Brann Mcgraw, Brannigan Mcgraw, Brantly Mcgraw, Brayan Mcgraw, Braydon Mcgraw, Breandan Mcgraw, Breh Mcgraw, Brence Mcgraw, Brenner Mcgraw, Brentwood Mcgraw, Bric
Mcgraw, Bridgette Mcgraw, Brig Mcgraw, Briggs Mcgraw, Brionne Mcgraw, Bronc Mcgraw, Brunson Mcgraw, Brycen Mcgraw, Jennifer Mcgraw, Amy Mcgraw, Melissa Mcgraw, Michelle Mcgraw, Kimberly Mcgraw, Lisa
Mcgraw, Angela Mcgraw, Heather Mcgraw, Stephanie Mcgraw, Nicole Mcgraw, Jessica Mcgraw, Elizabeth Mcgraw, Rebecca Mcgraw, Kelly Mcgraw, Mary Mcgraw, Christina Mcgraw, Amanda Mcgraw, Julie Mcgraw,
Sarah Mcgraw, Laura Mcgraw, Shannon Mcgraw, Christine Mcgraw, Tammy Mcgraw, Tracy Mcgraw, Karen Mcgraw, Dawn Mcgraw, Susan Mcgraw, Andrea Mcgraw, Tina Mcgraw, Patricia Mcgraw, Cynthia Mcgraw, Lori
Mcgraw, Rachel Mcgraw, April Mcgraw, Maria Mcgraw, Wendy Mcgraw, Crystal Mcgraw, Stacy Mcgraw, Erin Mcgraw, Jamie Mcgraw, Carrie Mcgraw, Tiffany Mcgraw, Tara Mcgraw, Sandra Mcgraw, Monica Mcgraw,
Danielle Mcgraw, Stacey Mcgraw, Pamela Mcgraw, Tonya Mcgraw, Sara Mcgraw, Michele Mcgraw, Teresa Mcgraw, Denise Mcgraw, Jill Mcgraw, Katherine Mcgraw, Melanie Mcgraw, Dana Mcgraw, Holly Mcgraw, Erica
Mcgraw, Brenda Mcgraw, Deborah Mcgraw, Tanya Mcgraw, Sharon Mcgraw, Donna Mcgraw, Amber Mcgraw, Emily Mcgraw, Linda Mcgraw, Robin Mcgraw, Kathleen Mcgraw, Leslie Mcgraw, Christy Mcgraw, Kristen
Mcgraw, Catherine Mcgraw, Kristin Mcgraw, Misty Mcgraw, Barbara Mcgraw, Heidi Mcgraw, Nancy Mcgraw, Cheryl Mcgraw, Theresa Mcgraw, Brandy Mcgraw, Alicia Mcgraw, Veronica Mcgraw, Gina Mcgraw,
Jacqueline Mcgraw, Rhonda Mcgraw, Anna Mcgraw, Renee Mcgraw, Megan Mcgraw, Tamara Mcgraw, Kathryn Mcgraw, Melinda Mcgraw, Debra Mcgraw, Sherry Mcgraw, Allison Mcgraw, Valerie Mcgraw, Diana Mcgraw,
Paula Mcgraw, Kristina Mcgraw, Ann Mcgraw, Margaret Mcgraw, Cindy Mcgraw, Victoria Mcgraw, Jodi Mcgraw, Natalie Mcgraw, Brandi Mcgraw, Kristi Mcgraw, Suzanne Mcgraw, Samantha Mcgraw, Beth Mcgraw,
Tracey Mcgraw, Regina Mcgraw, Vanessa Mcgraw, Kristy Mcgraw, Carolyn Mcgraw, Yolanda Mcgraw, Deanna Mcgraw, Carla Mcgraw, Sheila Mcgraw, Laurie Mcgraw, Anne Mcgraw, Shelly Mcgraw, Diane Mcgraw,
Sabrina Mcgraw, Janet Mcgraw, Katrina Mcgraw, Erika Mcgraw, Courtney Mcgraw, Colleen Mcgraw, Carol Mcgraw, Julia Mcgraw, Jenny Mcgraw, Jaime Mcgraw, Kathy Mcgraw, Felicia Mcgraw, Alison Mcgraw,
Lauren Mcgraw, Kelli Mcgraw, Leah Mcgraw, Ashley Mcgraw, Kim Mcgraw, Traci Mcgraw, Kristine Mcgraw, Tricia Mcgraw, Joy Mcgraw, Krista Mcgraw, Kara Mcgraw, Terri Mcgraw, Sonya Mcgraw, Aimee Mcgraw,
Natasha Mcgraw, Cassandra Mcgraw, Bridget Mcgraw, Anita Mcgraw, Kari Mcgraw, Nichole Mcgraw, Christie Mcgraw, Marie Mcgraw, Virginia Mcgraw, Connie Mcgraw, Martha Mcgraw, Carmen Mcgraw, Stacie
Mcgraw, Lynn Mcgraw, Monique Mcgraw, Katie Mcgraw, Kristie Mcgraw, Shelley Mcgraw, Sherri Mcgraw, Angel Mcgraw, Bonnie Mcgraw, Mandy Mcgraw, Jody Mcgraw, Shawna Mcgraw, Kerry Mcgraw, Annette Mcgraw,
Yvonne Mcgraw, Toni Mcgraw, Meredith Mcgraw, Molly Mcgraw, Kendra Mcgraw, Joanna Mcgraw, Sonia Mcgraw, Janice Mcgraw, Robyn Mcgraw, Brooke Mcgraw, Kerri Mcgraw, Sheri Mcgraw, Becky Mcgraw, Gloria
Mcgraw, Mindy Mcgraw, Tracie Mcgraw, Angie Mcgraw, Kellie Mcgraw, Claudia Mcgraw, Ruth Mcgraw, Wanda Mcgraw, Jeanette Mcgraw, Cathy Mcgraw, Adrienne Mcgraw, Kelley Mcgraw, Rachael Mcgraw, Beverly
Mcgraw, Candace Mcgraw, Sylvia Mcgraw, Penny Mcgraw, Charlene Mcgraw, Trisha Mcgraw, Charlotte Mcgraw, Belinda Mcgraw, Candice Mcgraw, Yvette Mcgraw, Lindsay Mcgraw, Keri Mcgraw, Melody Mcgraw,
Caroline Mcgraw, Rosa Mcgraw, Bethany Mcgraw, Trina Mcgraw, Joyce Mcgraw, Joanne Mcgraw, Tabitha Mcgraw, Vicki Mcgraw, Tamika Mcgraw, Gretchen Mcgraw, Ginger Mcgraw, Betty Mcgraw, Dorothy Mcgraw,
Debbie Mcgraw, Darlene Mcgraw, Helen Mcgraw, Latoya Mcgraw, Ellen Mcgraw, Leigh Mcgraw, Rose Mcgraw, Shawn Mcgraw, Karla Mcgraw, Frances Mcgraw, Ana Mcgraw, Alice Mcgraw, Jean Mcgraw, Hope Mcgraw,
Tasha Mcgraw, Latonya Mcgraw, Latasha Mcgraw, Nikki Mcgraw, Maureen Mcgraw, Bobbie Mcgraw, Rita Mcgraw, Peggy Mcgraw, Shirley Mcgraw, Marsha Mcgraw, Cara Mcgraw, Christa Mcgraw, Audrey Mcgraw, Norma
Mcgraw, Shana Mcgraw, Juanita Mcgraw, Leticia Mcgraw, Kimberley Mcgraw, Billie Mcgraw, Evelyn Mcgraw, Tonia Mcgraw, Desiree Mcgraw, Judith Mcgraw, Joann Mcgraw, Shanna Mcgraw, Elaine Mcgraw, Angelica
Mcgraw, Charity Mcgraw, Staci Mcgraw, Tami Mcgraw, Judy Mcgraw, Rebekah Mcgraw, Raquel Mcgraw, Tammie Mcgraw, Jane Mcgraw, Meghan Mcgraw, Jackie Mcgraw, Sally Mcgraw, Jana Mcgraw, Priscilla Mcgraw,
Sandy Mcgraw, Rochelle Mcgraw, Summer Mcgraw, Jodie Mcgraw, Marcia Mcgraw, Alisha Mcgraw, Keisha Mcgraw, Stefanie Mcgraw, Casey Mcgraw, Loretta Mcgraw, Eva Mcgraw, Dena Mcgraw, Cristina Mcgraw,
Janelle Mcgraw, Christi Mcgraw, Naomi Mcgraw, Rachelle Mcgraw, Marisa Mcgraw, Irene Mcgraw, Kirsten Mcgraw, Jeanne Mcgraw, Jenifer Mcgraw, Roxanne Mcgraw, Miranda Mcgraw, Roberta Mcgraw, Dina Mcgraw,
Shauna Mcgraw, Sonja Mcgraw, Marilyn Mcgraw, Eileen Mcgraw, Gwendolyn Mcgraw, Vickie Mcgraw, Katina Mcgraw, Teri Mcgraw, Olivia Mcgraw, Amie Mcgraw, Lora Mcgraw, Cheri Mcgraw, Jennie Mcgraw, Lindsey
Mcgraw, Lesley Mcgraw, Lara Mcgraw, Lydia Mcgraw, Alisa Mcgraw, Sheryl Mcgraw, Autumn Mcgraw, Jacquelyn Mcgraw, Lynette Mcgraw, Esther Mcgraw, Nina Mcgraw, Antoinette Mcgraw, Gail Mcgraw, Deana
Mcgraw, Jaclyn Mcgraw, Lorraine Mcgraw, Alexandra Mcgraw, Jami Mcgraw, Ronda Mcgraw, Candy Mcgraw, Bobbi Mcgraw, Ericka Mcgraw, Abigail Mcgraw, Chandra Mcgraw, Ebony Mcgraw, Latisha Mcgraw, Cherie
Mcgraw, Marla Mcgraw, Lee Mcgraw, Angelia Mcgraw, Kenya Mcgraw, Joan Mcgraw, Jeannie Mcgraw, Shari Mcgraw, Ruby Mcgraw, Miriam Mcgraw, Lakisha Mcgraw, Tameka Mcgraw, Marcy Mcgraw, Angelina Mcgraw,
Faith Mcgraw, Marisol Mcgraw, Krystal Mcgraw, Dianna Mcgraw, Adriana Mcgraw, Audra Mcgraw, Angelique Mcgraw, Marissa Mcgraw, Grace Mcgraw, Alyssa Mcgraw, Janine Mcgraw, Alexis Mcgraw, Elisa Mcgraw,
Jolene Mcgraw, Karin Mcgraw, Shelby Mcgraw, Wendi Mcgraw, Tania Mcgraw, Melisa Mcgraw, Bernadette Mcgraw, Lorena Mcgraw, Shelia Mcgraw, Rosemary Mcgraw, Ramona Mcgraw, Terry Mcgraw, Nora Mcgraw,
Marlene Mcgraw, Camille Mcgraw, Tanisha Mcgraw, Carey Mcgraw, Alma Mcgraw, Sherrie Mcgraw, Cecilia Mcgraw, Celeste Mcgraw, Kate Mcgraw, Betsy Mcgraw, Brandie Mcgraw, Annie Mcgraw, Darla Mcgraw,
Lillian Mcgraw, Maribel Mcgraw, Glenda Mcgraw, Chasity Mcgraw, Patrice Mcgraw, Amelia Mcgraw, Lorie Mcgraw, Tabatha Mcgraw, Elena Mcgraw, Lana Mcgraw, Jocelyn Mcgraw, Jeannette Mcgraw, Guadalupe
Mcgraw, Marcie Mcgraw, Johanna Mcgraw, Josephine Mcgraw, Tisha Mcgraw, Michael Mcgraw, Leanne Mcgraw, Vivian Mcgraw, Whitney Mcgraw, Darcy Mcgraw, Hilary Mcgraw, Marci Mcgraw, Doris Mcgraw, Selena
Mcgraw, Constance Mcgraw, Serena Mcgraw, Daphne Mcgraw, Hollie Mcgraw, Bridgette Mcgraw, Tammi Mcgraw, Jeanine Mcgraw, Terra Mcgraw, Margarita Mcgraw, Vicky Mcgraw, Allyson Mcgraw, Marianne Mcgraw,
Elisabeth Mcgraw, Paige Mcgraw, Iris Mcgraw, Lena Mcgraw, Lakeisha Mcgraw, Jillian Mcgraw, Arlene Mcgraw, Latanya Mcgraw, Emma Mcgraw, Aisha Mcgraw, Tia Mcgraw, Alissa Mcgraw, Sophia Mcgraw, Hannah
Mcgraw, Lawanda Mcgraw, Chrystal Mcgraw, Lea Mcgraw, Cari Mcgraw, Jessie Mcgraw, Mia Mcgraw, Chastity Mcgraw, Lynda Mcgraw, Cassie Mcgraw, Marjorie Mcgraw, Liza Mcgraw, Edith Mcgraw, Jenna Mcgraw,
Joni Mcgraw, Misti Mcgraw, Blanca Mcgraw, Dionne Mcgraw, Lashonda Mcgraw, Leann Mcgraw, Yesenia Mcgraw, Corinne Mcgraw, Jasmine Mcgraw, Malinda Mcgraw, Gabriela Mcgraw, Shonda Mcgraw, Laurel Mcgraw,
Hillary Mcgraw, Kimberlee Mcgraw, Karrie Mcgraw, Irma Mcgraw, Pauline Mcgraw, Claire Mcgraw, Dora Mcgraw, Isabel Mcgraw, Georgia Mcgraw, Esmeralda Mcgraw, Nadine Mcgraw, Lakesha Mcgraw, Marcella
Mcgraw, Luz Mcgraw, Ladonna Mcgraw, Katharine Mcgraw, Phyllis Mcgraw, Sue Mcgraw, Janie Mcgraw, Kisha Mcgraw, Dianne Mcgraw, Myra Mcgraw, Gabrielle Mcgraw, Ingrid Mcgraw, Maryann Mcgraw, Rene Mcgraw,
Lucy Mcgraw, Bridgett Mcgraw, Karyn Mcgraw, Janel Mcgraw, Mandi Mcgraw, Paulette Mcgraw, Lucinda Mcgraw, Carissa Mcgraw, Genevieve Mcgraw, Olga Mcgraw, Dolores Mcgraw, Catina Mcgraw, Randi Mcgraw,
Tera Mcgraw, Jeannine Mcgraw, Nakia Mcgraw, Janette Mcgraw, June Mcgraw, Alana Mcgraw, Annmarie Mcgraw, Lashawn Mcgraw, Noelle Mcgraw, Chelsea Mcgraw, Brittany Mcgraw, Clarissa Mcgraw, Shanda Mcgraw,
Cathleen Mcgraw, Susana Mcgraw, Susanne Mcgraw, Chanda Mcgraw, Edna Mcgraw, Catrina Mcgraw, Kerrie Mcgraw, Francine Mcgraw, Mona Mcgraw, Alyson Mcgraw, Doreen Mcgraw, Maggie Mcgraw, Lynne Mcgraw,
Tori Mcgraw, Clara Mcgraw, Demetria Mcgraw, Beatrice Mcgraw, Valarie Mcgraw, Jeri Mcgraw, Kasey Mcgraw, Silvia Mcgraw, Gena Mcgraw, Janna Mcgraw, Abby Mcgraw, Cristy Mcgraw, Deirdre Mcgraw, Deidre
Mcgraw, Tawana Mcgraw, Deena Mcgraw, Jan Mcgraw, Maricela Mcgraw, Tamra Mcgraw, Daisy Mcgraw, Sondra Mcgraw, Caryn Mcgraw, Nadia Mcgraw, Lorrie Mcgraw, Patty Mcgraw, Anissa Mcgraw, Farrah Mcgraw,
Bianca Mcgraw, Tiffani Mcgraw, Carly Mcgraw, Delia Mcgraw, Susie Mcgraw, Tessa Mcgraw, Latrice Mcgraw, Shellie Mcgraw, Renae Mcgraw, Corina Mcgraw, Deanne Mcgraw, Rena Mcgraw, Louise Mcgraw, Larissa
Mcgraw, Latosha Mcgraw, Lois Mcgraw, Maritza Mcgraw, Julianne Mcgraw, Mildred Mcgraw, Janell Mcgraw, Karina Mcgraw, Christal Mcgraw, Antonia Mcgraw, James Mcgraw, Ida Mcgraw, Marina Mcgraw, Lacey
Mcgraw, Kristal Mcgraw, Ami Mcgraw, Christopher Mcgraw, Gladys Mcgraw, Cori Mcgraw, Gwen Mcgraw, Eleanor Mcgraw, Elisha Mcgraw, Justine Mcgraw, Madeline Mcgraw, Ursula Mcgraw, Charmaine Mcgraw,
Rosemarie Mcgraw, Suzette Mcgraw, Elise Mcgraw, Bertha Mcgraw, Gayle Mcgraw, Lynnette Mcgraw, Sheree Mcgraw, Araceli Mcgraw, Chantel Mcgraw, Kesha Mcgraw, Dayna Mcgraw, Malissa Mcgraw, Dara Mcgraw,
Latonia Mcgraw, Jayme Mcgraw, Robert Mcgraw, Jerri Mcgraw, Carolina Mcgraw, Athena Mcgraw, Hilda Mcgraw, Ivy Mcgraw, Mara Mcgraw, Tosha Mcgraw, Marion Mcgraw, Elsa Mcgraw, Stella Mcgraw, Devon
Mcgraw, Marlo Mcgraw, Danette Mcgraw, Mayra Mcgraw, Kay Mcgraw, Bernice Mcgraw, Adrianne Mcgraw, Geneva Mcgraw, Colette Mcgraw, David Mcgraw, Anitra Mcgraw, Letitia Mcgraw, Marian Mcgraw, Windy
Mcgraw, Jason Mcgraw, Katy Mcgraw, Felecia Mcgraw, Elissa Mcgraw, Susanna Mcgraw, Lucia Mcgraw, Celia Mcgraw, Margie Mcgraw, Kami Mcgraw, Adrian Mcgraw, Holli Mcgraw, Delores Mcgraw, Sasha Mcgraw,
Tamera Mcgraw, Tonja Mcgraw, Taryn Mcgraw, Stacia Mcgraw, John Mcgraw, Tomeka Mcgraw, Anastasia Mcgraw, Marta Mcgraw, Patti Mcgraw, Jeanna Mcgraw, Georgina Mcgraw, Eugenia Mcgraw, Tawnya Mcgraw,
Corey Mcgraw, Leeann Mcgraw, Juliet Mcgraw, Chris Mcgraw, Dominique Mcgraw, Mitzi Mcgraw, Margo Mcgraw, Selina Mcgraw, Marnie Mcgraw, Casandra Mcgraw, Deann Mcgraw, Leanna Mcgraw, Vera Mcgraw,
Consuelo Mcgraw, Buffy Mcgraw, Lourdes Mcgraw, Florence Mcgraw, Geraldine Mcgraw, Tyra Mcgraw, Alecia Mcgraw, Greta Mcgraw, Carole Mcgraw, Francesca Mcgraw, Meagan Mcgraw, Jeanie Mcgraw, Rosalinda
Mcgraw, Trudy Mcgraw, Haley Mcgraw, Sandi Mcgraw, Corrie Mcgraw, Maura Mcgraw, Sharonda Mcgraw, Graciela Mcgraw, Michell Mcgraw, Terrie Mcgraw, Eve Mcgraw, Tiffanie Mcgraw, Rosalind Mcgraw, Marlena
Mcgraw, Shannan Mcgraw, Thelma Mcgraw, Carie Mcgraw, Lashanda Mcgraw, Shelli Mcgraw, Karie Mcgraw, Danelle Mcgraw, Tawanda Mcgraw, Ryan Mcgraw, Simone Mcgraw, Maya Mcgraw, Mollie Mcgraw, Deidra
Mcgraw, Josie Mcgraw, Noemi Mcgraw, Briana Mcgraw, Venus Mcgraw, Lauri Mcgraw, Richelle Mcgraw, Candi Mcgraw, Ella Mcgraw, Cora Mcgraw, Latricia Mcgraw, Marisela Mcgraw, Beatriz Mcgraw, Shanon
Mcgraw, Alejandra Mcgraw, Aubrey Mcgraw, Jeana Mcgraw, Nanette Mcgraw, Brianne Mcgraw, Leona Mcgraw, Helena Mcgraw, Kris Mcgraw, Christian Mcgraw, Annemarie Mcgraw, Andria Mcgraw, Mindi Mcgraw,
Rosalyn Mcgraw, Shanta Mcgraw, Mercedes Mcgraw, Polly Mcgraw, Juana Mcgraw, Rosie Mcgraw, Martina Mcgraw, Therese Mcgraw, Cory Mcgraw, Kayla Mcgraw, Shantel Mcgraw, Joey Mcgraw, Shalonda Mcgraw,
Mellissa Mcgraw, Ada Mcgraw, Brandee Mcgraw, Sunshine Mcgraw, Tamiko Mcgraw, Danita Mcgraw, Niki Mcgraw, Rosanna Mcgraw, Joelle Mcgraw, Monika Mcgraw, Trista Mcgraw, Chiquita Mcgraw, William Mcgraw,
Brook Mcgraw, Janis Mcgraw, Rae Mcgraw, Kizzy Mcgraw, Cortney Mcgraw, Cecelia Mcgraw, Jaimie Mcgraw, Christin Mcgraw, Christen Mcgraw, Tanika Mcgraw, Darci Mcgraw, Claudine Mcgraw, Sharla Mcgraw,
Toya Mcgraw, Racheal Mcgraw, Noel Mcgraw, Jada Mcgraw, Davina Mcgraw, Della Mcgraw, Kori Mcgraw, Tomika Mcgraw, Callie Mcgraw, Jamila Mcgraw, Aileen Mcgraw, Dusty Mcgraw, Rocio Mcgraw, Kia Mcgraw,
Benita Mcgraw, Jena Mcgraw, Aurora Mcgraw, Penelope Mcgraw, Sadie Mcgraw, Patsy Mcgraw, Valencia Mcgraw, Lisette Mcgraw, Tamela Mcgraw, Mari Mcgraw, Tana Mcgraw, Ivette Mcgraw, Eliza Mcgraw, Darcie
Mcgraw, Tawanna Mcgraw, Bonita Mcgraw, Debora Mcgraw, Lacy Mcgraw, Tarsha Mcgraw, Stephenie Mcgraw, Johnna Mcgraw, Kathrine Mcgraw, Lakeshia Mcgraw, Griselda Mcgraw, Charla Mcgraw, Lesa Mcgraw, Sunny
Mcgraw, Celina Mcgraw, Karri Mcgraw, Corinna Mcgraw, Dixie Mcgraw, Rosalie Mcgraw, Rhiannon Mcgraw, Eunice Mcgraw, Mariah Mcgraw, Angeline Mcgraw, Danna Mcgraw, Shayla Mcgraw, Brenna Mcgraw, Juliana
Mcgraw, Brianna Mcgraw, Octavia Mcgraw, Leila Mcgraw, Liliana Mcgraw, Shameka Mcgraw, Lissette Mcgraw, Cary Mcgraw, Chrissy Mcgraw, Corrine Mcgraw, Destiny Mcgraw, India Mcgraw, Adriane Mcgraw, Kandi
Mcgraw, Coleen Mcgraw, Angelita Mcgraw, Denice Mcgraw, Janeen Mcgraw, Suzanna Mcgraw, Dee Mcgraw, Magdalena Mcgraw, Lola Mcgraw, Brian Mcgraw, Nikole Mcgraw, Sofia Mcgraw, Nicolle Mcgraw, Lucretia
Mcgraw, Mellisa Mcgraw, Morgan Mcgraw, Alexa Mcgraw, Faye Mcgraw, Daniel Mcgraw, Antionette Mcgraw, Tennille Mcgraw, Ayanna Mcgraw, Maxine Mcgraw, Christel Mcgraw, Alysia Mcgraw, Alexandria Mcgraw,
Lenora Mcgraw, Aida Mcgraw, Kyla Mcgraw, Joseph Mcgraw, Bree Mcgraw, Kira Mcgraw, Renita Mcgraw, Sharron Mcgraw, Jenni Mcgraw, Ethel Mcgraw, Fatima Mcgraw, Hazel Mcgraw, Nellie Mcgraw, Juli Mcgraw,
Delilah Mcgraw, Pearl Mcgraw, Caren Mcgraw, Melonie Mcgraw, Robbie Mcgraw, Lindy Mcgraw, Tangela Mcgraw, Trena Mcgraw, Daniela Mcgraw, Michaela Mcgraw, Carri Mcgraw, Sommer Mcgraw, Alanna Mcgraw,
Tressa Mcgraw, Georgette Mcgraw, Nikita Mcgraw, Chanel Mcgraw, Aretha Mcgraw, Charisse Mcgraw, Laquita Mcgraw, Natalia Mcgraw, Josette Mcgraw, Roxanna Mcgraw, Kirstin Mcgraw, Shani Mcgraw, Nichol
Mcgraw, Regan Mcgraw, Salina Mcgraw, Anjanette Mcgraw, Susannah Mcgraw, Jammie Mcgraw, Kenyatta Mcgraw, Jasmin Mcgraw, Anthony Mcgraw, Ginny Mcgraw, Lillie Mcgraw, Missy Mcgraw, Starla Mcgraw,
Lucille Mcgraw, Lorna Mcgraw, Tiana Mcgraw, Melodie Mcgraw, Fawn Mcgraw, Candida Mcgraw, Carin Mcgraw, Elvira Mcgraw, Shonna Mcgraw, Carmela Mcgraw, Spring Mcgraw, Shara Mcgraw, Yolonda Mcgraw,
Josefina Mcgraw, Evette Mcgraw, Carisa Mcgraw, Sherita Mcgraw, Roseann Mcgraw, Saundra Mcgraw, Myrna Mcgraw, Bambi Mcgraw, Layla Mcgraw, Lilia Mcgraw, Mechelle Mcgraw, Cathryn Mcgraw, Jayne Mcgraw,
Lolita Mcgraw, Rosario Mcgraw, Eric Mcgraw, Valeria Mcgraw, Jade Mcgraw, Nicki Mcgraw, Maranda Mcgraw, Felisha Mcgraw, Raina Mcgraw, Tracee Mcgraw, Cindi Mcgraw, Freda Mcgraw, Gillian Mcgraw, Damaris
Mcgraw, Ava Mcgraw, Brigitte Mcgraw, Nicola Mcgraw, Esperanza Mcgraw, Leilani Mcgraw, Sharlene Mcgraw, Marni Mcgraw, Claudette Mcgraw, Keely Mcgraw, Hayley Mcgraw, Chantelle Mcgraw, Joleen Mcgraw,
Elsie Mcgraw, Rebeca Mcgraw, Mendy Mcgraw, Adria Mcgraw, Harmony Mcgraw, Francisca Mcgraw, Jolie Mcgraw, Vikki Mcgraw, Nikia Mcgraw, Tamatha Mcgraw, Keli Mcgraw, Keshia Mcgraw, Charissa Mcgraw,
Imelda Mcgraw, Francis Mcgraw, Richard Mcgraw, Shantell Mcgraw, Shawanda Mcgraw, Caitlin Mcgraw, Frankie Mcgraw, Shea Mcgraw, Cherry Mcgraw, Sheena Mcgraw, Brigette Mcgraw, Keesha Mcgraw, Cristi
Mcgraw, Justina Mcgraw, Rashida Mcgraw, Emilie Mcgraw, Ashlee Mcgraw, Shane Mcgraw, Cherise Mcgraw, Qiana Mcgraw, Latarsha Mcgraw, Crissy Mcgraw, Celena Mcgraw, Kylie Mcgraw, Rolanda Mcgraw, Crista
Mcgraw, Roslyn Mcgraw, Angella Mcgraw, Shamika Mcgraw, Wilma Mcgraw, Violet Mcgraw, Kyra Mcgraw, Kevin Mcgraw, Marybeth Mcgraw, Alesha Mcgraw, Matthew Mcgraw, Shanika Mcgraw, Shawnda Mcgraw, Cristal
Mcgraw, Julissa Mcgraw, Lila Mcgraw, Alycia Mcgraw, Camilla Mcgraw, Johnnie Mcgraw, Dori Mcgraw, Nichelle Mcgraw, Venessa Mcgraw, Anika Mcgraw, Lily Mcgraw, Tamica Mcgraw, Rhoda Mcgraw, Alethea
Mcgraw, Carina Mcgraw, Roxann Mcgraw, Latrina Mcgraw, Charles Mcgraw, Bessie Mcgraw, Julianna Mcgraw, Marquita Mcgraw, Gayla Mcgraw, Kandy Mcgraw, Estella Mcgraw, Marguerite Mcgraw, Chana Mcgraw,
Robbin Mcgraw, Cristin Mcgraw, Raven Mcgraw, Tanesha Mcgraw, Alina Mcgraw, Juliette Mcgraw, Milagros Mcgraw, Inez Mcgraw, Vonda Mcgraw, Jenelle Mcgraw, Elaina Mcgraw, Dedra Mcgraw, Verna Mcgraw,
Starr Mcgraw, Bobby Mcgraw, Lizette Mcgraw, Dawna Mcgraw, Danyelle Mcgraw, Petra Mcgraw, Stephaine Mcgraw, Angelic Mcgraw, Brittney Mcgraw, Corrina Mcgraw, Britt Mcgraw, Stephany Mcgraw, Lorri
Mcgraw, Shayna Mcgraw, Flora Mcgraw, Jesse Mcgraw, Velma Mcgraw, Racquel Mcgraw, Stefani Mcgraw, Phoebe Mcgraw, Yadira Mcgraw, Liana Mcgraw, Milissa Mcgraw, Pam Mcgraw, Zoe Mcgraw, Libby Mcgraw,
Shasta Mcgraw, Luisa Mcgraw, Barbra Mcgraw, Janae Mcgraw, Shandra Mcgraw, Thomas Mcgraw, Kelsey Mcgraw, Shante Mcgraw, Bettina Mcgraw, Lashunda Mcgraw, Melina Mcgraw, Mattie Mcgraw, Meaghan Mcgraw,
Timothy Mcgraw, Rosetta Mcgraw, Mistie Mcgraw, Steven Mcgraw, Patrica Mcgraw, Shay Mcgraw, Takisha Mcgraw, Geri Mcgraw, Teressa Mcgraw, Aaron Mcgraw, Clare Mcgraw, Noreen Mcgraw, Adina Mcgraw,
Rosanne Mcgraw, Luciana Mcgraw, Nakisha Mcgraw, Trinity Mcgraw, Cicely Mcgraw, Minerva Mcgraw, Devin Mcgraw, Gabriella Mcgraw, Lela Mcgraw, Kendall Mcgraw, Treva Mcgraw, Hallie Mcgraw, Kristan
Mcgraw, Lia Mcgraw, Viola Mcgraw, Portia Mcgraw, Jonna Mcgraw, Kellee Mcgraw, Danica Mcgraw, Kristyn Mcgraw, Lina Mcgraw, Jewel Mcgraw, Tamala Mcgraw, Jenniffer Mcgraw, Melony Mcgraw, Kacey Mcgraw,
Quiana Mcgraw, Shawnna Mcgraw, Rana Mcgraw, Dannielle Mcgraw, Andra Mcgraw, Lenore Mcgraw, Carlene Mcgraw, Malia Mcgraw, Lanette Mcgraw, Jacquline Mcgraw, Willie Mcgraw, Precious Mcgraw, Collette
Mcgraw, Mariana Mcgraw, Ayana Mcgraw, Daniella Mcgraw, Thea Mcgraw, Viviana Mcgraw, Kindra Mcgraw, Gracie Mcgraw, Louisa Mcgraw, Adele Mcgraw, Annamarie Mcgraw, Kiesha Mcgraw, Sherie Mcgraw, Althea
Mcgraw, Catalina Mcgraw, January Mcgraw, Ernestine Mcgraw, Corie Mcgraw, Lesli Mcgraw, Shanita Mcgraw, Tresa Mcgraw, Maryanne Mcgraw, Tamar Mcgraw, Rayna Mcgraw, Siobhan Mcgraw, Demetra Mcgraw,
Kimberlie Mcgraw, Rhea Mcgraw, Elva Mcgraw, Elvia Mcgraw, Karissa Mcgraw, Maryellen Mcgraw, Valorie Mcgraw, Adrianna Mcgraw, Nicolette Mcgraw, Daniele Mcgraw, Trish Mcgraw, Twila Mcgraw, Abbie
Mcgraw, Towanda Mcgraw, Lorinda Mcgraw, Marcela Mcgraw, Kyle Mcgraw, Carmelita Mcgraw, Helene Mcgraw, Mark Mcgraw, Jeffrey Mcgraw, Lidia Mcgraw, Kenyetta Mcgraw, Lupe Mcgraw, Cami Mcgraw, Cher
Mcgraw, Minnie Mcgraw, Toby Mcgraw, Karena Mcgraw, Katheryn Mcgraw, Renea Mcgraw, Kathie Mcgraw, Laverne Mcgraw, Shona Mcgraw, Linette Mcgraw, Kecia Mcgraw, Elana Mcgraw, Eboni Mcgraw, Loren Mcgraw,
Tabetha Mcgraw, Dione Mcgraw, Glenna Mcgraw, Marti Mcgraw, Gia Mcgraw, Tena Mcgraw, Cameron Mcgraw, Roxana Mcgraw, Alberta Mcgraw, Dolly Mcgraw, Pamala Mcgraw, Talia Mcgraw, Tobi Mcgraw, Latesha
Mcgraw, Larhonda Mcgraw, Sybil Mcgraw, Kandice Mcgraw, Yasmin Mcgraw, Christiana Mcgraw, Sydney Mcgraw, Chantal Mcgraw, Estela Mcgraw, Jaimee Mcgraw, Salena Mcgraw, Tamekia Mcgraw, Dalia Mcgraw,
Twana Mcgraw, Jacklyn Mcgraw, Scarlett Mcgraw, Tanja Mcgraw, Kenisha Mcgraw, Daria Mcgraw, Agnes Mcgraw, Mandie Mcgraw, Lani Mcgraw, May Mcgraw, Sharee Mcgraw, Tenisha Mcgraw, Twyla Mcgraw, Chantell
Mcgraw, Shiela Mcgraw, Isela Mcgraw, Karey Mcgraw, Alaina Mcgraw, Loriann Mcgraw, Star Mcgraw, Samara Mcgraw, Janene Mcgraw, Gerri Mcgraw, Lasonya Mcgraw, Lavonne Mcgraw, Rikki Mcgraw, Meridith
Mcgraw, Lorene Mcgraw, Scott Mcgraw, Stephani Mcgraw, Kimberli Mcgraw, Zandra Mcgraw, Sallie Mcgraw, Cecily Mcgraw, Rachele Mcgraw, Melynda Mcgraw, Reagan Mcgraw, Carmella Mcgraw, Danyell Mcgraw,
Ivonne Mcgraw, Jamey Mcgraw, Shawana Mcgraw, Aja Mcgraw, Casie Mcgraw, Alena Mcgraw, Barbie Mcgraw, Clarice Mcgraw, Kassandra Mcgraw, Tiffiny Mcgraw, Madelyn Mcgraw, Cherish Mcgraw, Lavonda Mcgraw,
Malisa Mcgraw, Stormy Mcgraw, Amalia Mcgraw, Machelle Mcgraw, Abbey Mcgraw, Evangelina Mcgraw, Serina Mcgraw, Fannie Mcgraw, Joi Mcgraw, Tarra Mcgraw, Dale Mcgraw, Shaunna Mcgraw, Jeremy Mcgraw,
Sarina Mcgraw, Harriet Mcgraw, Holley Mcgraw, Reyna Mcgraw, Ayesha Mcgraw, Natosha Mcgraw, Princess Mcgraw, Shavon Mcgraw, Ashleigh Mcgraw, Laticia Mcgraw, Cristine Mcgraw, Asia Mcgraw, Antonette
Mcgraw, Dinah Mcgraw, Jordan Mcgraw, Cherilyn Mcgraw, Mackenzie Mcgraw, Giselle Mcgraw, Edwina Mcgraw, Mae Mcgraw, Kathi Mcgraw, Shalanda Mcgraw, Donita Mcgraw, Melissia Mcgraw, Cristie Mcgraw, Mirna
Mcgraw, Cheyenne Mcgraw, Leslee Mcgraw, Jeni Mcgraw, Jacinda Mcgraw, Consuela Mcgraw, Evangeline Mcgraw, Joanie Mcgraw, Nickie Mcgraw, Jolynn Mcgraw, Malika Mcgraw, Nathalie Mcgraw, Mamie Mcgraw,
Paul Mcgraw, Davida Mcgraw, Erinn Mcgraw, Laronda Mcgraw, Latina Mcgraw, Liberty Mcgraw, Nicol Mcgraw, Danyel Mcgraw, Tommie Mcgraw, Joshua Mcgraw, Shaun Mcgraw, Jodee Mcgraw, Teena Mcgraw, Shanell
Mcgraw, Calandra Mcgraw, Iesha Mcgraw, Devona Mcgraw, Aracely Mcgraw, Brandon Mcgraw, Dona Mcgraw, Micah Mcgraw, Adela Mcgraw, Lashon Mcgraw, Reina Mcgraw, Sierra Mcgraw, Addie Mcgraw, Carlotta
Mcgraw, Nia Mcgraw, Ariana Mcgraw, Chante Mcgraw, Tamisha Mcgraw, Ali Mcgraw, Deedee Mcgraw, Sarita Mcgraw, Jeniffer Mcgraw, Kiley Mcgraw, Ronnie Mcgraw, Cherri Mcgraw, Nickole Mcgraw, Hattie Mcgraw,
Syreeta Mcgraw, Chaya Mcgraw, Dani Mcgraw, Catharine Mcgraw, Tamie Mcgraw, Perla Mcgraw, Billiejo Mcgraw, Deloris Mcgraw, Lashawnda Mcgraw, Meggan Mcgraw, Temeka Mcgraw, Dorothea Mcgraw, Marianna
Mcgraw, Lou Mcgraw, Tesha Mcgraw, Karmen Mcgraw, Luann Mcgraw, Rona Mcgraw, Elyse Mcgraw, Latashia Mcgraw, Dacia Mcgraw, Nita Mcgraw, Tashia Mcgraw, Kenneth Mcgraw, Mireya Mcgraw, Carleen Mcgraw,
Alfreda Mcgraw, Candie Mcgraw, Delicia Mcgraw, Lakiesha Mcgraw, Renata Mcgraw, Juliann Mcgraw, Patience Mcgraw, Alexia Mcgraw, Lakeesha Mcgraw, Petrina Mcgraw, Liane Mcgraw, Ester Mcgraw, Kimberely
Mcgraw, Millie Mcgraw, Roni Mcgraw, Reba Mcgraw, Tory Mcgraw, Maryjo Mcgraw, Aundrea Mcgraw, Kasandra Mcgraw, Melba Mcgraw, Tiffaney Mcgraw, Maisha Mcgraw, Kenna Mcgraw, Detra Mcgraw, Jerry Mcgraw,
Meghann Mcgraw, Sanjuanita Mcgraw, Emilia Mcgraw, Denita Mcgraw, Shaunda Mcgraw, Shavonne Mcgraw, Jose Mcgraw, Roseanne Mcgraw, Kiersten Mcgraw, Sophie Mcgraw, Maren Mcgraw, Giovanna Mcgraw, Bethann
Mcgraw, Ilene Mcgraw, Dorian Mcgraw, Lanita Mcgraw, Ariel Mcgraw, Piper Mcgraw, Shawnee Mcgraw, Cammie Mcgraw, Joellen Mcgraw, Millicent Mcgraw, Rachell Mcgraw, Billy Mcgraw, Felica Mcgraw, Jacinta
Mcgraw, Krissy Mcgraw, Letisha Mcgraw, Pennie Mcgraw, Delisa Mcgraw, Evonne Mcgraw, Keysha Mcgraw, Migdalia Mcgraw, Opal Mcgraw, Taunya Mcgraw, Farah Mcgraw, Ina Mcgraw, Samatha Mcgraw, Linnea
Mcgraw, Lyn Mcgraw, Merry Mcgraw, Patrina Mcgraw, Terese Mcgraw, Leandra Mcgraw, Cornelia Mcgraw, Manda Mcgraw, Lissa Mcgraw, Coretta Mcgraw, Gidget Mcgraw, Demetrice Mcgraw, Mabel Mcgraw, Necole
Mcgraw, Torrie Mcgraw, Kandace Mcgraw, Loraine Mcgraw, Tarah Mcgraw, Cody Mcgraw, Cinnamon Mcgraw, Shonta Mcgraw, Rosalia Mcgraw, Delana Mcgraw, Nannette Mcgraw, Amee Mcgraw, Ofelia Mcgraw, Shakira
Mcgraw, Aletha Mcgraw, Kacy Mcgraw, Rosita Mcgraw, Talisha Mcgraw, Moira Mcgraw, Sebrina Mcgraw, Katrice Mcgraw, Eden Mcgraw, Jaqueline Mcgraw, Kirstie Mcgraw, Gregory Mcgraw, Letha Mcgraw, Apryl
Mcgraw, Kristel Mcgraw, Fabiola Mcgraw, Steffanie Mcgraw, Elicia Mcgraw, Nedra Mcgraw, Maia Mcgraw, Sharita Mcgraw, Tammara Mcgraw, Charise Mcgraw, Cheree Mcgraw, Andrew Mcgraw, Honey Mcgraw, Velvet
Mcgraw, Anglea Mcgraw, Brett Mcgraw, Henrietta Mcgraw, Jenine Mcgraw, Meg Mcgraw, Khalilah Mcgraw, Sheronda Mcgraw, Dyan Mcgraw, Marty Mcgraw, Suzan Mcgraw, Liz Mcgraw, Brande Mcgraw, Flor Mcgraw,
Tameika Mcgraw, Manuela Mcgraw, Stephen Mcgraw, Kamilah Mcgraw, Karolyn Mcgraw, Demetrius Mcgraw, Lacie Mcgraw, Lakeysha Mcgraw, Jannette Mcgraw, Vonetta Mcgraw, Gale Mcgraw, Shiloh Mcgraw, Shira
Mcgraw, Gisela Mcgraw, Jacquelin Mcgraw, Natisha Mcgraw, Billi Mcgraw, Charleen Mcgraw, Brigid Mcgraw, Dorinda Mcgraw, Jacki Mcgraw, Keena Mcgraw, Lamonica Mcgraw, Shena Mcgraw, Michel Mcgraw, Nanci
Mcgraw, Donya Mcgraw, Leisa Mcgraw, Rosalba Mcgraw, Anabel Mcgraw, Annalisa Mcgraw, Amity Mcgraw, Mimi Mcgraw, Erma Mcgraw, Lyndsey Mcgraw, Marlana Mcgraw, Skye Mcgraw, Chloe Mcgraw, Inga Mcgraw,
Jonathan Mcgraw, Melaine Mcgraw, Lorelei Mcgraw, Marivel Mcgraw, Nelly Mcgraw, Amiee Mcgraw, Cathrine Mcgraw, Jonelle Mcgraw, Sacha Mcgraw, Jennette Mcgraw, Lilly Mcgraw, Shawnta Mcgraw, Tisa Mcgraw,
Judi Mcgraw, Cherrie Mcgraw, Annetta Mcgraw, Charlette Mcgraw, Christiane Mcgraw, Jesica Mcgraw, Pilar Mcgraw, Elida Mcgraw, Twanna Mcgraw, Madonna Mcgraw, Tausha Mcgraw, Carletta Mcgraw, Leesa
Mcgraw, Matilda Mcgraw, Nakita Mcgraw, Natashia Mcgraw, Kaci Mcgraw, Mica Mcgraw, Dionna Mcgraw, Margot Mcgraw, Maribeth Mcgraw, Darby Mcgraw, Shondra Mcgraw, Socorro Mcgraw, Misha Mcgraw, Taylor
Mcgraw, Becki Mcgraw, Kiana Mcgraw, Valentina Mcgraw, Ashlie Mcgraw, Desirae Mcgraw, Lizabeth Mcgraw, Alesia Mcgraw, Roxane Mcgraw, Donielle Mcgraw, Kelle Mcgraw, Larisa Mcgraw, Paola Mcgraw, Aleta
Mcgraw, Dorene Mcgraw, Kitty Mcgraw, Shenita Mcgraw, Tatiana Mcgraw, Felisa Mcgraw, Yashica Mcgraw, Jeanetta Mcgraw, Tony Mcgraw, Latrisha Mcgraw, Nyree Mcgraw, Ronald Mcgraw, Romona Mcgraw, Shawanna
Mcgraw, Edward Mcgraw, Justin Mcgraw, Tuesday Mcgraw, Avis Mcgraw, Edie Mcgraw, Aurelia Mcgraw, Nova Mcgraw, Ruthie Mcgraw, Estelle Mcgraw, Mikki Mcgraw, Jamee Mcgraw, Micaela Mcgraw, Phaedra Mcgraw,
Sean Mcgraw, Fiona Mcgraw, Shilo Mcgraw, Trudi Mcgraw, Blythe Mcgraw, Julee Mcgraw, Alia Mcgraw, Lachelle Mcgraw, Roseanna Mcgraw, Donald Mcgraw, Nydia Mcgraw, Effie Mcgraw, Jenell Mcgraw, Carolee
Mcgraw, Karma Mcgraw, Violeta Mcgraw, Ilana Mcgraw, Leonor Mcgraw, Talitha Mcgraw, Deedra Mcgraw, Jimmie Mcgraw, Buffie Mcgraw, Iva Mcgraw, Joycelyn Mcgraw, Randa Mcgraw, Tawny Mcgraw, Bonny Mcgraw,
Dallas Mcgraw, Felice Mcgraw, Lawana Mcgraw, Corine Mcgraw, Kizzie Mcgraw, Ladawn Mcgraw, Torri Mcgraw, Marva Mcgraw, Sabina Mcgraw, Taneka Mcgraw, Deandra Mcgraw, Kary Mcgraw, Juliane Mcgraw, Asha
Mcgraw, Mika Mcgraw, Twanda Mcgraw, Dawne Mcgraw, Ellie Mcgraw, Lawanna Mcgraw, Rowena Mcgraw, Caron Mcgraw, Chad Mcgraw, Ronna Mcgraw, Tatum Mcgraw, Caprice Mcgraw, Cinthia Mcgraw, Debby Mcgraw,
Deeann Mcgraw, Shalon Mcgraw, Ernestina Mcgraw, Andre Mcgraw, Annamaria Mcgraw, Latoshia Mcgraw, Shemeka Mcgraw, Stacee Mcgraw, Tyesha Mcgraw, Annabelle Mcgraw, Germaine Mcgraw, Risa Mcgraw, Latoria
Mcgraw, Patrick Mcgraw, Wendie Mcgraw, Laureen Mcgraw, Cristen Mcgraw, Kati Mcgraw, Niccole Mcgraw, Sabra Mcgraw, Shoshana Mcgraw, Tequila Mcgraw, Melva Mcgraw, Micki Mcgraw, Darcey Mcgraw, Coral
Mcgraw, Fonda Mcgraw, Lashaunda Mcgraw, Pepper Mcgraw, Trinette Mcgraw, Eloisa Mcgraw, Goldie Mcgraw, Tijuana Mcgraw, Venita Mcgraw, Danae Mcgraw, Tessie Mcgraw, Rebecka Mcgraw, Taisha Mcgraw, Heide
Mcgraw, Joslyn Mcgraw, Rebbecca Mcgraw, Magda Mcgraw, Roxie Mcgraw, Daniell Mcgraw, Kellye Mcgraw, Timika Mcgraw, Dannette Mcgraw, Lakita Mcgraw, Shawnte Mcgraw, Dusti Mcgraw, Shenika Mcgraw, Amberly
Mcgraw, Koren Mcgraw, Carolynn Mcgraw, Thomasina Mcgraw, Bobbiejo Mcgraw, Candance Mcgraw, Jacque Mcgraw, Nikisha Mcgraw, Torie Mcgraw, Cyndi Mcgraw, Diann Mcgraw, Marylou Mcgraw, Nelda Mcgraw,
Prudence Mcgraw, Sharyn Mcgraw, Carman Mcgraw, Coreen Mcgraw, Amina Mcgraw, Neva Mcgraw, Scarlet Mcgraw, Ammie Mcgraw, Haydee Mcgraw, Jaymie Mcgraw, Rina Mcgraw, Lyndsay Mcgraw, Shantelle Mcgraw,
Alyce Mcgraw, Danika Mcgraw, Leeanne Mcgraw, Tamesha Mcgraw, Iliana Mcgraw, Kylee Mcgraw, Lisha Mcgraw, Mercy Mcgraw, Nona Mcgraw, Teresita Mcgraw, Xochitl Mcgraw, Carmel Mcgraw, Jessi Mcgraw,
Kourtney Mcgraw, Elia Mcgraw, Lasandra Mcgraw, Alisia Mcgraw, Candis Mcgraw, Nicholle Mcgraw, Nissa Mcgraw, Micheal Mcgraw, Kali Mcgraw, Tajuana Mcgraw, Aleshia Mcgraw, Daina Mcgraw, Fay Mcgraw,
Rivka Mcgraw, Jennefer Mcgraw, Kena Mcgraw, Cassidy Mcgraw, Toshia Mcgraw, Larry Mcgraw, Travis Mcgraw, Chaka Mcgraw, Anya Mcgraw, Leia Mcgraw, Nola Mcgraw, Rosalva Mcgraw, Arica Mcgraw, Hanna
Mcgraw, Kenda Mcgraw, Tiesha Mcgraw, Jewell Mcgraw, Roshanda Mcgraw, Arianne Mcgraw, Eryn Mcgraw, Denna Mcgraw, George Mcgraw, Jerilyn Mcgraw, Lakia Mcgraw, Melani Mcgraw, Raeann Mcgraw, Season
Mcgraw, Sherilyn Mcgraw, Lizbeth Mcgraw, Myesha Mcgraw, Suzy Mcgraw, Emilee Mcgraw, Kristian Mcgraw, Leighann Mcgraw, Akilah Mcgraw, Britney Mcgraw, Juan Mcgraw, Bobbijo Mcgraw, Dalila Mcgraw, Ileana
Mcgraw, Contessa Mcgraw, Darleen Mcgraw, Deshawn Mcgraw, Audrea Mcgraw, Kawana Mcgraw, Lona Mcgraw, Margret Mcgraw, Melany Mcgraw, Michal Mcgraw, Odessa Mcgraw, Reva Mcgraw, Lula Mcgraw, Mariann
Mcgraw, Kanika Mcgraw, Paris Mcgraw, Shaneka Mcgraw, Sherrell Mcgraw, Suzann Mcgraw, Chelsey Mcgraw, Laquisha Mcgraw, Willow Mcgraw, Kaye Mcgraw, Muriel Mcgraw, Britta Mcgraw, Dania Mcgraw, Jesenia
Mcgraw, Nilda Mcgraw, Dottie Mcgraw, Nereida Mcgraw, Kellyann Mcgraw, Alishia Mcgraw, Denisha Mcgraw, Marietta Mcgraw, Merideth Mcgraw, Enid Mcgraw, Isis Mcgraw, Rosio Mcgraw, America Mcgraw, Belen
Mcgraw, Randy Mcgraw, Karol Mcgraw, Kasie Mcgraw, Julieann Mcgraw, Sari Mcgraw, Bronwyn Mcgraw, Concepcion Mcgraw, Criselda Mcgraw, Natacha Mcgraw, Ariane Mcgraw, Breanna Mcgraw, Shayne Mcgraw,
Alysha Mcgraw, Gary Mcgraw, Rasheeda Mcgraw, Sherice Mcgraw, Tonisha Mcgraw, Katrena Mcgraw, Shemika Mcgraw, Sirena Mcgraw, Breanne Mcgraw, Simona Mcgraw, Lakenya Mcgraw, Winona Mcgraw, Adam Mcgraw,
Caridad Mcgraw, Deonna Mcgraw, Kimber Mcgraw, Lashaun Mcgraw, Maile Mcgraw, Tomekia Mcgraw, Bernadine Mcgraw, Miesha Mcgraw, Adelina Mcgraw, Tiara Mcgraw, Selene Mcgraw, Shae Mcgraw, Tomi Mcgraw,
Laila Mcgraw, Keith Mcgraw, Lilian Mcgraw, Sherika Mcgraw, Mariela Mcgraw, Moriah Mcgraw, Page Mcgraw, Raegan Mcgraw, Timeka Mcgraw, Tonda Mcgraw, Danya Mcgraw, Kacie Mcgraw, Sabrena Mcgraw, Shala
Mcgraw, Yalonda Mcgraw, Jermaine Mcgraw, Ranee Mcgraw, Sherron Mcgraw, Tameca Mcgraw, Winter Mcgraw, Yulanda Mcgraw, Gabriel Mcgraw, Johna Mcgraw, Keya Mcgraw, Tobie Mcgraw, Johnetta Mcgraw, Rochell
Mcgraw, Celestine Mcgraw, Christinia Mcgraw, Kymberly Mcgraw, Vernita Mcgraw, Jina Mcgraw, Lashawna Mcgraw, Malena Mcgraw, Gilda Mcgraw, Somer Mcgraw, Aliza Mcgraw, Christene Mcgraw, Marilee Mcgraw,
Ricki Mcgraw, Tiffiney Mcgraw, Winifred Mcgraw, Antonio Mcgraw, Cassondra Mcgraw, Suzie Mcgraw, Etta Mcgraw, Toi Mcgraw, Ashli Mcgraw, Charis Mcgraw, Isabelle Mcgraw, Jayna Mcgraw, Nena Mcgraw, Nisha
Mcgraw, Jenise Mcgraw, Lizzie Mcgraw, Maegan Mcgraw, Contina Mcgraw, Joe Mcgraw, Sunday Mcgraw, Tanna Mcgraw, Eleni Mcgraw, Mirella Mcgraw, Concetta Mcgraw, Jeneen Mcgraw, Nerissa Mcgraw, Bethanie
Mcgraw, Latoyia Mcgraw, Maryjane Mcgraw, Treena Mcgraw, Benjamin Mcgraw, Keturah Mcgraw, Paulina Mcgraw, Rasheda Mcgraw, Shaunta Mcgraw, Vickey Mcgraw, Ashanti Mcgraw, Georgiana Mcgraw, Noelia
Mcgraw, Georgianna Mcgraw, Troy Mcgraw, Yulonda Mcgraw, Carley Mcgraw, Stephine Mcgraw, Oralia Mcgraw, Pricilla Mcgraw, Ranae Mcgraw, Lory Mcgraw, Africa Mcgraw, Lekisha Mcgraw, Rosaura Mcgraw,
Shelita Mcgraw, Louann Mcgraw, Shanti Mcgraw, Takesha Mcgraw, Tonie Mcgraw, Santa Mcgraw, Zulema Mcgraw, Dahlia Mcgraw, Jacquelynn Mcgraw, Karon Mcgraw, Teisha Mcgraw, Carlos Mcgraw, Jerrie Mcgraw,
Shyla Mcgraw, Jacqulyn Mcgraw, Janee Mcgraw, Lupita Mcgraw, Ora Mcgraw, Pia Mcgraw, Dodie Mcgraw, Mariam Mcgraw, Meegan Mcgraw, Shanelle Mcgraw, Chanelle Mcgraw, Janetta Mcgraw, Chanell Mcgraw, Hiedi
Mcgraw, Lesia Mcgraw, Nikkia Mcgraw, Ivory Mcgraw, Karry Mcgraw, Marcey Mcgraw, Theodora Mcgraw, Lonnie Mcgraw, Kai Mcgraw, Mina Mcgraw, Seana Mcgraw, Kathlene Mcgraw, Melvina Mcgraw, Temika Mcgraw,
Kala Mcgraw, Leta Mcgraw, Shaunte Mcgraw, Sheron Mcgraw, Tameeka Mcgraw, Eloise Mcgraw, Tatanisha Mcgraw, Catrice Mcgraw, Malaika Mcgraw, Raymond Mcgraw, Sharhonda Mcgraw, Tandra Mcgraw, Tess Mcgraw,
Linsey Mcgraw, Lottie Mcgraw, Miki Mcgraw, Ronni Mcgraw, Tiffini Mcgraw, Jenee Mcgraw, Alta Mcgraw, Elba Mcgraw, Jamillah Mcgraw, Jannie Mcgraw, Loree Mcgraw, Roselyn Mcgraw, Winnie Mcgraw, Alvina
Mcgraw, Dorcas Mcgraw, Laci Mcgraw, Merri Mcgraw, Mickie Mcgraw, Penni Mcgraw, Allegra Mcgraw, Dreama Mcgraw, Nidia Mcgraw, Pattie Mcgraw, Gianna Mcgraw, Kimberlyn Mcgraw, Roshonda Mcgraw, Joel
Mcgraw, Nettie Mcgraw, Blair Mcgraw, Terina Mcgraw, Ebonie Mcgraw, Kera Mcgraw, Ruthann Mcgraw, Randee Mcgraw, Bryan Mcgraw, Jolyn Mcgraw, Stephannie Mcgraw, Delta Mcgraw, Janella Mcgraw, Laquanda
Mcgraw, Lonna Mcgraw, Marcelle Mcgraw, Tavia Mcgraw, Terrell Mcgraw, Chasidy Mcgraw, Danell Mcgraw, Khadijah Mcgraw, Neely Mcgraw, Nicholas Mcgraw, Nicky Mcgraw, Jennine Mcgraw, Tomica Mcgraw,
Xiomara Mcgraw, Deneen Mcgraw, Magaly Mcgraw, Zoraida Mcgraw, Marquetta Mcgraw, Tianna Mcgraw, Annabel Mcgraw, Brie Mcgraw, Caterina Mcgraw, Dennise Mcgraw, Madeleine Mcgraw, Marika Mcgraw, Angelena
Mcgraw, Doretha Mcgraw, Quintina Mcgraw, Rashonda Mcgraw, Carlie Mcgraw, Krysta Mcgraw, Lianne Mcgraw, Martine Mcgraw, Nacole Mcgraw, Regena Mcgraw, Jeanene Mcgraw, Kassie Mcgraw, Larae Mcgraw, Leana
Mcgraw, Ronica Mcgraw, Tenesha Mcgraw, Deeanna Mcgraw, Echo Mcgraw, Melita Mcgraw, Johnny Mcgraw, Lorianne Mcgraw, Makeba Mcgraw, Nakesha Mcgraw, Zenaida Mcgraw, Ophelia Mcgraw, Gennifer Mcgraw,
Rasheedah Mcgraw, Jenette Mcgraw, Lashana Mcgraw, Lashandra Mcgraw, Blanche Mcgraw, Kina Mcgraw, Dia Mcgraw, Emmy Mcgraw, Janina Mcgraw, Krishna Mcgraw, Shawnette Mcgraw, Wenona Mcgraw, Hailey
Mcgraw, Marita Mcgraw, Charita Mcgraw, Diedra Mcgraw, Annika Mcgraw, Gay Mcgraw, Karima Mcgraw, Keila Mcgraw, Lajuana Mcgraw, Marya Mcgraw, Omayra Mcgraw, Carl Mcgraw, Gricelda Mcgraw, Lynsey Mcgraw,
Jama Mcgraw, Lynelle Mcgraw, Telisha Mcgraw, Todd Mcgraw, Camie Mcgraw, Luanne Mcgraw, Mira Mcgraw, Raylene Mcgraw, Rosana Mcgraw, Nakeisha Mcgraw, Brea Mcgraw, Leisha Mcgraw, Maira Mcgraw, Shera
Mcgraw, Tangie Mcgraw, Tari Mcgraw, Treasa Mcgraw, Angle Mcgraw, Jaimi Mcgraw, Joie Mcgraw, Ryann Mcgraw, Sueann Mcgraw, Tiffney Mcgraw, Alida Mcgraw, Jackeline Mcgraw, Tonika Mcgraw, Lakeitha
Mcgraw, Lezlie Mcgraw, Stephania Mcgraw, Zulma Mcgraw, Candra Mcgraw, Charlie Mcgraw, Aleisha Mcgraw, Andreana Mcgraw, Berta Mcgraw, Deon Mcgraw, Gigi Mcgraw, Meri Mcgraw, Cherice Mcgraw, Mya Mcgraw,
Rani Mcgraw, Sunni Mcgraw, Felicity Mcgraw, Marcus Mcgraw, Myisha Mcgraw, Eureka Mcgraw, Gertrude Mcgraw, Lavette Mcgraw, Karisa Mcgraw, Lacresha Mcgraw, Olympia Mcgraw, Bryn Mcgraw, Frank Mcgraw,
Jovita Mcgraw, Leola Mcgraw, Augusta Mcgraw, Karlene Mcgraw, Shaina Mcgraw, Tenille Mcgraw, Fanny Mcgraw, Katerina Mcgraw, Lainie Mcgraw, Natalee Mcgraw, Tandy Mcgraw, Thalia Mcgraw, Elda Mcgraw,
Jacalyn Mcgraw, Beckie Mcgraw, Chenoa Mcgraw, Leeanna Mcgraw, Sharie Mcgraw, Tyisha Mcgraw, Zenobia Mcgraw, Diedre Mcgraw, Charline Mcgraw, Janey Mcgraw, Shanan Mcgraw, Keva Mcgraw, Samuel Mcgraw,
Sandie Mcgraw, Zabrina Mcgraw, Earlene Mcgraw, Jacy Mcgraw, Janessa Mcgraw, Sarai Mcgraw, Sharice Mcgraw, Tambra Mcgraw, Cammy Mcgraw, Lashun Mcgraw, Leonora Mcgraw, Dennis Mcgraw, Lelia Mcgraw,
Cherita Mcgraw, Courtenay Mcgraw, Keira Mcgraw, Tommy Mcgraw, Genia Mcgraw, Jordana Mcgraw, Kaylene Mcgraw, Loni Mcgraw, Mickey Mcgraw, Sheba Mcgraw, Sherell Mcgraw, Anisa Mcgraw, Camelia Mcgraw,
Domenica Mcgraw, Ines Mcgraw, Nathan Mcgraw, Sherrill Mcgraw, Delma Mcgraw, Essie Mcgraw, Lavon Mcgraw, Marilynn Mcgraw, Myrtle Mcgraw, Takiyah Mcgraw, Alba Mcgraw, Delena Mcgraw, Leyla Mcgraw,
Lynell Mcgraw, Tayna Mcgraw, Jazmin Mcgraw, Vannessa Mcgraw, Awilda Mcgraw, Carma Mcgraw, Derrick Mcgraw, Kasi Mcgraw, Malka Mcgraw, Carli Mcgraw, Cordelia Mcgraw, Jesus Mcgraw, Kristeen Mcgraw,
Chrissie Mcgraw, Guillermina Mcgraw, Joana Mcgraw, Lauralee Mcgraw, Serenity Mcgraw, Alethia Mcgraw, Dulce Mcgraw, Lavinia Mcgraw, Nilsa Mcgraw, Tinisha Mcgraw, Veda Mcgraw, Demetris Mcgraw, Jamilah
Mcgraw, Janean Mcgraw, Soledad Mcgraw, Talisa Mcgraw, Yael Mcgraw, Adeline Mcgraw, Amparo Mcgraw, Shannah Mcgraw, Anel Mcgraw, Corissa Mcgraw, Myriam Mcgraw, September Mcgraw, Shawntel Mcgraw,
Camisha Mcgraw, Cathie Mcgraw, Reanna Mcgraw, Shavonda Mcgraw, Azure Mcgraw, Darnell Mcgraw, Delphine Mcgraw, Keren Mcgraw, Lashondra Mcgraw, Yajaira Mcgraw, Yessenia Mcgraw, Adelaida Mcgraw, Aura
Mcgraw, Dorthy Mcgraw, Karly Mcgraw, Latonja Mcgraw, Mendi Mcgraw, Wende Mcgraw, Adriene Mcgraw, Arin Mcgraw, Cinda Mcgraw, Jeanmarie Mcgraw, Khristina Mcgraw, Rory Mcgraw, Shanel Mcgraw, Zena
Mcgraw, Astrid Mcgraw, Darice Mcgraw, Melia Mcgraw, Nell Mcgraw, Sage Mcgraw, Tedra Mcgraw, Tyler Mcgraw, Inger Mcgraw, Lekesha Mcgraw, Celine Mcgraw, Marchelle Mcgraw, Shaundra Mcgraw, Bethanne
Mcgraw, Kathryne Mcgraw, Yanira Mcgraw, Jacquetta Mcgraw, Jessika Mcgraw, Niesha Mcgraw, Shantae Mcgraw, Tira Mcgraw, Elke Mcgraw, Freida Mcgraw, Gretta Mcgraw, Kerstin Mcgraw, San Mcgraw, Suzannah
Mcgraw, Tamira Mcgraw, Zina Mcgraw, Charmain Mcgraw, Cherisse Mcgraw, Kylene Mcgraw, Lacretia Mcgraw, Mavis Mcgraw, Telisa Mcgraw, Alysa Mcgraw, Carlita Mcgraw, Christianne Mcgraw, Franchesca Mcgraw,
Joetta Mcgraw, Lashelle Mcgraw, Rosamaria Mcgraw, Seema Mcgraw, Youlanda Mcgraw, Clarisa Mcgraw, Donella Mcgraw, Markita Mcgraw, Mesha Mcgraw, Poppy Mcgraw, Arleen Mcgraw, Cecile Mcgraw, Krisha
Mcgraw, Mckenzie Mcgraw, Sharmaine Mcgraw, Willa Mcgraw, Anessa Mcgraw, Donetta Mcgraw, Donnetta Mcgraw, Leora Mcgraw, Christena Mcgraw, Dorie Mcgraw, Eddie Mcgraw, Herlinda Mcgraw, Lise Mcgraw,
Rosalee Mcgraw, Temple Mcgraw, Tonette Mcgraw, Cheryle Mcgraw, Cleo Mcgraw, Elma Mcgraw, Erlinda Mcgraw, Peter Mcgraw, Sparkle Mcgraw, Evie Mcgraw, Lavina Mcgraw, Marna Mcgraw, Meranda Mcgraw,
Tynisha Mcgraw, Alyse Mcgraw, Anisha Mcgraw, Bevin Mcgraw, Glory Mcgraw, Jonnie Mcgraw, Marinda Mcgraw, Natascha Mcgraw, Shanetta Mcgraw, Angelle Mcgraw, Dion Mcgraw, Quinn Mcgraw, Shanette Mcgraw,
Shaniqua Mcgraw, Shantay Mcgraw, Tamarra Mcgraw, Diona Mcgraw, Hortencia Mcgraw, Latara Mcgraw, Lisamarie Mcgraw, Sidney Mcgraw, Alessandra Mcgraw, Idalia Mcgraw, Malina Mcgraw, Sindy Mcgraw,
Antonella Mcgraw, Kasha Mcgraw, Laquinta Mcgraw, Tysha Mcgraw, Adena Mcgraw, Aide Mcgraw, Bernita Mcgraw, Bettie Mcgraw, Delinda Mcgraw, Lashundra Mcgraw, Sanjuana Mcgraw, Susann Mcgraw, Vilma
Mcgraw, Deandrea Mcgraw, Devorah Mcgraw, Marisha Mcgraw, Reena Mcgraw, Takeisha Mcgraw, Tora Mcgraw, Carlyn Mcgraw, Carrieann Mcgraw, Chanta Mcgraw, China Mcgraw, Rhianna Mcgraw, Sharleen Mcgraw,
Jessenia Mcgraw, Kemberly Mcgraw, Rosalina Mcgraw, Rosella Mcgraw, Latrese Mcgraw, Mecca Mcgraw, Nneka Mcgraw, Alva Mcgraw, Douglas Mcgraw, Mandee Mcgraw, Sherese Mcgraw, Taya Mcgraw, Tenika Mcgraw,
Donelle Mcgraw, Frieda Mcgraw, Joye Mcgraw, Kayce Mcgraw, Sammie Mcgraw, Shereen Mcgraw, Tomiko Mcgraw, Tommi Mcgraw, Marleen Mcgraw, Shonte Mcgraw, Avery Mcgraw, Delfina Mcgraw, Dollie Mcgraw,
Kaycee Mcgraw, Lakesia Mcgraw, Penney Mcgraw, Tinika Mcgraw, Tywanda Mcgraw, Betina Mcgraw, Debrah Mcgraw, Latia Mcgraw, Lynetta Mcgraw, Michaelle Mcgraw, Shunta Mcgraw, Soraya Mcgraw, Anastacia
Mcgraw, Charisma Mcgraw, Jonie Mcgraw, Shunda Mcgraw, Terah Mcgraw, Trinidad Mcgraw, Valeri Mcgraw, Charlena Mcgraw, Elizebeth Mcgraw, Joella Mcgraw, Johnette Mcgraw, Nikol Mcgraw, Rosalynn Mcgraw,
Aime Mcgraw, Aviva Mcgraw, Joselyn Mcgraw, Lashanna Mcgraw, Tarrah Mcgraw, Candyce Mcgraw, Estrella Mcgraw, Mable Mcgraw, Renay Mcgraw, Shontel Mcgraw, Wendee Mcgraw, Angelene Mcgraw, Karli Mcgraw,
Maricruz Mcgraw, Minda Mcgraw, Robbi Mcgraw, Towana Mcgraw, Charolette Mcgraw, Kandis Mcgraw, Keeley Mcgraw, Kimbley Mcgraw, Kristene Mcgraw, Lashell Mcgraw, Merissa Mcgraw, Taneshia Mcgraw, Tristan
Mcgraw, Vida Mcgraw, Zelda Mcgraw, Angele Mcgraw, Beronica Mcgraw, Damita Mcgraw, Debbra Mcgraw, Lanetta Mcgraw, Priya Mcgraw, Tamia Mcgraw, Arlinda Mcgraw, Armida Mcgraw, Chere Mcgraw, Errin Mcgraw,
Kamala Mcgraw, Lashan Mcgraw, Luana Mcgraw, Yalanda Mcgraw, Antonietta Mcgraw, Colby Mcgraw, Jona Mcgraw, Keitha Mcgraw, Dava Mcgraw, Herminia Mcgraw, Allie Mcgraw, Dawnmarie Mcgraw, Eula Mcgraw,
Monet Mcgraw, Shirlene Mcgraw, Tarina Mcgraw, Yasmine Mcgraw, Cassaundra Mcgraw, Chelsie Mcgraw, Fern Mcgraw, Rosina Mcgraw, Dyana Mcgraw, Lavern Mcgraw, Mario Mcgraw, Ollie Mcgraw, Rodney Mcgraw,
Bridgit Mcgraw, Julieta Mcgraw, Korie Mcgraw, Sumer Mcgraw, Arianna Mcgraw, Carmon Mcgraw, Chassidy Mcgraw, Ivana Mcgraw, Jerusha Mcgraw, Marne Mcgraw, Melisha Mcgraw, Regenia Mcgraw, Sarena Mcgraw,
Shanae Mcgraw, Tona Mcgraw, Cherese Mcgraw, Deitra Mcgraw, Dustin Mcgraw, Kory Mcgraw, Suzana Mcgraw, Synthia Mcgraw, Zakia Mcgraw, Lisbeth Mcgraw, Matilde Mcgraw, Terresa Mcgraw, Tiffeny Mcgraw,
Toinette Mcgraw, Charmin Mcgraw, Julienne Mcgraw, Marcelina Mcgraw, Kama Mcgraw, Katisha Mcgraw, Tiffiany Mcgraw, Tobey Mcgraw, Adelita Mcgraw, Atiya Mcgraw, Bradley Mcgraw, Cathi Mcgraw, Dominica
Mcgraw, Jeanelle Mcgraw, Jeffery Mcgraw, Rachal Mcgraw, Sammi Mcgraw, Anamaria Mcgraw, Arcelia Mcgraw, Isabella Mcgraw, Janay Mcgraw, Janett Mcgraw, Jonell Mcgraw, Lakeya Mcgraw, Leigha Mcgraw,
Alexander Mcgraw, Denae Mcgraw, Dustie Mcgraw, Felicita Mcgraw, Freedom Mcgraw, Harriett Mcgraw, Katia Mcgraw, Londa Mcgraw, Marlyn Mcgraw, Tereasa Mcgraw, Analisa Mcgraw, Annissa Mcgraw, August
Mcgraw, Jenae Mcgraw, Jenice Mcgraw, Kamie Mcgraw, Shanee Mcgraw, Sherise Mcgraw, Tamieka Mcgraw, Karianne Mcgraw, Naima Mcgraw, Natarsha Mcgraw, Vita Mcgraw, Anjali Mcgraw, Ena Mcgraw, Jaci Mcgraw,
Jolanda Mcgraw, Kimi Mcgraw, Celestina Mcgraw, Devonna Mcgraw, Latrece Mcgraw, Leatrice Mcgraw, Lucila Mcgraw, Mignon Mcgraw, Stefany Mcgraw, Tamyra Mcgraw, Tracye Mcgraw, Beulah Mcgraw, Danetta
Mcgraw, Dede Mcgraw, Dwana Mcgraw, Julisa Mcgraw, Kanisha Mcgraw, Lanae Mcgraw, Ronette Mcgraw, Sandee Mcgraw, Veronique Mcgraw, Deirdra Mcgraw, Ember Mcgraw, Kerin Mcgraw, Lekeisha Mcgraw, Letisia
Mcgraw, Pandora Mcgraw, Zaneta Mcgraw, Earline Mcgraw, Evon Mcgraw, Lisandra Mcgraw, Lura Mcgraw, Raelene Mcgraw, Shontell Mcgraw, Ila Mcgraw, Kamisha Mcgraw, Tinamarie Mcgraw, Tish Mcgraw, Jay
Mcgraw, Kayleen Mcgraw, Maryam Mcgraw, Phillip Mcgraw, Santina Mcgraw, Taneisha Mcgraw, Zakiya Mcgraw, Brandice Mcgraw, Jennifier Mcgraw, Mariaelena Mcgraw, Mirian Mcgraw, Shalene Mcgraw, Tammera
Mcgraw, Tesa Mcgraw, Zaida Mcgraw, Anjelica Mcgraw, Arika Mcgraw, Babette Mcgraw, Cameo Mcgraw, Dwan Mcgraw, Kiera Mcgraw, Michella Mcgraw, Nekia Mcgraw, Rashelle Mcgraw, Robynn Mcgraw, Sheilah
Mcgraw, Velia Mcgraw, Betsey Mcgraw, Chiara Mcgraw, Kortney Mcgraw, Tressie Mcgraw, Tywana Mcgraw, Valery Mcgraw, Arnetta Mcgraw, Carlee Mcgraw, Craig Mcgraw, Debbi Mcgraw, Genesis Mcgraw, Genny
Mcgraw, Jori Mcgraw, Kandie Mcgraw, Lorin Mcgraw, Melodee Mcgraw, Cherlyn Mcgraw, Karleen Mcgraw, Windi Mcgraw, Beverley Mcgraw, Loreen Mcgraw, Malynda Mcgraw, Nada Mcgraw, Nichola Mcgraw, Sakinah
Mcgraw, Steffany Mcgraw, Tunisia Mcgraw, Vanita Mcgraw, Gemma Mcgraw, Jennell Mcgraw, Lakecia Mcgraw, Shameeka Mcgraw, Towanna Mcgraw, Tywanna Mcgraw, Vanesa Mcgraw, Dea Mcgraw, Desire Mcgraw, Donnie
Mcgraw, Kiki Mcgraw, Makesha Mcgraw, Rebeccah Mcgraw, Anglia Mcgraw, Branda Mcgraw, Kawanna Mcgraw, Tawnia Mcgraw, Debi Mcgraw, Elysia Mcgraw, Lorretta Mcgraw, Renetta Mcgraw, Tamu Mcgraw, Anneliese
Mcgraw, Devan Mcgraw, Rochel Mcgraw, Vernetta Mcgraw, Anetra Mcgraw, Brynn Mcgraw, Claribel Mcgraw, Denielle Mcgraw, Eugena Mcgraw, Fara Mcgraw, Genie Mcgraw, Joya Mcgraw, Keisa Mcgraw, Loria Mcgraw,
Tahira Mcgraw, Tawna Mcgraw, Angila Mcgraw, Chara Mcgraw, Cybil Mcgraw, Jimmy Mcgraw, Kendal Mcgraw, Serita Mcgraw, Tova Mcgraw, Vernice Mcgraw, Cruz Mcgraw, Jameelah Mcgraw, Karan Mcgraw, Kyna
Mcgraw, Larita Mcgraw, Maricella Mcgraw, Mariel Mcgraw, Elyssa Mcgraw, Kathyrn Mcgraw, Raechel Mcgraw, Jolee Mcgraw, Earnestine Mcgraw, Livia Mcgraw, Nechama Mcgraw, Shawndra Mcgraw, Antonina Mcgraw,
Chassity Mcgraw, Codi Mcgraw, Georgetta Mcgraw, Gwyn Mcgraw, Jacqualine Mcgraw, Latania Mcgraw, Tawni Mcgraw, Tiwana Mcgraw, Toyia Mcgraw, Willette Mcgraw, Anette Mcgraw, Brandye Mcgraw, Danny
Mcgraw, Enedina Mcgraw, Korina Mcgraw, Latisa Mcgraw, Vasiliki Mcgraw, Victor Mcgraw, Daneen Mcgraw, Elnora Mcgraw, Shareen Mcgraw, Amey Mcgraw, Carroll Mcgraw, Channon Mcgraw, Charletta Mcgraw,
Deserie Mcgraw, Jovan Mcgraw, Latrica Mcgraw, Lorina Mcgraw, Love Mcgraw, Carolann Mcgraw, Fran Mcgraw, Ilona Mcgraw, Kandra Mcgraw, Keela Mcgraw, Lasonia Mcgraw, Leatha Mcgraw, Lizzette Mcgraw, Macy
Mcgraw, Magan Mcgraw, Rebekka Mcgraw, Vanetta Mcgraw, Azucena Mcgraw, Brigit Mcgraw, Jamaica Mcgraw, Jameka Mcgraw, Sheridan Mcgraw, Aliya Mcgraw, Chantay Mcgraw, Giuseppina Mcgraw, Lyndi Mcgraw,
Ragan Mcgraw, Retha Mcgraw, Bari Mcgraw, Daryl Mcgraw, Elonda Mcgraw, Grisel Mcgraw, Joline Mcgraw, Korey Mcgraw, Milinda Mcgraw, Rubi Mcgraw, Sabine Mcgraw, Silvana Mcgraw, Turkessa Mcgraw, Myla
Mcgraw, Sharese Mcgraw, Cheron Mcgraw, Joell Mcgraw, Lachandra Mcgraw, Lateefah Mcgraw, Sanya Mcgraw, Wesley Mcgraw, Amira Mcgraw, Imani Mcgraw, Joette Mcgraw, Kimmie Mcgraw, Lynna Mcgraw, Sheneka
Mcgraw, Sunnie Mcgraw, Andi Mcgraw, Angelyn Mcgraw, Maurice Mcgraw, Melodi Mcgraw, Nelida Mcgraw, Sherida Mcgraw, Shronda Mcgraw, Takia Mcgraw, Toshiba Mcgraw, Trishia Mcgraw, Alonda Mcgraw, Araseli
Mcgraw, Laketa Mcgraw, Lesha Mcgraw, Magen Mcgraw, Neisha Mcgraw, Selma Mcgraw, Shannen Mcgraw, Toria Mcgraw, Aline Mcgraw, Aubree Mcgraw, Delaina Mcgraw, Joannie Mcgraw, Mai Mcgraw, Niya Mcgraw,
Dimitra Mcgraw, Karl Mcgraw, Kimberle Mcgraw, Quanda Mcgraw, Alica Mcgraw, Chaundra Mcgraw, Cherly Mcgraw, Donnell Mcgraw, Ellyn Mcgraw, Lanell Mcgraw, Lawrence Mcgraw, Tonita Mcgraw, Vania Mcgraw,
Genea Mcgraw, Kathaleen Mcgraw, Laquetta Mcgraw, Lorien Mcgraw, Mallory Mcgraw, Margery Mcgraw, Porsha Mcgraw, Agatha Mcgraw, Alayna Mcgraw, Barbi Mcgraw, Fatimah Mcgraw, Jaclynn Mcgraw, Kalisha
Mcgraw, Liesl Mcgraw, Alita Mcgraw, Janita Mcgraw, Ola Mcgraw, Shawne Mcgraw, Talena Mcgraw, Angla Mcgraw, Autum Mcgraw, Bertina Mcgraw, Samira Mcgraw, Shalena Mcgraw, Consuella Mcgraw, Corin Mcgraw,
Delora Mcgraw, Lanna Mcgraw, Pat Mcgraw, Roshunda Mcgraw, Jonette Mcgraw, Latishia Mcgraw, Reginia Mcgraw, Ronnette Mcgraw, Sherree Mcgraw, Tamicka Mcgraw, Alejandrina Mcgraw, Carlisa Mcgraw, Jene
Mcgraw, Loralee Mcgraw, Lorenza Mcgraw, Luis Mcgraw, Yasmeen Mcgraw, Zakiyyah Mcgraw, Ashly Mcgraw, Cassy Mcgraw, Chari Mcgraw, Indira Mcgraw, Ivelisse Mcgraw, Jaymi Mcgraw, Karalee Mcgraw, Lauretta
Mcgraw, Raelynn Mcgraw, Tifany Mcgraw, Torrey Mcgraw, Amyjo Mcgraw, Annalee Mcgraw, Hester Mcgraw, Josefa Mcgraw, Latresa Mcgraw, Lita Mcgraw, Marysol Mcgraw, Mindie Mcgraw, Shan Mcgraw, Susy Mcgraw,
Teneka Mcgraw, Arnita Mcgraw, Charice Mcgraw, Cleopatra Mcgraw, Cotina Mcgraw, Dawnette Mcgraw, Julene Mcgraw, Lashay Mcgraw, Latrell Mcgraw, Lucrecia Mcgraw, Ricky Mcgraw, Shamekia Mcgraw, Camellia
Mcgraw, Cassi Mcgraw, Curtis Mcgraw, Hana Mcgraw, Isha Mcgraw, Jacquelyne Mcgraw, Jenica Mcgraw, Norah Mcgraw, Oneida Mcgraw, Subrina Mcgraw, Tyrone Mcgraw, Angi Mcgraw, Anjeanette Mcgraw, Bella
Mcgraw, Donica Mcgraw, Kareen Mcgraw, Laina Mcgraw, Luci Mcgraw, Mischelle Mcgraw, Tanita Mcgraw, Heaven Mcgraw, Madelin Mcgraw, Michaele Mcgraw, Rashell Mcgraw, Tamarah Mcgraw, Viki Mcgraw, Candee
Mcgraw, Danyale Mcgraw, Karis Mcgraw, Mishelle Mcgraw, Shanequa Mcgraw, Shellee Mcgraw, Despina Mcgraw, Dodi Mcgraw, Katherin Mcgraw, Mechele Mcgraw, Pebbles Mcgraw, Roshelle Mcgraw, Carry Mcgraw,
Denine Mcgraw, Dory Mcgraw, Jolinda Mcgraw, Julieanne Mcgraw, Karlyn Mcgraw, Kathern Mcgraw, Kerensa Mcgraw, Kerrin Mcgraw, Lafonda Mcgraw, Launa Mcgraw, Marin Mcgraw, Ona Mcgraw, Sheli Mcgraw,
Trixie Mcgraw, Yahaira Mcgraw, Casi Mcgraw, Ciara Mcgraw, Darline Mcgraw, Drema Mcgraw, Francie Mcgraw, Gerald Mcgraw, Khara Mcgraw, Lashone Mcgraw, Meloney Mcgraw, Melyssa Mcgraw, Merrie Mcgraw,
Neysa Mcgraw, Queen Mcgraw, Rashawn Mcgraw, Robbyn Mcgraw, Sharilyn Mcgraw, Sharyl Mcgraw, Tamitha Mcgraw, Tresha Mcgraw, Amaris Mcgraw, Bernetta Mcgraw, Channel Mcgraw, Corrin Mcgraw, Deadra Mcgraw,
Filomena Mcgraw, Jaye Mcgraw, Kerriann Mcgraw, Kysha Mcgraw, Lanie Mcgraw, Lashauna Mcgraw, Lashona Mcgraw, Laurinda Mcgraw, Monifa Mcgraw, Sharri Mcgraw, Shontae Mcgraw, Terrica Mcgraw, Thresa
Mcgraw, Tondra Mcgraw, Geralyn Mcgraw, Ginamarie Mcgraw, Jesseca Mcgraw, Lasonja Mcgraw, Marica Mcgraw, Marlen Mcgraw, Virgie Mcgraw, Genna Mcgraw, Kriste Mcgraw, Kyndra Mcgraw, Tequilla Mcgraw,
Vernell Mcgraw, Zoila Mcgraw, Aiesha Mcgraw, Andera Mcgraw, Blake Mcgraw, Carlena Mcgraw, Deangela Mcgraw, Lavonia Mcgraw, Lesly Mcgraw, Savannah Mcgraw, Danisha Mcgraw, Dierdre Mcgraw, Hermelinda
Mcgraw, Kam Mcgraw, Milena Mcgraw, Omega Mcgraw, Rosaria Mcgraw, Russell Mcgraw, Shaquita Mcgraw, Tashika Mcgraw, Arlena Mcgraw, Delisha Mcgraw, Demetric Mcgraw, Lakendra Mcgraw, Marilu Mcgraw, Neha
Mcgraw, Terria Mcgraw, Carita Mcgraw, Kallie Mcgraw, Makisha Mcgraw, Martie Mcgraw, Mitzie Mcgraw, Stephane Mcgraw, Donette Mcgraw, Joli Mcgraw, Kathlyn Mcgraw, Keona Mcgraw, Lakishia Mcgraw, Laurice
Mcgraw, Melena Mcgraw, Nan Mcgraw, Rafaela Mcgraw, Taura Mcgraw, Wynette Mcgraw, Arielle Mcgraw, Carlette Mcgraw, Cyrstal Mcgraw, Dewanda Mcgraw, Kimmy Mcgraw, Pauletta Mcgraw, Shannel Mcgraw, Shelbi
Mcgraw, Camillia Mcgraw, Charlyn Mcgraw, Christianna Mcgraw, Gerry Mcgraw, Jamy Mcgraw, Karoline Mcgraw, Katrinia Mcgraw, Lorissa Mcgraw, Magdalene Mcgraw, Mariza Mcgraw, Markisha Mcgraw, Mindee
Mcgraw, Terica Mcgraw, Terisa Mcgraw, Tyronda Mcgraw, Aracelis Mcgraw, Betzaida Mcgraw, Charon Mcgraw, Chastidy Mcgraw, Henry Mcgraw, Jamika Mcgraw, Janiece Mcgraw, Kariann Mcgraw, Kawanda Mcgraw,
Saprina Mcgraw, Chera Mcgraw, Genoveva Mcgraw, Jacey Mcgraw, Jannifer Mcgraw, Jaylene Mcgraw, Meisha Mcgraw, Daphney Mcgraw, Delois Mcgraw, Denelle Mcgraw, Diamond Mcgraw, Felita Mcgraw, Hilarie
Mcgraw, Jamica Mcgraw, Joelene Mcgraw, Kisa Mcgraw, Lilliana Mcgraw, Sulema Mcgraw, Wendolyn Mcgraw, Calista Mcgraw, Darrell Mcgraw, Elinor Mcgraw, Faviola Mcgraw, Jenean Mcgraw, Lalena Mcgraw,
Mischa Mcgraw, Tamikia Mcgraw, Alanda Mcgraw, Ashia Mcgraw, Bryna Mcgraw, Serene Mcgraw, Shauntel Mcgraw, Sheria Mcgraw, Talya Mcgraw, Thersa Mcgraw, Walter Mcgraw, Yevette Mcgraw, Yolande Mcgraw,
Ashlea Mcgraw, Gaylene Mcgraw, Khadija Mcgraw, Latitia Mcgraw, Micheline Mcgraw, Monalisa Mcgraw, Sakina Mcgraw, Stacye Mcgraw, Taffy Mcgraw, Tifani Mcgraw, Wilhelmina Mcgraw, Amara Mcgraw, Gypsy
Mcgraw, Joely Mcgraw, Kameron Mcgraw, Keasha Mcgraw, Louella Mcgraw, Riki Mcgraw, Torey Mcgraw, Adrien Mcgraw, Arwen Mcgraw, Crysta Mcgraw, Delaine Mcgraw, Denyse Mcgraw, Georgie Mcgraw, Janea
Mcgraw, Medina Mcgraw, Meka Mcgraw, Nika Mcgraw, Rashanda Mcgraw, Corena Mcgraw, Denee Mcgraw, Kimya Mcgraw, Lanisha Mcgraw, Luella Mcgraw, Marylin Mcgraw, Merritt Mcgraw, Mysti Mcgraw, Shawntay
Mcgraw, Shirelle Mcgraw, Siri Mcgraw, Cayce Mcgraw, Chava Mcgraw, Jovanna Mcgraw, Lorine Mcgraw, Maryelizabeth Mcgraw, Sonji Mcgraw, Titania Mcgraw, Brent Mcgraw, Jamye Mcgraw, Kela Mcgraw, Korin
Mcgraw, Monette Mcgraw, Rebbeca Mcgraw, Rebeka Mcgraw, Stormie Mcgraw, Angelo Mcgraw, Camela Mcgraw, Darlena Mcgraw, Dietra Mcgraw, Gwendolen Mcgraw, Kenia Mcgraw, Kesia Mcgraw, Kolleen Mcgraw,
Leighanne Mcgraw, Lindi Mcgraw, Marcee Mcgraw, Miya Mcgraw, Sami Mcgraw, Tameko Mcgraw, Tamora Mcgraw, Tanaya Mcgraw, Tiki Mcgraw, Toniann Mcgraw, Jeaneen Mcgraw, Lizeth Mcgraw, Manuel Mcgraw,
Marlina Mcgraw, Monisha Mcgraw, Nichele Mcgraw, Rechelle Mcgraw, Richele Mcgraw, Shenna Mcgraw, Sky Mcgraw, Tahnee Mcgraw, Brita Mcgraw, Erikka Mcgraw, Jani Mcgraw, Kattie Mcgraw, Kimbra Mcgraw,
Quincy Mcgraw, Shaquana Mcgraw, Sherelle Mcgraw, Sherryl Mcgraw, Shireen Mcgraw, Theressa Mcgraw, Yecenia Mcgraw, Albert Mcgraw, Ansley Mcgraw, Clair Mcgraw, Danille Mcgraw, Evelia Mcgraw, Jacob
Mcgraw, Jon Mcgraw, Katrin Mcgraw, Kiva Mcgraw, Mirta Mcgraw, Pearlie Mcgraw, Roger Mcgraw, Shawntell Mcgraw, Terrance Mcgraw, Vincent Mcgraw, Aimie Mcgraw, Alise Mcgraw, Caryl Mcgraw, Devora Mcgraw,
Dian Mcgraw, Genelle Mcgraw, Johannah Mcgraw, Lanora Mcgraw, Memory Mcgraw, Mila Mcgraw, Shandy Mcgraw, Shella Mcgraw, Shila Mcgraw, Tahisha Mcgraw, Channa Mcgraw, Christeen Mcgraw, Janise Mcgraw,
Kristee Mcgraw, Melaney Mcgraw, Tanis Mcgraw, Teasha Mcgraw, Arthur Mcgraw, Derek Mcgraw, Ginette Mcgraw, Kamesha Mcgraw, Keriann Mcgraw, Lateisha Mcgraw, Merrilee Mcgraw, Nikesha Mcgraw, Sherrita
Mcgraw, Shontay Mcgraw, Bess Mcgraw, Candelaria Mcgraw, Charo Mcgraw, Dominga Mcgraw, Kateri Mcgraw, Kelsie Mcgraw, Kenyata Mcgraw, Lane Mcgraw, Lateshia Mcgraw, Lili Mcgraw, Linnette Mcgraw, Odette
Mcgraw, Sharika Mcgraw, Sherra Mcgraw, Tama Mcgraw, Cicily Mcgraw, Clarinda Mcgraw, Corri Mcgraw, Donnita Mcgraw, Duana Mcgraw, Gisele Mcgraw, Laquitta Mcgraw, Mahogany Mcgraw, Seanna Mcgraw, Sena
Mcgraw, Shelbie Mcgraw, Steffani Mcgraw, Tauna Mcgraw, Zara Mcgraw, Aminah Mcgraw, Amye Mcgraw, Damon Mcgraw, Doria Mcgraw, Frederica Mcgraw, Hollis Mcgraw, Katey Mcgraw, Nila Mcgraw, Pamella Mcgraw,
Ray Mcgraw, Taina Mcgraw, Zenia Mcgraw, Ananda Mcgraw, Becca Mcgraw, Dessie Mcgraw, Donnette Mcgraw, Marney Mcgraw, Nekisha Mcgraw, Sharell Mcgraw, Stefania Mcgraw, Tandi Mcgraw, Allana Mcgraw,
Andree Mcgraw, Arlette Mcgraw, Denisa Mcgraw, Ieshia Mcgraw, Latrecia Mcgraw, Meryl Mcgraw, Shakia Mcgraw, Shaye Mcgraw, Clover Mcgraw, Cresta Mcgraw, Francina Mcgraw, Fredericka Mcgraw, Kimbery
Mcgraw, Shameika Mcgraw, Sherene Mcgraw, Armanda Mcgraw, Cally Mcgraw, Cydney Mcgraw, Dasha Mcgraw, Gwynne Mcgraw, Josephina Mcgraw, Lavonna Mcgraw, Melessa Mcgraw, Nycole Mcgraw, Nyoka Mcgraw, Pansy
Mcgraw, Shakita Mcgraw, Sonjia Mcgraw, Wanita Mcgraw, Xenia Mcgraw, Any Mcgraw, Bobi Mcgraw, Bonni Mcgraw, Chevelle Mcgraw, Chrystie Mcgraw, Genine Mcgraw, Iona Mcgraw, Kezia Mcgraw, Latresha Mcgraw,
Lucie Mcgraw, Malikah Mcgraw, Micha Mcgraw, Reginald Mcgraw, Ronita Mcgraw, Sharina Mcgraw, Sharise Mcgraw, Stormi Mcgraw, Tammey Mcgraw, Angelika Mcgraw, Anja Mcgraw, Carianne Mcgraw, Dalene Mcgraw,
Florinda Mcgraw, Irasema Mcgraw, Karren Mcgraw, Kedra Mcgraw, Koreen Mcgraw, Leena Mcgraw, Libra Mcgraw, Moncia Mcgraw, Rosaline Mcgraw, Roshawn Mcgraw, Rusty Mcgraw, Sonda Mcgraw, Talina Mcgraw,
Trudie Mcgraw, Audry Mcgraw, Celinda Mcgraw, Deona Mcgraw, Jennipher Mcgraw, Jolena Mcgraw, Kimerly Mcgraw, Kym Mcgraw, Laquana Mcgraw, Madalyn Mcgraw, Renate Mcgraw, Una Mcgraw, Barrie Mcgraw,
Chanin Mcgraw, Deven Mcgraw, Hadley Mcgraw, Hether Mcgraw, Ninfa Mcgraw, Sha Mcgraw, Sloane Mcgraw, Tamla Mcgraw, Amada Mcgraw, Belynda Mcgraw, Celesta Mcgraw, Chelsa Mcgraw, Corene Mcgraw, Dawana
Mcgraw, Elly Mcgraw, Laure Mcgraw, Pascha Mcgraw, Shenell Mcgraw, Tashana Mcgraw, Trecia Mcgraw, Angelea Mcgraw, Dominque Mcgraw, Frederick Mcgraw, Ira Mcgraw, Karine Mcgraw, Lachanda Mcgraw, Lyla
Mcgraw, Melanee Mcgraw, Nikkole Mcgraw, Ronetta Mcgraw, Shanise Mcgraw, Shiree Mcgraw, Vincenza Mcgraw, Yolando Mcgraw, Adelle Mcgraw, Ara Mcgraw, Bria Mcgraw, Cheryll Mcgraw, Delila Mcgraw, Delina
Mcgraw, Jonetta Mcgraw, Kamela Mcgraw, Lacinda Mcgraw, Lashanta Mcgraw, Laurene Mcgraw, Lenita Mcgraw, Martin Mcgraw, Neda Mcgraw, Sharen Mcgraw, Tanishia Mcgraw, Bronwen Mcgraw, Jamilla Mcgraw,
Jennelle Mcgraw, Lavetta Mcgraw, Makeda Mcgraw, Omaira Mcgraw, Randie Mcgraw, Ricci Mcgraw, Romy Mcgraw, Sherlyn Mcgraw, Sonal Mcgraw, Tierra Mcgraw, Tonyia Mcgraw, Caressa Mcgraw, Carmina Mcgraw,
Dann Mcgraw, Dawnelle Mcgraw, Isa Mcgraw, Lameka Mcgraw, Lavada Mcgraw, Lavita Mcgraw, Lessie Mcgraw, Miracle Mcgraw, Mylinda Mcgraw, Nancie Mcgraw, Raeanne Mcgraw, Shanique Mcgraw, Sharolyn Mcgraw,
Alex Mcgraw, Cheyanne Mcgraw, Christan Mcgraw, Halima Mcgraw, Karlee Mcgraw, Katherina Mcgraw, Lisabeth Mcgraw, Marylynn Mcgraw, Naimah Mcgraw, Ria Mcgraw, Tierney Mcgraw, Velda Mcgraw, Abbe Mcgraw,
Alane Mcgraw, Belia Mcgraw, Falisha Mcgraw, Kaylee Mcgraw, Latausha Mcgraw, Lynnea Mcgraw, Myranda Mcgraw, Nyla Mcgraw, Rima Mcgraw, Seneca Mcgraw, Aprille Mcgraw, Carlina Mcgraw, Christle Mcgraw,
Corliss Mcgraw, Crystle Mcgraw, Desha Mcgraw, Kea Mcgraw, Kristena Mcgraw, Kristiana Mcgraw, Lari Mcgraw, Lorita Mcgraw, Marlaina Mcgraw, Marquitta Mcgraw, Ondrea Mcgraw, Tashara Mcgraw, Devonne
Mcgraw, Domonique Mcgraw, Jannet Mcgraw, Kila Mcgraw, Krysten Mcgraw, Leslye Mcgraw, Lynnae Mcgraw, Shela Mcgraw, Sherica Mcgraw, Timberly Mcgraw, Arminda Mcgraw, Essence Mcgraw, Gala Mcgraw, Hellen
Mcgraw, Jannine Mcgraw, Kamille Mcgraw, Lalita Mcgraw, Layne Mcgraw, Letecia Mcgraw, Panagiota Mcgraw, Philana Mcgraw, Trisa Mcgraw, Akisha Mcgraw, Audria Mcgraw, Berenice Mcgraw, Bette Mcgraw,
Denese Mcgraw, Donia Mcgraw, Johana Mcgraw, Kamika Mcgraw, Katelyn Mcgraw, Laine Mcgraw, Lajuan Mcgraw, Lasondra Mcgraw, Marketta Mcgraw, Patrizia Mcgraw, Sascha Mcgraw, Shamara Mcgraw, Shamica
Mcgraw, Tirzah Mcgraw, Trinia Mcgraw, Tyla Mcgraw, Venice Mcgraw, Carline Mcgraw, Deyanira Mcgraw, Isabell Mcgraw, Jackqueline Mcgraw, Kendria Mcgraw, Marlee Mcgraw, Sharisse Mcgraw, Shawntae Mcgraw,
Sholanda Mcgraw, Thomasine Mcgraw, Tikisha Mcgraw, Tyna Mcgraw, Aria Mcgraw, Colene Mcgraw, Daphine Mcgraw, Dorthea Mcgraw, Genise Mcgraw, Keia Mcgraw, Leasa Mcgraw, Nohemi Mcgraw, Rosann Mcgraw,
Ruthanne Mcgraw, Shundra Mcgraw, Taria Mcgraw, Valinda Mcgraw, Adrain Mcgraw, Brunilda Mcgraw, Caralee Mcgraw, Charlee Mcgraw, Katasha Mcgraw, Krystina Mcgraw, Krystyna Mcgraw, Lovie Mcgraw,
Marquette Mcgraw, Marybel Mcgraw, Melvin Mcgraw, Nathaniel Mcgraw, Rickie Mcgraw, Romana Mcgraw, Tene Mcgraw, Tinesha Mcgraw, Tyann Mcgraw, Wednesday Mcgraw, Abra Mcgraw, Adia Mcgraw, Casondra
Mcgraw, Charmane Mcgraw, Chevonne Mcgraw, Gwenn Mcgraw, Laketha Mcgraw, Louis Mcgraw, Marvin Mcgraw, Micole Mcgraw, Novella Mcgraw, Oriana Mcgraw, Raena Mcgraw, Sharone Mcgraw, Suellen Mcgraw, Thais
Mcgraw, Trenda Mcgraw, Zuleika Mcgraw, Aundria Mcgraw, Ayisha Mcgraw, Catrena Mcgraw, Corinthia Mcgraw, Dorine Mcgraw, Dulcie Mcgraw, Dyanna Mcgraw, Kimyatta Mcgraw, Lekeshia Mcgraw, Lenette Mcgraw,
Nesha Mcgraw, Reta Mcgraw, Rian Mcgraw, Rori Mcgraw, Samia Mcgraw, Shelonda Mcgraw, Sima Mcgraw, Adriann Mcgraw, Beata Mcgraw, Dewanna Mcgraw, Elisia Mcgraw, Lida Mcgraw, Myriah Mcgraw, Porsche
Mcgraw, Regine Mcgraw, Roma Mcgraw, Shandi Mcgraw, Sharae Mcgraw, Brady Mcgraw, Denitra Mcgraw, Erik Mcgraw, Jeane Mcgraw, Jinny Mcgraw, Karah Mcgraw, Mechell Mcgraw, Mikel Mcgraw, Nikkie Mcgraw,
Nisa Mcgraw, Nubia Mcgraw, Peaches Mcgraw, Philip Mcgraw, Ramie Mcgraw, Romelia Mcgraw, Roshell Mcgraw, Tai Mcgraw, Tracia Mcgraw, Wonda Mcgraw, Alix Mcgraw, Aprile Mcgraw, Ilda Mcgraw, Kacee Mcgraw,
Kathrin Mcgraw, Lettie Mcgraw, Michaelene Mcgraw, Rania Mcgraw, Shilpa Mcgraw, Teddi Mcgraw, Tegan Mcgraw, Terrilyn Mcgraw, Albertina Mcgraw, Aryn Mcgraw, Augustina Mcgraw, Belle Mcgraw, Britton
Mcgraw, Irena Mcgraw, Jacquie Mcgraw, Katrine Mcgraw, Lorenda Mcgraw, Magali Mcgraw, Mikaela Mcgraw, Remona Mcgraw, Rossana Mcgraw, Shree Mcgraw, Adrea Mcgraw, Ashaki Mcgraw, Beatris Mcgraw, Elisabet
Mcgraw, Gene Mcgraw, Greer Mcgraw, Hillery Mcgraw, Jillene Mcgraw, Keyona Mcgraw, Latifa Mcgraw, Marshell Mcgraw, Miguel Mcgraw, Resa Mcgraw, Roy Mcgraw, Tarnisha Mcgraw, Tekisha Mcgraw, Teshia
Mcgraw, Tomara Mcgraw, Vashti Mcgraw, Brienne Mcgraw, Calvin Mcgraw, Corrinne Mcgraw, Dorina Mcgraw, Errica Mcgraw, Eulalia Mcgraw, Jacelyn Mcgraw, Jimi Mcgraw, Kavita Mcgraw, Keyana Mcgraw, Kiara
Mcgraw, Lahoma Mcgraw, Merilee Mcgraw, Milisa Mcgraw, Pollyanna Mcgraw, Rabecca Mcgraw, Shalisa Mcgraw, Tyree Mcgraw, Baby Mcgraw, Cali Mcgraw, Cherron Mcgraw, Conswella Mcgraw, Dayle Mcgraw, Deniece
Mcgraw, Elesha Mcgraw, Eugene Mcgraw, Fredricka Mcgraw, Letty Mcgraw, Marlisa Mcgraw, Meadow Mcgraw, Nana Mcgraw, Ricardo Mcgraw, Sunita Mcgraw, Tahirah Mcgraw, Aleasha Mcgraw, Alene Mcgraw, Jerica
Mcgraw, Jonni Mcgraw, Jorie Mcgraw, Keenya Mcgraw, Phillis Mcgraw, Scottie Mcgraw, Tamy Mcgraw, Adelaide Mcgraw, Akia Mcgraw, Alona Mcgraw, Brandilyn Mcgraw, Davita Mcgraw, Deondra Mcgraw, Guinevere
Mcgraw, Jaquetta Mcgraw, Joyelle Mcgraw, Kaitlin Mcgraw, Kammy Mcgraw, Lashonna Mcgraw, Manisha Mcgraw, Nyesha Mcgraw, Phebe Mcgraw, Shadonna Mcgraw, Tamila Mcgraw, Treasure Mcgraw, Wayne Mcgraw,
Charmayne Mcgraw, Corry Mcgraw, Doni Mcgraw, Jannell Mcgraw, Kaia Mcgraw, Karna Mcgraw, Kinya Mcgraw, Lasha Mcgraw, Lecia Mcgraw, Letricia Mcgraw, Paulita Mcgraw, Sharnell Mcgraw, Sheniqua Mcgraw,
Toy Mcgraw, Unique Mcgraw, Wynona Mcgraw, Audrie Mcgraw, Cambria Mcgraw, Crissie Mcgraw, Deseree Mcgraw, Deva Mcgraw, Jenney Mcgraw, Jessamyn Mcgraw, Leda Mcgraw, Lorry Mcgraw, Marjory Mcgraw, Marnee
Mcgraw, Monigue Mcgraw, Nakeya Mcgraw, Passion Mcgraw, Ryanne Mcgraw, Sharona Mcgraw, Shelina Mcgraw, Tabbatha Mcgraw, Tameshia Mcgraw, Adelia Mcgraw, Anica Mcgraw, Calli Mcgraw, Carolyne Mcgraw,
Chiffon Mcgraw, Elizabet Mcgraw, Ermelinda Mcgraw, Freddie Mcgraw, Jamia Mcgraw, Korrie Mcgraw, Latrenda Mcgraw, Makia Mcgraw, Marcel Mcgraw, Meta Mcgraw, Noni Mcgraw, Sigrid Mcgraw, Zita Mcgraw,
Angelee Mcgraw, Balinda Mcgraw, Brandyn Mcgraw, Christia Mcgraw, Eneida Mcgraw, Irish Mcgraw, Jackelyn Mcgraw, Jacqui Mcgraw, Kammie Mcgraw, Kendell Mcgraw, Kiya Mcgraw, Mellanie Mcgraw, Sherina
Mcgraw, Syretta Mcgraw, Teia Mcgraw, Teneshia Mcgraw, Tomasa Mcgraw, Tonna Mcgraw, Virgina Mcgraw, Zelma Mcgraw, Adella Mcgraw, Alea Mcgraw, Annita Mcgraw, Denetra Mcgraw, Georgeann Mcgraw, Gwyneth
Mcgraw, Jenea Mcgraw, Kalli Mcgraw, Khristine Mcgraw, Krissie Mcgraw, Lanelle Mcgraw, Letrice Mcgraw, Meshell Mcgraw, Michon Mcgraw, Nekita Mcgraw, Raelyn Mcgraw, Shaleen Mcgraw, Sharanda Mcgraw, Sol
Mcgraw, Suzi Mcgraw, Ulanda Mcgraw, Ame Mcgraw, Aprill Mcgraw, Brooks Mcgraw, Carlye Mcgraw, Francisco Mcgraw, Gaynell Mcgraw, Gwenda Mcgraw, Heidy Mcgraw, Helga Mcgraw, Janese Mcgraw, Kathrina
Mcgraw, Katonya Mcgraw, Kodi Mcgraw, Leshawn Mcgraw, London Mcgraw, Lyndee Mcgraw, Marc Mcgraw, Miyoshi Mcgraw, Onika Mcgraw, Peyton Mcgraw, Shandell Mcgraw, Terena Mcgraw, Acacia Mcgraw, Aleida
Mcgraw, Antoine Mcgraw, Arletha Mcgraw, Carlin Mcgraw, Danni Mcgraw, Davette Mcgraw, December Mcgraw, Jenene Mcgraw, Karalyn Mcgraw, Kimbely Mcgraw, Lakisa Mcgraw, Lynae Mcgraw, Margaux Mcgraw,
Michela Mcgraw, Renne Mcgraw, Tanzania Mcgraw, Tenia Mcgraw, Toyna Mcgraw, Veronika Mcgraw, Amal Mcgraw, Camila Mcgraw, Chi Mcgraw, Dominic Mcgraw, Macie Mcgraw, Maida Mcgraw, Malisha Mcgraw, Milagro
Mcgraw, Mitchell Mcgraw, Shondell Mcgraw, Taira Mcgraw, Argelia Mcgraw, Casaundra Mcgraw, Courtnie Mcgraw, Dalena Mcgraw, Donyell Mcgraw, Ericia Mcgraw, Kelsi Mcgraw, Madelene Mcgraw, Mikelle Mcgraw,
Moria Mcgraw, Salome Mcgraw, Starlet Mcgraw, Stepanie Mcgraw, Stephnie Mcgraw, Teesha Mcgraw, Allisa Mcgraw, Aneesah Mcgraw, Breann Mcgraw, Chevon Mcgraw, Deatrice Mcgraw, Delaney Mcgraw, Jodene
Mcgraw, Julian Mcgraw, Lorilee Mcgraw, Margarette Mcgraw, Marshall Mcgraw, Meleah Mcgraw, Nadene Mcgraw, Nefertiti Mcgraw, Rebekkah Mcgraw, Shakina Mcgraw, Sibyl Mcgraw, Teria Mcgraw, Briget Mcgraw,
Cortina Mcgraw, Gennie Mcgraw, Georgeanna Mcgraw, Jael Mcgraw, Katharina Mcgraw, Kennetha Mcgraw, Kerrianne Mcgraw, Meggin Mcgraw, Rochele Mcgraw, Santos Mcgraw, Senta Mcgraw, Shanie Mcgraw, Sona
Mcgraw, Yana Mcgraw, Alicea Mcgraw, Capri Mcgraw, Cassey Mcgraw, Daun Mcgraw, Denette Mcgraw, Harold Mcgraw, Jacie Mcgraw, Jennica Mcgraw, Krissa Mcgraw, Lawonda Mcgraw, Lenna Mcgraw, Maija Mcgraw,
Ninette Mcgraw, Tomeika Mcgraw, Trenna Mcgraw, Amberlee Mcgraw, Avril Mcgraw, Charnell Mcgraw, Chenita Mcgraw, Crystel Mcgraw, Eugenie Mcgraw, Ginnie Mcgraw, Jerome Mcgraw, Karee Mcgraw, Latunya
Mcgraw, Lianna Mcgraw, Lisaann Mcgraw, Maribell Mcgraw, Marylyn Mcgraw, Molli Mcgraw, Raye Mcgraw, Risha Mcgraw, Sherrice Mcgraw, Taralyn Mcgraw, Tika Mcgraw, Tyanna Mcgraw, Wilda Mcgraw, Brandalyn
Mcgraw, Camesha Mcgraw, Ericca Mcgraw, Ernest Mcgraw, Glynis Mcgraw, Larena Mcgraw, Latecia Mcgraw, Lindsy Mcgraw, Melea Mcgraw, Monic Mcgraw, Suanne Mcgraw, Tammra Mcgraw, Tyeisha Mcgraw, Allen
Mcgraw, Ambra Mcgraw, Cindee Mcgraw, Deja Mcgraw, Denika Mcgraw, Easter Mcgraw, Emerald Mcgraw, Jann Mcgraw, Jeannetta Mcgraw, Keyla Mcgraw, Lataya Mcgraw, Natividad Mcgraw, Nikola Mcgraw, Quanita
Mcgraw, Shalyn Mcgraw, Temekia Mcgraw, Tishia Mcgraw, Allena Mcgraw, Andrena Mcgraw, Aurea Mcgraw, Buffi Mcgraw, Darren Mcgraw, Dovie Mcgraw, Evelina Mcgraw, Fatina Mcgraw, Ivey Mcgraw, Jennafer
Mcgraw, Kimberlea Mcgraw, Kishia Mcgraw, Lacrecia Mcgraw, Laney Mcgraw, Latora Mcgraw, Robinette Mcgraw, Shalita Mcgraw, Shanin Mcgraw, Terasa Mcgraw, Tiphanie Mcgraw, Tiwanna Mcgraw, Ashlyn Mcgraw,
Breana Mcgraw, Emely Mcgraw, Evan Mcgraw, Evangelia Mcgraw, Halle Mcgraw, Lynnetta Mcgraw, Mandisa Mcgraw, Mardi Mcgraw, Neomi Mcgraw, Peggie Mcgraw, Senaida Mcgraw, Shantee Mcgraw, Alondra Mcgraw,
Bettyjo Mcgraw, Dawnell Mcgraw, Jaunita Mcgraw, Jaymee Mcgraw, Jeneane Mcgraw, Kamara Mcgraw, Korinne Mcgraw, Lacosta Mcgraw, Latrena Mcgraw, Letticia Mcgraw, Luanna Mcgraw, Meighan Mcgraw, Merinda
Mcgraw, Raechelle Mcgraw, Randall Mcgraw, Shermaine Mcgraw, Tacy Mcgraw, Telicia Mcgraw, Trica Mcgraw, Unknown Mcgraw, Venetia Mcgraw, Virgen Mcgraw, Chanita Mcgraw, Christe Mcgraw, Collene Mcgraw,
Deni Mcgraw, Eliana Mcgraw, Evelin Mcgraw, Jessa Mcgraw, Katarina Mcgraw, Kersten Mcgraw, Krysti Mcgraw, Lanise Mcgraw, Lannette Mcgraw, Latacha Mcgraw, Lavenia Mcgraw, Magnolia Mcgraw, Mayte Mcgraw,
Missie Mcgraw, Renda Mcgraw, Sapna Mcgraw, Shawan Mcgraw, Starlene Mcgraw, Stasia Mcgraw, Tamaria Mcgraw, Tiffanee Mcgraw, Yazmin Mcgraw, Ammy Mcgraw, Blossom Mcgraw, Cherokee Mcgraw, Chrissa Mcgraw,
Dean Mcgraw, Donell Mcgraw, Jannelle Mcgraw, Jenessa Mcgraw, Jenne Mcgraw, Kimiko Mcgraw, Lenise Mcgraw, Lotoya Mcgraw, Mariko Mcgraw, Myeshia Mcgraw, Nadja Mcgraw, Shalana Mcgraw, Tangee Mcgraw,
Waynette Mcgraw, Alan Mcgraw, Bennie Mcgraw, Berlinda Mcgraw, Brinda Mcgraw, Brooklyn Mcgraw, Bruce Mcgraw, Daisha Mcgraw, Devina Mcgraw, Donisha Mcgraw, Judie Mcgraw, Khristy Mcgraw, Margarett
Mcgraw, Markesha Mcgraw, Melannie Mcgraw, Nickol Mcgraw, Pamelia Mcgraw, Roselle Mcgraw, Shirleen Mcgraw, Terrence Mcgraw, Trenise Mcgraw, Alda Mcgraw, Ariella Mcgraw, Aubrie Mcgraw, Aziza Mcgraw,
Carrol Mcgraw, Charnita Mcgraw, Deshanna Mcgraw, Donyale Mcgraw, Dorrie Mcgraw, Elayne Mcgraw, Feather Mcgraw, Gregoria Mcgraw, Kanesha Mcgraw, Kirsti Mcgraw, Maryalice Mcgraw, Maude Mcgraw, Millissa
Mcgraw, Monya Mcgraw, Shareese Mcgraw, Shereka Mcgraw, Tenaya Mcgraw, Vonnie Mcgraw, Bunny Mcgraw, Carmin Mcgraw, Cissy Mcgraw, Damika Mcgraw, Dayla Mcgraw, Dewana Mcgraw, Jorge Mcgraw, Kaila Mcgraw,
Kaisha Mcgraw, Kaylynn Mcgraw, Kemba Mcgraw, Lael Mcgraw, Lysa Mcgraw, Roberto Mcgraw, Tanasha Mcgraw, Tynisa Mcgraw, Vina Mcgraw, Angelette Mcgraw, Angenette Mcgraw, Aquila Mcgraw, Carrianne Mcgraw,
Charlynn Mcgraw, Denean Mcgraw, Dondi Mcgraw, Enriqueta Mcgraw, Georganna Mcgraw, Glinda Mcgraw, Jenel Mcgraw, Larinda Mcgraw, Latika Mcgraw, Lequita Mcgraw, Licia Mcgraw, Marilou Mcgraw, Sylvie
Mcgraw, Aaliyah Mcgraw, Cozette Mcgraw, Damian Mcgraw, Ginna Mcgraw, Janalee Mcgraw, Jenniefer Mcgraw, Kierstin Mcgraw, Lexie Mcgraw, Maja Mcgraw, Michelene Mcgraw, Norine Mcgraw, Sana Mcgraw, Sarrah
Mcgraw, Sharna Mcgraw, Sherre Mcgraw, Shoshanna Mcgraw, Tangi Mcgraw, Tillie Mcgraw, Aishia Mcgraw, Allene Mcgraw, Antoniette Mcgraw, Brittani Mcgraw, Candise Mcgraw, Carlota Mcgraw, Debroah Mcgraw,
Demeka Mcgraw, Dondra Mcgraw, Franca Mcgraw, Gaye Mcgraw, Jeanell Mcgraw, Lark Mcgraw, Laurin Mcgraw, Lourie Mcgraw, Machell Mcgraw, Pasha Mcgraw, Shelena Mcgraw, Tangelia Mcgraw, Tani Mcgraw, Twilla
Mcgraw, Tynesha Mcgraw, Vena Mcgraw, Vernessa Mcgraw, Aleesha Mcgraw, Cassia Mcgraw, Catheryn Mcgraw, Chavon Mcgraw, Codie Mcgraw, Conchita Mcgraw, Felipa Mcgraw, Jeanice Mcgraw, Kelleen Mcgraw,
Latona Mcgraw, Loreal Mcgraw, Lutricia Mcgraw, Natanya Mcgraw, Perry Mcgraw, Samaria Mcgraw, Sharene Mcgraw, Shavone Mcgraw, Stavroula Mcgraw, Tatia Mcgraw, Terrina Mcgraw, Veroncia Mcgraw, Alaine
Mcgraw, Carmalita Mcgraw, Chirstina Mcgraw, Dawanna Mcgraw, Delonda Mcgraw, Ebonee Mcgraw, Jamela Mcgraw, Kerie Mcgraw, Lanika Mcgraw, Latressa Mcgraw, Priscella Mcgraw, Sharifa Mcgraw, Shenequa
Mcgraw, Shon Mcgraw, Taresa Mcgraw, Viva Mcgraw, Annabell Mcgraw, Arelis Mcgraw, Carra Mcgraw, Char Mcgraw, Chekesha Mcgraw, Cherylann Mcgraw, Crystall Mcgraw, Diandra Mcgraw, Dwayne Mcgraw, Elishia
Mcgraw, Emi Mcgraw, Jennife Mcgraw, Kassi Mcgraw, Keyonna Mcgraw, Kresta Mcgraw, Kriston Mcgraw, Laree Mcgraw, Lashea Mcgraw, Lauryn Mcgraw, Lennie Mcgraw, Nailah Mcgraw, Neena Mcgraw, Nelia Mcgraw,
Reiko Mcgraw, Sejal Mcgraw, Shanea Mcgraw, Sharman Mcgraw, Talesha Mcgraw, Teresia Mcgraw, Tunya Mcgraw, Annett Mcgraw, Catherina Mcgraw, Charlita Mcgraw, Dandrea Mcgraw, Darcel Mcgraw, Eleanore
Mcgraw, Falicia Mcgraw, Glynda Mcgraw, Karra Mcgraw, Kerra Mcgraw, Ladona Mcgraw, Loretha Mcgraw, Lus Mcgraw, Marketa Mcgraw, Megen Mcgraw, Shalini Mcgraw, Sheela Mcgraw, Tally Mcgraw, Thuy Mcgraw,
Tristen Mcgraw, Yashika Mcgraw, Aleah Mcgraw, Arline Mcgraw, Catarina Mcgraw, Cris Mcgraw, Dawnita Mcgraw, Kelliann Mcgraw, Kristle Mcgraw, Lanesha Mcgraw, Malanie Mcgraw, Nekesha Mcgraw, Rhona
Mcgraw, Roslynn Mcgraw, Sariah Mcgraw, Takeshia Mcgraw, Theda Mcgraw, Altagracia Mcgraw, Anneke Mcgraw, Aspen Mcgraw, Azalea Mcgraw, Jamelle Mcgraw, Kalyn Mcgraw, Lavonya Mcgraw, Malea Mcgraw,
Marykate Mcgraw, Meliss Mcgraw, Noelani Mcgraw, Patina Mcgraw, Riann Mcgraw, Rolonda Mcgraw, Shaney Mcgraw, Shanice Mcgraw, Wynne Mcgraw, Yocheved Mcgraw, Aysha Mcgraw, Cindie Mcgraw, Daphanie
Mcgraw, Darcee Mcgraw, Demitra Mcgraw, Francene Mcgraw, Ian Mcgraw, Janaya Mcgraw, Jera Mcgraw, Jeremiah Mcgraw, Karlie Mcgraw, Kiwana Mcgraw, Krystin Mcgraw, Larina Mcgraw, Marylee Mcgraw, Maurine
Mcgraw, Mistee Mcgraw, Rainey Mcgraw, Raynette Mcgraw, Robyne Mcgraw, Ronelle Mcgraw, Sharma Mcgraw, Shelene Mcgraw, Stevie Mcgraw, Wynter Mcgraw, Ylonda Mcgraw, Zondra Mcgraw, Aesha Mcgraw, Alpha
Mcgraw, Angala Mcgraw, Che Mcgraw, Courtnay Mcgraw, Deatra Mcgraw, Genell Mcgraw, Jerrica Mcgraw, Jozette Mcgraw, Kerianne Mcgraw, Krissi Mcgraw, Millisa Mcgraw, Minna Mcgraw, Naeemah Mcgraw,
Shantrell Mcgraw, Theresia Mcgraw, Tomeko Mcgraw, Vanda Mcgraw, Annelise Mcgraw, Bettye Mcgraw, Brina Mcgraw, Carmelina Mcgraw, Charlotta Mcgraw, Charmine Mcgraw, Darbi Mcgraw, Devra Mcgraw,
Donnamarie Mcgraw, Enza Mcgraw, Jack Mcgraw, Lamika Mcgraw, Lashaundra Mcgraw, Latavia Mcgraw, Mauri Mcgraw, Nevada Mcgraw, Patrisha Mcgraw, Rashel Mcgraw, Rayann Mcgraw, Shanay Mcgraw, Shareka
Mcgraw, Sharlyn Mcgraw, Sharra Mcgraw, Suzzanne Mcgraw, Tascha Mcgraw, Tomeca Mcgraw, Annisa Mcgraw, Anthea Mcgraw, Bethel Mcgraw, Charlesetta Mcgraw, Davetta Mcgraw, Deonne Mcgraw, Emmie Mcgraw,
Fanta Mcgraw, Felicitas Mcgraw, Flavia Mcgraw, Glenn Mcgraw, Gwendolynn Mcgraw, Hollye Mcgraw, Jalene Mcgraw, Jamel Mcgraw, Jenita Mcgraw, Jonda Mcgraw, Kameka Mcgraw, Lagina Mcgraw, Lateasha Mcgraw,
Louanne Mcgraw, Ludivina Mcgraw, Ramonita Mcgraw, Reshonda Mcgraw, Romina Mcgraw, Samone Mcgraw, Shauntae Mcgraw, Shelle Mcgraw, Tasheka Mcgraw, Tashima Mcgraw, Tekesha Mcgraw, Trini Mcgraw, Tyese
Mcgraw, Alesa Mcgraw, Brigett Mcgraw, Camella Mcgraw, Charee Mcgraw, Coby Mcgraw, Constantina Mcgraw, Contrina Mcgraw, Danise Mcgraw, Delanie Mcgraw, Denia Mcgraw, Detrice Mcgraw, Donyelle Mcgraw,
Estee Mcgraw, Freya Mcgraw, Gabriele Mcgraw, Heidie Mcgraw, Idella Mcgraw, Iraida Mcgraw, Jonica Mcgraw, Kaley Mcgraw, Kirsty Mcgraw, Laurette Mcgraw, Nataki Mcgraw, Patrisia Mcgraw, Ronnetta Mcgraw,
Signe Mcgraw, Tarita Mcgraw, Trenia Mcgraw, Annessa Mcgraw, Celene Mcgraw, Cerissa Mcgraw, Chinita Mcgraw, Danah Mcgraw, Dannelle Mcgraw, Dawnielle Mcgraw, Doretta Mcgraw, Dwanna Mcgraw, Jinger
Mcgraw, Karilyn Mcgraw, Loida Mcgraw, Miko Mcgraw, Mirtha Mcgraw, Myia Mcgraw, Perri Mcgraw, Ragina Mcgraw, Rea Mcgraw, Saralyn Mcgraw, Saskia Mcgraw, Sharry Mcgraw, Tiare Mcgraw, Tonica Mcgraw,
Waleska Mcgraw, Xochilt Mcgraw, Zipporah Mcgraw, Delynn Mcgraw, Detria Mcgraw, Earl Mcgraw, Enedelia Mcgraw, Jeanett Mcgraw, Jonita Mcgraw, Kenita Mcgraw, Laural Mcgraw, Leonard Mcgraw, Ligia Mcgraw,
Linnie Mcgraw, Lowanda Mcgraw, Ltanya Mcgraw, Monquie Mcgraw, Paloma Mcgraw, Rafael Mcgraw, Steve Mcgraw, Torina Mcgraw, Yolunda Mcgraw, Cariann Mcgraw, Carmita Mcgraw, Cerise Mcgraw, Dalana Mcgraw,
Dawnya Mcgraw, Deshannon Mcgraw, Deshonda Mcgraw, Desirea Mcgraw, Eartha Mcgraw, Emelia Mcgraw, Francoise Mcgraw, Gerrie Mcgraw, Gitty Mcgraw, Jared Mcgraw, Kianna Mcgraw, Lamanda Mcgraw, Laraine
Mcgraw, Latonda Mcgraw, Latrease Mcgraw, Leaann Mcgraw, Liv Mcgraw, Maha Mcgraw, Meliza Mcgraw, Shawnetta Mcgraw, Shontelle Mcgraw, Sloan Mcgraw, Taja Mcgraw, Tammye Mcgraw, Terika Mcgraw, Vinita
Mcgraw, Charese Mcgraw, Charisa Mcgraw, Danella Mcgraw, Halley Mcgraw, Jara Mcgraw, Jawana Mcgraw, Jobina Mcgraw, Juniper Mcgraw, Leondra Mcgraw, Letonya Mcgraw, Marianela Mcgraw, Mariella Mcgraw,
Marisel Mcgraw, Marlinda Mcgraw, Maurita Mcgraw, Merrill Mcgraw, Nalani Mcgraw, Natika Mcgraw, Ralph Mcgraw, Shajuana Mcgraw, Sharmon Mcgraw, Tanda Mcgraw, Clarence Mcgraw, Enjoli Mcgraw, Hali
Mcgraw, Isaura Mcgraw, Joellyn Mcgraw, Kaylyn Mcgraw, Ketra Mcgraw, Latayna Mcgraw, Melisse Mcgraw, Natoshia Mcgraw, Niko Mcgraw, Shalina Mcgraw, Sharrie Mcgraw, Silva Mcgraw, Taren Mcgraw, Tawnie
Mcgraw, Venetta Mcgraw, Chenelle Mcgraw, Chonda Mcgraw, Drucilla Mcgraw, Genene Mcgraw, Jaquita Mcgraw, Kally Mcgraw, Laury Mcgraw, Lucina Mcgraw, Lynde Mcgraw, Marley Mcgraw, Scherrie Mcgraw, Shaila
Mcgraw, Shalimar Mcgraw, Shannyn Mcgraw, Shantina Mcgraw, Shirl Mcgraw, Takita Mcgraw, Tineka Mcgraw, Valisa Mcgraw, Adrina Mcgraw, Ambre Mcgraw, Arlisa Mcgraw, Cecila Mcgraw, Demetrica Mcgraw, Dyann
Mcgraw, Erina Mcgraw, Haven Mcgraw, Hector Mcgraw, Janika Mcgraw, Johnnetta Mcgraw, Kellyanne Mcgraw, Lacrisha Mcgraw, Lamar Mcgraw, Litisha Mcgraw, Malkia Mcgraw, Marcellina Mcgraw, Otilia Mcgraw,
Pascale Mcgraw, Ramonda Mcgraw, Safiya Mcgraw, Sebrena Mcgraw, Shifra Mcgraw, Sina Mcgraw, Tashina Mcgraw, Teffany Mcgraw, Trevor Mcgraw, Veleka Mcgraw, Zanetta Mcgraw, Arie Mcgraw, Azalia Mcgraw,
Cedric Mcgraw, Chasta Mcgraw, Cherelle Mcgraw, Cyndee Mcgraw, Darnetta Mcgraw, Dinora Mcgraw, Dorena Mcgraw, Howard Mcgraw, Joyel Mcgraw, July Mcgraw, Lance Mcgraw, Lore Mcgraw, Michelina Mcgraw,
Olive Mcgraw, Reem Mcgraw, Rica Mcgraw, Shaneen Mcgraw, Sharlotte Mcgraw, Shavonna Mcgraw, Shawnie Mcgraw, Sherria Mcgraw, Sian Mcgraw, Stephanee Mcgraw, Susette Mcgraw, Terilyn Mcgraw, Tiajuana
Mcgraw, Tristin Mcgraw, Vada Mcgraw, Vesta Mcgraw, Yara Mcgraw, Alyssia Mcgraw, Arian Mcgraw, Chianti Mcgraw, Chriselda Mcgraw, Darian Mcgraw, Dawnetta Mcgraw, Desarae Mcgraw, Dinorah Mcgraw, Djuana
Mcgraw, Elka Mcgraw, Golda Mcgraw, Honor Mcgraw, Jamille Mcgraw, Jesusita Mcgraw, Junita Mcgraw, Kenyada Mcgraw, Konnie Mcgraw, Kwanza Mcgraw, Lissett Mcgraw, Marlon Mcgraw, Marlys Mcgraw, Marshelle
Mcgraw, Meagen Mcgraw, Myka Mcgraw, Mylissa Mcgraw, Patrece Mcgraw, Raguel Mcgraw, Sharia Mcgraw, Shawnell Mcgraw, Teanna Mcgraw, Adrena Mcgraw, Brad Mcgraw, Charlisa Mcgraw, Dannie Mcgraw, Fernanda
Mcgraw, Jeania Mcgraw, Kareema Mcgraw, Karyl Mcgraw, Kellianne Mcgraw, Lilah Mcgraw, Lorelle Mcgraw, Lynnell Mcgraw, Quana Mcgraw, Rondi Mcgraw, Rosiland Mcgraw, Ruben Mcgraw, Sharmin Mcgraw,
Taniesha Mcgraw, Tashonda Mcgraw, Valecia Mcgraw, Alnisa Mcgraw, Cherity Mcgraw, Coletta Mcgraw, Damary Mcgraw, Darcia Mcgraw, Dashawn Mcgraw, Garnet Mcgraw, Hadassah Mcgraw, Javon Mcgraw, Kareem
Mcgraw, Keara Mcgraw, Kerryann Mcgraw, Loleta Mcgraw, Marijo Mcgraw, Maritsa Mcgraw, Marvella Mcgraw, Miriah Mcgraw, Necia Mcgraw, Priscila Mcgraw, Ramon Mcgraw, Shalawn Mcgraw, Sidra Mcgraw, Sundee
Mcgraw, Tasia Mcgraw, Tiwanda Mcgraw, Xavier Mcgraw, Alayne Mcgraw, Anesha Mcgraw, Anndrea Mcgraw, Astra Mcgraw, Breezy Mcgraw, Carletha Mcgraw, Chantele Mcgraw, Damara Mcgraw, Denell Mcgraw, Dessa
Mcgraw, Marcell Mcgraw, Maretta Mcgraw, Marline Mcgraw, Melania Mcgraw, Modesta Mcgraw, Montoya Mcgraw, Rashidah Mcgraw, Rusti Mcgraw, Shannell Mcgraw, Shaune Mcgraw, Sheresa Mcgraw, Stasha Mcgraw,
Talaya Mcgraw, Taletha Mcgraw, Tashawna Mcgraw, Terrah Mcgraw, Yessica Mcgraw, Yolander Mcgraw, Ani Mcgraw, Cyndy Mcgraw, Dannell Mcgraw, Edythe Mcgraw, Elodia Mcgraw, Felissa Mcgraw, Jamison Mcgraw,
Janalyn Mcgraw, Jodell Mcgraw, Josetta Mcgraw, Karmin Mcgraw, Kenesha Mcgraw, Keyna Mcgraw, Kimara Mcgraw, Kiran Mcgraw, Ladonya Mcgraw, Latorya Mcgraw, Marizol Mcgraw, Nadya Mcgraw, Pamula Mcgraw,
Pleshette Mcgraw, Tamaka Mcgraw, Tifanie Mcgraw, Tunisha Mcgraw, Verona Mcgraw, Veronda Mcgraw, Allicia Mcgraw, Allisha Mcgraw, Allissa Mcgraw, Allyn Mcgraw, Andriana Mcgraw, Anneka Mcgraw, Bridgitte
Mcgraw, Brigida Mcgraw, Chaunte Mcgraw, Christol Mcgraw, Elita Mcgraw, Jacquelene Mcgraw, Jenefer Mcgraw, Jessyca Mcgraw, Kammi Mcgraw, Kasondra Mcgraw, Katricia Mcgraw, Katrinka Mcgraw, Lakasha
Mcgraw, Maryrose Mcgraw, Mirinda Mcgraw, Natoya Mcgraw, Omar Mcgraw, Santana Mcgraw, Shakima Mcgraw, Shatina Mcgraw, Shawnita Mcgraw, Sherine Mcgraw, Starlett Mcgraw, Tanga Mcgraw, Teal Mcgraw, Teara
Mcgraw, Tiffanny Mcgraw, Vivien Mcgraw, Altovise Mcgraw, Conni Mcgraw, Dawanda Mcgraw, Ellena Mcgraw, Janica Mcgraw, Jeralyn Mcgraw, Kameelah Mcgraw, Kasia Mcgraw, Katharyn Mcgraw, Lamesha Mcgraw,
Laurieann Mcgraw, Leon Mcgraw, Margherita Mcgraw, Marguerita Mcgraw, Mariama Mcgraw, Mariea Mcgraw, Maryfrances Mcgraw, Mayda Mcgraw, Meena Mcgraw, Mikal Mcgraw, Nicholette Mcgraw, Nitza Mcgraw,
Norman Mcgraw, Raychelle Mcgraw, Riva Mcgraw, Ronit Mcgraw, Rosenda Mcgraw, Royce Mcgraw, Saira Mcgraw, Samona Mcgraw, Toye Mcgraw, Zinnia Mcgraw, Alfred Mcgraw, Amita Mcgraw, Aqueelah Mcgraw, Blenda
Mcgraw, Charron Mcgraw, Chrisie Mcgraw, Ema Mcgraw, Endia Mcgraw, Georgine Mcgraw, Gretel Mcgraw, Johnita Mcgraw, Joscelyn Mcgraw, Kiona Mcgraw, Lamont Mcgraw, Lateesha Mcgraw, Liisa Mcgraw, Lizet
Mcgraw, Marny Mcgraw, Martisha Mcgraw, Marvette Mcgraw, Natonya Mcgraw, Oona Mcgraw, Philomena Mcgraw, Sahar Mcgraw, Shantia Mcgraw, Starlette Mcgraw, Talana Mcgraw, Tereza Mcgraw, Torsha Mcgraw,
Tyeshia Mcgraw, Aislinn Mcgraw, Alexandrea Mcgraw, Artisha Mcgraw, Assunta Mcgraw, Belva Mcgraw, Britany Mcgraw, Bryanna Mcgraw, Chauntel Mcgraw, Cristel Mcgraw, Damali Mcgraw, Daphane Mcgraw,
Dorianne Mcgraw, Imogene Mcgraw, Janai Mcgraw, Laveda Mcgraw, Lucero Mcgraw, Lyssa Mcgraw, Marielle Mcgraw, Natalya Mcgraw, Pammy Mcgraw, Patches Mcgraw, Rhiana Mcgraw, Sakeena Mcgraw, Tanea Mcgraw,
Tiffinie Mcgraw, Xaviera Mcgraw, Arla Mcgraw, Collen Mcgraw, Deane Mcgraw, Don Mcgraw, Gari Mcgraw, Jazmine Mcgraw, Joani Mcgraw, Jude Mcgraw, Lacee Mcgraw, Lakeasha Mcgraw, Leandrea Mcgraw, Marvina
Mcgraw, Minette Mcgraw, Nico Mcgraw, Noell Mcgraw, Ouida Mcgraw, Rainbow Mcgraw, Shania Mcgraw, Shemekia Mcgraw, Velinda Mcgraw, Yetta Mcgraw, Adell Mcgraw, Alyshia Mcgraw, Arlana Mcgraw, Athina
Mcgraw, Calley Mcgraw, Carson Mcgraw, Cassandre Mcgraw, Channing Mcgraw, Cherith Mcgraw, Darnella Mcgraw, Deepa Mcgraw, Eboney Mcgraw, Florentina Mcgraw, Fredia Mcgraw, Jerilynn Mcgraw, Kamila
Mcgraw, Keeli Mcgraw, Korena Mcgraw, Korri Mcgraw, Lakeia Mcgraw, Lulu Mcgraw, Madelaine Mcgraw, Marja Mcgraw, Maryhelen Mcgraw, Mei Mcgraw, Quintella Mcgraw, Rainy Mcgraw, Rashunda Mcgraw, Rheanna
Mcgraw, Saran Mcgraw, Shanese Mcgraw, Shelisa Mcgraw, Tamecia Mcgraw, Tanica Mcgraw, Tawnee Mcgraw, Van Mcgraw, Vivienne Mcgraw, Yoland Mcgraw, Yvonna Mcgraw, Abbi Mcgraw, Afrika Mcgraw, Anise
Mcgraw, Aron Mcgraw, Carrieanne Mcgraw, Chauna Mcgraw, Daena Mcgraw, Elin Mcgraw, Jerrilyn Mcgraw, Landa Mcgraw, Latoyna Mcgraw, Leane Mcgraw, Lilliam Mcgraw, Marquisha Mcgraw, Marry Mcgraw, Matasha
Mcgraw, Matina Mcgraw, Megin Mcgraw, Mitchelle Mcgraw, Nakeshia Mcgraw, Pearline Mcgraw, Rayne Mcgraw, Reeshemah Mcgraw, Romaine Mcgraw, Shameca Mcgraw, Sterling Mcgraw, Suzzette Mcgraw, Tangy
Mcgraw, Tonetta Mcgraw, Trenice Mcgraw, Zahra Mcgraw, Abena Mcgraw, Aliesha Mcgraw, Andreia Mcgraw, Beryl Mcgraw, Cathey Mcgraw, Crystalyn Mcgraw, Davonna Mcgraw, Delphia Mcgraw, Denisse Mcgraw,
Elina Mcgraw, Elisheva Mcgraw, Evangela Mcgraw, Felina Mcgraw, Gisella Mcgraw, Ilka Mcgraw, Janeth Mcgraw, Katara Mcgraw, Laquesha Mcgraw, Lysandra Mcgraw, Malana Mcgraw, Morgen Mcgraw, Natali
Mcgraw, Niambi Mcgraw, Niketa Mcgraw, Pamla Mcgraw, Rebeckah Mcgraw, Revonda Mcgraw, Sarabeth Mcgraw, Sharmila Mcgraw, Shylo Mcgraw, Takiya Mcgraw, Trevia Mcgraw, Aleatha Mcgraw, Anabelle Mcgraw,
Blima Mcgraw, Bliss Mcgraw, Christyn Mcgraw, Chrysta Mcgraw, Cristela Mcgraw, Deliah Mcgraw, Ellisa Mcgraw, Genice Mcgraw, Isadora Mcgraw, Jawanna Mcgraw, Kamaria Mcgraw, Ladena Mcgraw, Larenda
Mcgraw, Laurissa Mcgraw, Leela Mcgraw, Lovina Mcgraw, Lyra Mcgraw, Maryanna Mcgraw, Marybell Mcgraw, Meika Mcgraw, Meshelle Mcgraw, Moana Mcgraw, Nicoletta Mcgraw, Radiah Mcgraw, Rupal Mcgraw,
Shannin Mcgraw, Shyra Mcgraw, Stanley Mcgraw, Sundae Mcgraw, Taunia Mcgraw, Tomorrow Mcgraw, Abbigail Mcgraw, Amoreena Mcgraw, Anamarie Mcgraw, Chani Mcgraw, Dayanara Mcgraw, Delane Mcgraw, Demika
Mcgraw, Desi Mcgraw, Desma Mcgraw, Eugina Mcgraw, Florine Mcgraw, Johanne Mcgraw, Karman Mcgraw, Katheryne Mcgraw, Kema Mcgraw, Kenitra Mcgraw, Lareina Mcgraw, Laryssa Mcgraw, Lasaundra Mcgraw,
Lorriane Mcgraw, Lurdes Mcgraw, Meredyth Mcgraw, Mireille Mcgraw, Naomie Mcgraw, Rewa Mcgraw, Rosette Mcgraw, Rukiya Mcgraw, Shanica Mcgraw, Sherill Mcgraw, Shonette Mcgraw, Adonica Mcgraw, Aquilla
Mcgraw, Billijo Mcgraw, Bretta Mcgraw, Charmian Mcgraw, Chondra Mcgraw, Coty Mcgraw, Danine Mcgraw, Davena Mcgraw, Davia Mcgraw, Falana Mcgraw, Fredrica Mcgraw, Harry Mcgraw, Jae Mcgraw, Jawanda
Mcgraw, Keana Mcgraw, Krisann Mcgraw, Laqueta Mcgraw, Lashannon Mcgraw, Lavelle Mcgraw, Layna Mcgraw, Linde Mcgraw, Lorisa Mcgraw, Lyndie Mcgraw, Macey Mcgraw, Maricel Mcgraw, Nadina Mcgraw, Nadiyah
Mcgraw, Nereyda Mcgraw, Sequoia Mcgraw, Stephene Mcgraw, Tahesha Mcgraw, Takeya Mcgraw, Tali Mcgraw, Tiera Mcgraw, Tomasina Mcgraw, Verena Mcgraw, Yvonda Mcgraw, Aracelia Mcgraw, Arletta Mcgraw,
Augustine Mcgraw, Barry Mcgraw, Brittny Mcgraw, Christella Mcgraw, Courtnee Mcgraw, Dakota Mcgraw, Denesha Mcgraw, Donnella Mcgraw, Fotini Mcgraw, Gita Mcgraw, Holland Mcgraw, Jacquiline Mcgraw,
Jamilyn Mcgraw, Kafi Mcgraw, Kalena Mcgraw, Katya Mcgraw, Kit Mcgraw, Laniece Mcgraw, Lauree Mcgraw, Leslea Mcgraw, Maite Mcgraw, Mitzy Mcgraw, Myria Mcgraw, Njeri Mcgraw, Oscar Mcgraw, Rain Mcgraw,
Rema Mcgraw, Ronee Mcgraw, Shevon Mcgraw, Shirlee Mcgraw, Sissy Mcgraw, Sonali Mcgraw, Sundra Mcgraw, Taralee Mcgraw, Trace Mcgraw, Tyana Mcgraw, Agustina Mcgraw, Akua Mcgraw, Allyssa Mcgraw, Andrina
Mcgraw, Angell Mcgraw, Antwanette Mcgraw, Carmelia Mcgraw, Chelsi Mcgraw, Cheronda Mcgraw, Clorinda Mcgraw, Crissa Mcgraw, Cyrena Mcgraw, Dante Mcgraw, Destini Mcgraw, Emiko Mcgraw, Era Mcgraw,
Gentry Mcgraw, Gini Mcgraw, Hanan Mcgraw, Haylee Mcgraw, Hollee Mcgraw, Jannett Mcgraw, Javier Mcgraw, Jenay Mcgraw, Jennier Mcgraw, Kandee Mcgraw, Kerilyn Mcgraw, Kissy Mcgraw, Konstantina Mcgraw,
Ladana Mcgraw, Logan Mcgraw, Lovette Mcgraw, Manal Mcgraw, Marykay Mcgraw, Phoenix Mcgraw, Rosalin Mcgraw, Shatara Mcgraw, Shawnya Mcgraw, Sheralyn Mcgraw, Sheretta Mcgraw, Suann Mcgraw, Tava Mcgraw,
Bobette Mcgraw, Carrissa Mcgraw, Coralee Mcgraw, Daneille Mcgraw, Edwin Mcgraw, Emelda Mcgraw, Flecia Mcgraw, Jame Mcgraw, Janeane Mcgraw, Jelena Mcgraw, Katja Mcgraw, Kelia Mcgraw, Lanee Mcgraw,
Laverna Mcgraw, Lorra Mcgraw, Marlin Mcgraw, Mieke Mcgraw, Miroslava Mcgraw, Nekeisha Mcgraw, Nikeya Mcgraw, Noami Mcgraw, Saida Mcgraw, Sienna Mcgraw, Sundi Mcgraw, Tylene Mcgraw, Yakima Mcgraw,
Alicha Mcgraw, Amani Mcgraw, Armandina Mcgraw, Bethani Mcgraw, Carine Mcgraw, Carmencita Mcgraw, Chantee Mcgraw, Chavonne Mcgraw, Chinyere Mcgraw, Clarisse Mcgraw, Evett Mcgraw, Felisia Mcgraw,
Flossie Mcgraw, Indra Mcgraw, Inge Mcgraw, Keiko Mcgraw, Kristia Mcgraw, Lady Mcgraw, Latrelle Mcgraw, Lean Mcgraw, Lewanda Mcgraw, Marit Mcgraw, Melodye Mcgraw, Moneka Mcgraw, Naisha Mcgraw, Najah
Mcgraw, Ronisha Mcgraw, Shamona Mcgraw, Shantal Mcgraw, Sharay Mcgraw, Shawndell Mcgraw, Shekita Mcgraw, Shelah Mcgraw, Shenise Mcgraw, Soraida Mcgraw, Sujey Mcgraw, Suni Mcgraw, Tiona Mcgraw, Tuwana
Mcgraw, Wynetta Mcgraw, Angelie Mcgraw, Annalise Mcgraw, Carlen Mcgraw, Cherryl Mcgraw, Donnelle Mcgraw, Florencia Mcgraw, Gema Mcgraw, Genita Mcgraw, Happy Mcgraw, Jessy Mcgraw, Karimah Mcgraw,
Lanitra Mcgraw, Louanna Mcgraw, Maisie Mcgraw, Marco Mcgraw, Margaretta Mcgraw, Marguita Mcgraw, Marice Mcgraw, Marygrace Mcgraw, Retta Mcgraw, Roderick Mcgraw, Ronny Mcgraw, Shalom Mcgraw, Sherena
Mcgraw, Tamura Mcgraw, Threasa Mcgraw, Ulonda Mcgraw, Zachary Mcgraw, Aletta Mcgraw, Amaryllis Mcgraw, Amethyst Mcgraw, Antoinetta Mcgraw, Aricka Mcgraw, Bobbyjo Mcgraw, Camile Mcgraw, Charyl Mcgraw,
Elysa Mcgraw, Jameela Mcgraw, Jeannett Mcgraw, Jyl Mcgraw, Kinda Mcgraw, Kristianne Mcgraw, Latifah Mcgraw, Lavera Mcgraw, Madaline Mcgraw, Marea Mcgraw, Melaina Mcgraw, Natesha Mcgraw, Oliva Mcgraw,
Renell Mcgraw, Salli Mcgraw, Satina Mcgraw, Shanyn Mcgraw, Sharelle Mcgraw, Sharena Mcgraw, Shenetta Mcgraw, Sherlonda Mcgraw, Sun Mcgraw, Teah Mcgraw, Teana Mcgraw, Tramaine Mcgraw, Wenda Mcgraw,
Wyndi Mcgraw, Yoko Mcgraw, Adrean Mcgraw, Aishah Mcgraw, Aundra Mcgraw, Betsaida Mcgraw, Chrisann Mcgraw, Clarita Mcgraw, Correna Mcgraw, Cynitha Mcgraw, Cythia Mcgraw, Damien Mcgraw, Dedria Mcgraw,
Divina Mcgraw, Dody Mcgraw, Elanda Mcgraw, Elizbeth Mcgraw, Estrellita Mcgraw, Eydie Mcgraw, Gerilyn Mcgraw, Hedy Mcgraw, Jacklynn Mcgraw, Kimberleigh Mcgraw, Kristinia Mcgraw, Lakeeta Mcgraw, Larie
Mcgraw, Latrisa Mcgraw, Lesleigh Mcgraw, Marella Mcgraw, Martinique Mcgraw, Melessia Mcgraw, Nicolasa Mcgraw, Quantina Mcgraw, Remy Mcgraw, Robina Mcgraw, Roya Mcgraw, Saba Mcgraw, Sacheen Mcgraw,
Shalee Mcgraw, Sharlet Mcgraw, Shereese Mcgraw, Sherronda Mcgraw, Shewanda Mcgraw, Skyla Mcgraw, Tallie Mcgraw, Tanganyika Mcgraw, Taressa Mcgraw, Tarshia Mcgraw, Timi Mcgraw, Zola Mcgraw, Ainsley
Mcgraw, Alivia Mcgraw, Ardith Mcgraw, Artina Mcgraw, Ashely Mcgraw, Athanasia Mcgraw, Chantale Mcgraw, Chimene Mcgraw, Corinn Mcgraw, Daffney Mcgraw, Danene Mcgraw, Daphnie Mcgraw, Deeanne Mcgraw,
Erynn Mcgraw, Glennis Mcgraw, Kalee Mcgraw, Kandyce Mcgraw, Kathren Mcgraw, Kelcey Mcgraw, Laticha Mcgraw, Lenee Mcgraw, Luvenia Mcgraw, Machele Mcgraw, Maris Mcgraw, Marisella Mcgraw, Nakeia Mcgraw,
Nickey Mcgraw, Nikkita Mcgraw, Noella Mcgraw, Peri Mcgraw, Raushanah Mcgraw, Renisha Mcgraw, Shahidah Mcgraw, Shakisha Mcgraw, Shevonne Mcgraw, Tashema Mcgraw, Tondalaya Mcgraw, Trinita Mcgraw, Akiko
Mcgraw, Alfredia Mcgraw, Bathsheba Mcgraw, Billye Mcgraw, Carter Mcgraw, Clementina Mcgraw, Danyele Mcgraw, Darnisha Mcgraw, Darryl Mcgraw, Eda Mcgraw, Emy Mcgraw, Faustina Mcgraw, Hayden Mcgraw,
Hortensia Mcgraw, Ivon Mcgraw, Javonna Mcgraw, Keita Mcgraw, Lakeisa Mcgraw, Lan Mcgraw, Leota Mcgraw, Letasha Mcgraw, Maggi Mcgraw, Monita Mcgraw, Nadirah Mcgraw, Nikka Mcgraw, Noemy Mcgraw, Roxy
Mcgraw, Sabrinia Mcgraw, Sanda Mcgraw, Sharan Mcgraw, Sharnette Mcgraw, Swati Mcgraw, Tiny Mcgraw, Tonjia Mcgraw, Trella Mcgraw, Venecia Mcgraw, Zana Mcgraw, Zorana Mcgraw, Adrienna Mcgraw, Ama
Mcgraw, Angelisa Mcgraw, Beckey Mcgraw, Bobbye Mcgraw, Cammi Mcgraw, Chequita Mcgraw, Clinton Mcgraw, Dama Mcgraw, Danee Mcgraw, Dari Mcgraw, Deandre Mcgraw, Deetta Mcgraw, Devonda Mcgraw, Edelmira
Mcgraw, Ieasha Mcgraw, Joylynn Mcgraw, Krisinda Mcgraw, Kristol Mcgraw, Kwana Mcgraw, Lacresia Mcgraw, Ladawna Mcgraw, Lavanda Mcgraw, Lavena Mcgraw, Leaha Mcgraw, Massiel Mcgraw, Michiko Mcgraw,
Naketa Mcgraw, Persephone Mcgraw, Raynell Mcgraw, Ronell Mcgraw, Shalynn Mcgraw, Shimeka Mcgraw, Shiquita Mcgraw, Tanaka Mcgraw, Tomasita Mcgraw, Trixy Mcgraw, Tyson Mcgraw, Andy Mcgraw, Arden
Mcgraw, Carlissa Mcgraw, Chery Mcgraw, Duane Mcgraw, Ereka Mcgraw, Gracia Mcgraw, Halona Mcgraw, Jamal Mcgraw, Kechia Mcgraw, Kelda Mcgraw, Latessa Mcgraw, Lynett Mcgraw, Mikala Mcgraw, Naida Mcgraw,
Nancee Mcgraw, Renna Mcgraw, Resha Mcgraw, Rifka Mcgraw, Rolinda Mcgraw, Shaneika Mcgraw, Syliva Mcgraw, Tamikka Mcgraw, Tamyka Mcgraw, Tarika Mcgraw, Tawania Mcgraw, Trinda Mcgraw, Trinetta Mcgraw,
Valrie Mcgraw, Anabela Mcgraw, Andretta Mcgraw, Capricia Mcgraw, Chistina Mcgraw, Coree Mcgraw, Darrah Mcgraw, Diahann Mcgraw, Faigy Mcgraw, Georgiann Mcgraw, Hillarie Mcgraw, Ieisha Mcgraw, Joeann
Mcgraw, Jovana Mcgraw, Kesa Mcgraw, Ladeana Mcgraw, Laresa Mcgraw, Lisset Mcgraw, Lynnann Mcgraw, Lysette Mcgraw, Melenie Mcgraw, Natina Mcgraw, Nicolina Mcgraw, Queena Mcgraw, Tameaka Mcgraw, Tansy
Mcgraw, Terressa Mcgraw, Topeka Mcgraw, Alicen Mcgraw, Aneka Mcgraw, Arlyn Mcgraw, Bethzaida Mcgraw, Caralyn Mcgraw, Cartina Mcgraw, Cathlene Mcgraw, Chaquita Mcgraw, Clifford Mcgraw, Danitra Mcgraw,
Dayana Mcgraw, Delania Mcgraw, Dennie Mcgraw, Edina Mcgraw, Electra Mcgraw, Elouise Mcgraw, Ginelle Mcgraw, Heath Mcgraw, Kassia Mcgraw, Katrenia Mcgraw, Keidra Mcgraw, Laconda Mcgraw, Loura Mcgraw,
Marchell Mcgraw, Mayela Mcgraw, Merle Mcgraw, Merlinda Mcgraw, Mychelle Mcgraw, Nannie Mcgraw, Nikko Mcgraw, Nori Mcgraw, Prescilla Mcgraw, Rika Mcgraw, Ronika Mcgraw, Sahara Mcgraw, Sandrea Mcgraw,
Senora Mcgraw, Shawntee Mcgraw, Sheryll Mcgraw, Shina Mcgraw, Terre Mcgraw, Tinna Mcgraw, Tonnette Mcgraw, Torre Mcgraw, Tya Mcgraw, Vianey Mcgraw, Zora Mcgraw, Adonna Mcgraw, Alberto Mcgraw, Almee
Mcgraw, Ashanta Mcgraw, Bridgid Mcgraw, Cindia Mcgraw, Cricket Mcgraw, Darah Mcgraw, Dineen Mcgraw, Elan Mcgraw, Iman Mcgraw, Jacquelynne Mcgraw, Jennilyn Mcgraw, Jessicca Mcgraw, Julio Mcgraw,
Kenosha Mcgraw, Kimyata Mcgraw, Lanessa Mcgraw, Leesha Mcgraw, Letoya Mcgraw, Nkenge Mcgraw, Priti Mcgraw, Raizel Mcgraw, Rosaisela Mcgraw, Sameerah Mcgraw, Sera Mcgraw, Shallon Mcgraw, Sharmane
Mcgraw, Shauntay Mcgraw, Sherhonda Mcgraw, Shonnie Mcgraw, Sueanne Mcgraw, Taneesha Mcgraw, Telina Mcgraw, Tenecia Mcgraw, Tyanne Mcgraw, Alexsandra Mcgraw, Angeles Mcgraw, Charlean Mcgraw, Danel
Mcgraw, Davi Mcgraw, Daysha Mcgraw, Demaris Mcgraw, Dorenda Mcgraw, Dorita Mcgraw, Dorlisa Mcgraw, Genee Mcgraw, Jamella Mcgraw, Kathyjo Mcgraw, Kaya Mcgraw, Latondra Mcgraw, Leshia Mcgraw, Mahala
Mcgraw, Marija Mcgraw, Maudie Mcgraw, Megann Mcgraw, Phuong Mcgraw, Reesa Mcgraw, Ronya Mcgraw, Selinda Mcgraw, Shama Mcgraw, Shirell Mcgraw, Shonita Mcgraw, Taiwana Mcgraw, Takenya Mcgraw, Talonda
Mcgraw, Tassie Mcgraw, Telena Mcgraw, Theodore Mcgraw, Violetta Mcgraw, Willetta Mcgraw, Adora Mcgraw, Alvin Mcgraw, Chauncey Mcgraw, Chrisy Mcgraw, Clancy Mcgraw, Claressa Mcgraw, Corrinna Mcgraw,
Darra Mcgraw, Deshawna Mcgraw, Donise Mcgraw, Elona Mcgraw, Gara Mcgraw, Georgene Mcgraw, Gila Mcgraw, Jania Mcgraw, Jocelynn Mcgraw, Joei Mcgraw, Kassy Mcgraw, Liset Mcgraw, Maleka Mcgraw, Mashell
Mcgraw, Melida Mcgraw, Michalene Mcgraw, Moya Mcgraw, Nyisha Mcgraw, Rayanne Mcgraw, Renia Mcgraw, Salima Mcgraw, Samanthia Mcgraw, Sammy Mcgraw, Santosha Mcgraw, Shaneeka Mcgraw, Shawntelle Mcgraw,
Shwanda Mcgraw, Sita Mcgraw, Tarin Mcgraw, Tawn Mcgraw, Tosca Mcgraw, Valisha Mcgraw, Vangie Mcgraw, Vernon Mcgraw, Zella Mcgraw, Amarilis Mcgraw, Amelie Mcgraw, Aubri Mcgraw, Bobbette Mcgraw, Byron
Mcgraw, Chance Mcgraw, Deshon Mcgraw, Eraina Mcgraw, Jaycee Mcgraw, Joyell Mcgraw, Keirsten Mcgraw, Kendi Mcgraw, Kendrea Mcgraw, Keyanna Mcgraw, Khristie Mcgraw, Kimberlyann Mcgraw, Kimyetta Mcgraw,
Kinberly Mcgraw, Lakeita Mcgraw, Lakina Mcgraw, Leida Mcgraw, Lenae Mcgraw, Mariette Mcgraw, Marymargaret Mcgraw, Meera Mcgraw, Micky Mcgraw, Naja Mcgraw, Oneka Mcgraw, Pedro Mcgraw, Rennie Mcgraw,
Sharlee Mcgraw, Sharonne Mcgraw, Shonya Mcgraw, Solange Mcgraw, Tashawn Mcgraw, Amorette Mcgraw, Andrienne Mcgraw, Ari Mcgraw, Carolin Mcgraw, Chalonda Mcgraw, Cinthya Mcgraw, Drusilla Mcgraw,
Erendira Mcgraw, Franki Mcgraw, Fredrick Mcgraw, Isaac Mcgraw, Jacklin Mcgraw, Jamese Mcgraw, Jeani Mcgraw, Josi Mcgraw, Jullie Mcgraw, Kadie Mcgraw, Kennetta Mcgraw, Keonna Mcgraw, Krisanne Mcgraw,
Lakrisha Mcgraw, Lakysha Mcgraw, Lasheka Mcgraw, Levette Mcgraw, Lexi Mcgraw, Lezlee Mcgraw, Lovely Mcgraw, Marah Mcgraw, Markeeta Mcgraw, Marlow Mcgraw, Meloni Mcgraw, Natausha Mcgraw, Nate Mcgraw,
Neoma Mcgraw, Nicci Mcgraw, Niema Mcgraw, Nykia Mcgraw, Oma Mcgraw, Quandra Mcgraw, Rasha Mcgraw, Shenelle Mcgraw, Sheryle Mcgraw, Sylena Mcgraw, Tasheba Mcgraw, Tennile Mcgraw, Terriann Mcgraw,
Tesia Mcgraw, Tuere Mcgraw, Tynika Mcgraw, Valena Mcgraw, Beverlee Mcgraw, Carriann Mcgraw, Chaney Mcgraw, Chantae Mcgraw, Charie Mcgraw, Cherell Mcgraw, Chrystina Mcgraw, Cookie Mcgraw, Delphina
Mcgraw, Denys Mcgraw, Digna Mcgraw, Doriann Mcgraw, Elecia Mcgraw, Elmira Mcgraw, Irina Mcgraw, Jacky Mcgraw, Jimmi Mcgraw, Joslin Mcgraw, Katty Mcgraw, Kimberlin Mcgraw, Laquan Mcgraw, Latonga
Mcgraw, Lauran Mcgraw, Lei Mcgraw, Maralee Mcgraw, Marilena Mcgraw, Markeisha Mcgraw, Mena Mcgraw, Mike Mcgraw, Nadeen Mcgraw, Ranelle Mcgraw, Renika Mcgraw, Ritu Mcgraw, Roshawnda Mcgraw, Sandria
Mcgraw, Shann Mcgraw, Shellene Mcgraw, Shemica Mcgraw, Shenandoah Mcgraw, Sherrye Mcgraw, Sholonda Mcgraw, Teka Mcgraw, Temica Mcgraw, Thomasena Mcgraw, Vallerie Mcgraw, Velva Mcgraw, Annjanette
Mcgraw, Arnitra Mcgraw, Asa Mcgraw, Barri Mcgraw, Deborha Mcgraw, Deshaun Mcgraw, Disa Mcgraw, Doree Mcgraw, Enrique Mcgraw, Fredrika Mcgraw, Gisel Mcgraw, Jemima Mcgraw, Kellene Mcgraw, Kinsey
Mcgraw, Lanice Mcgraw, Leonore Mcgraw, Liesel Mcgraw, Loredana Mcgraw, Lynita Mcgraw, Maeve Mcgraw, Michale Mcgraw, Michalle Mcgraw, Nadira Mcgraw, Orlando Mcgraw, Radonna Mcgraw, Rivkah Mcgraw,
Sanita Mcgraw, Shaketa Mcgraw, Shatika Mcgraw, Stepheny Mcgraw, Tashanda Mcgraw, Tashanna Mcgraw, Thera Mcgraw, Vernette Mcgraw, Virna Mcgraw, Wanetta Mcgraw, Andee Mcgraw, Aparna Mcgraw, Armando
Mcgraw, Brynne Mcgraw, Carolynne Mcgraw, Danyal Mcgraw, Drena Mcgraw, Faydra Mcgraw, Felishia Mcgraw, Gwendalyn Mcgraw, Jennyfer Mcgraw, Jesika Mcgraw, Johnell Mcgraw, Kalen Mcgraw, Keelie Mcgraw,
Kirby Mcgraw, Kristopher Mcgraw, Lynnett Mcgraw, Mikka Mcgraw, Moraima Mcgraw, Nasha Mcgraw, Nechelle Mcgraw, Nelson Mcgraw, Rashawnda Mcgraw, Rianna Mcgraw, Rogina Mcgraw, Rudy Mcgraw, Schwanda
Mcgraw, Serafina Mcgraw, Seth Mcgraw, Shantrice Mcgraw, Shenee Mcgraw, Sherrel Mcgraw, Shirin Mcgraw, Soila Mcgraw, Sujata Mcgraw, Tannia Mcgraw, Tiya Mcgraw, Tonnie Mcgraw, Tuwanda Mcgraw, Young
Mcgraw, Anjela Mcgraw, Aquanetta Mcgraw, Ardis Mcgraw, Brigitta Mcgraw, Carmilla Mcgraw, Catherin Mcgraw, Charna Mcgraw, Chaunda Mcgraw, Chela Mcgraw, Cherylyn Mcgraw, Clifton Mcgraw, Danice Mcgraw,
Daylene Mcgraw, Delecia Mcgraw, Dell Mcgraw, Eleonora Mcgraw, Gesenia Mcgraw, Giana Mcgraw, Iisha Mcgraw, Jacqualyn Mcgraw, Jule Mcgraw, Kanita Mcgraw, Karrin Mcgraw, Keishia Mcgraw, Kerith Mcgraw,
Kima Mcgraw, Letica Mcgraw, Letta Mcgraw, Lucianna Mcgraw, Mala Mcgraw, Maresa Mcgraw, Michaeline Mcgraw, Mitra Mcgraw, Norene Mcgraw, Raul Mcgraw, Robi Mcgraw, Sativa Mcgraw, Shakeema Mcgraw,
Shalisha Mcgraw, Sherece Mcgraw, Sherlene Mcgraw, Talicia Mcgraw, Tinia Mcgraw, Toneka Mcgraw, Tuwanna Mcgraw, Tymeka Mcgraw, Vesna Mcgraw, Yvetta Mcgraw, Akeisha Mcgraw, Almeta Mcgraw, Alvita
Mcgraw, Ameerah Mcgraw, Amena Mcgraw, Anntoinette Mcgraw, Asusena Mcgraw, Benetta Mcgraw, Bernard Mcgraw, Bibiana Mcgraw, Bracha Mcgraw, Branden Mcgraw, Carrisa Mcgraw, Channell Mcgraw, Christee
Mcgraw, Daisey Mcgraw, Dandra Mcgraw, Dene Mcgraw, Destinee Mcgraw, Emmalee Mcgraw, Evamarie Mcgraw, Felicha Mcgraw, Hally Mcgraw, Harmonie Mcgraw, Ivone Mcgraw, Janira Mcgraw, Jennfier Mcgraw,
Josalyn Mcgraw, Karley Mcgraw, Kellyjo Mcgraw, Kinshasa Mcgraw, Laurena Mcgraw, Lewis Mcgraw, Louvenia Mcgraw, Lynley Mcgraw, Maleah Mcgraw, Marceline Mcgraw, Maree Mcgraw, Maxie Mcgraw, Melora
Mcgraw, Minta Mcgraw, Nella Mcgraw, Nikea Mcgraw, Orlanda Mcgraw, Radhika Mcgraw, Renelle Mcgraw, Robynne Mcgraw, Rosland Mcgraw, Saudia Mcgraw, Shannette Mcgraw, Sharetta Mcgraw, Sharnita Mcgraw,
Shequita Mcgraw, Tanessa Mcgraw, Tannis Mcgraw, Thu Mcgraw, Toiya Mcgraw, Umeka Mcgraw, Yadhira Mcgraw, Birdie Mcgraw, Bradi Mcgraw, Carola Mcgraw, Chelle Mcgraw, Chermaine Mcgraw, Christelle Mcgraw,
Cortnie Mcgraw, Dalinda Mcgraw, Daphene Mcgraw, Demetri Mcgraw, Ennifer Mcgraw, Esta Mcgraw, Francheska Mcgraw, Griselle Mcgraw, Ife Mcgraw, Jahaira Mcgraw, Jammi Mcgraw, Javonne Mcgraw, Jenie
Mcgraw, Jocelin Mcgraw, Karole Mcgraw, Kelee Mcgraw, Khrista Mcgraw, Leya Mcgraw, Luwanda Mcgraw, Makayla Mcgraw, Marly Mcgraw, Nzinga Mcgraw, Paquita Mcgraw, Ramanda Mcgraw, Rayleen Mcgraw, Rinda
Mcgraw, Shauntell Mcgraw, Tamecka Mcgraw, Tanji Mcgraw, Tela Mcgraw, Teneisha Mcgraw, Valicia Mcgraw, Willona Mcgraw, Alejandro Mcgraw, Ameka Mcgraw, Ashante Mcgraw, Brisa Mcgraw, Charli Mcgraw,
Charmagne Mcgraw, Charrise Mcgraw, Crystol Mcgraw, Darya Mcgraw, Destiney Mcgraw, Elidia Mcgraw, France Mcgraw, Galadriel Mcgraw, Genifer Mcgraw, Gladis Mcgraw, Glori Mcgraw, Harriette Mcgraw, Ioanna
Mcgraw, Jeanann Mcgraw, Kittie Mcgraw, Ladon Mcgraw, Latesa Mcgraw, Linn Mcgraw, Loralie Mcgraw, Mazie Mcgraw, Mekisha Mcgraw, Nivia Mcgraw, Paulene Mcgraw, Renette Mcgraw, Saadia Mcgraw, Shakena
Mcgraw, Sheritta Mcgraw, Stepahnie Mcgraw, Taliah Mcgraw, Teretha Mcgraw, Tessy Mcgraw, Wendelyn Mcgraw, Zonia Mcgraw, Ailsa Mcgraw, Alonna Mcgraw, Andres Mcgraw, Asma Mcgraw, Austin Mcgraw, Brandis
Mcgraw, Corinda Mcgraw, Deetra Mcgraw, Dyanne Mcgraw, Eliabeth Mcgraw, Evelynn Mcgraw, Geana Mcgraw, Geeta Mcgraw, Georgeanne Mcgraw, Glynnis Mcgraw, Hilari Mcgraw, Hindy Mcgraw, Jackquline Mcgraw,
Joaquina Mcgraw, Jolynne Mcgraw, Kirsta Mcgraw, Latissa Mcgraw, Lucile Mcgraw, Makeisha Mcgraw, Marli Mcgraw, Meeghan Mcgraw, Michaelyn Mcgraw, Monte Mcgraw, Omeka Mcgraw, Patra Mcgraw, Ragen Mcgraw,
Reshma Mcgraw, Ronnell Mcgraw, Shakera Mcgraw, Shantella Mcgraw, Shar Mcgraw, Shelagh Mcgraw, Sherisse Mcgraw, Slyvia Mcgraw, Stephenia Mcgraw, Tarisha Mcgraw, Tria Mcgraw, Tynetta Mcgraw, Veena
Mcgraw, Aleksandra Mcgraw, Analia Mcgraw, Aneshia Mcgraw, Anesia Mcgraw, Anissia Mcgraw, Caroll Mcgraw, Celisa Mcgraw, Chariti Mcgraw, Charnette Mcgraw, Cheria Mcgraw, Dabney Mcgraw, Davon Mcgraw,
Demita Mcgraw, Denny Mcgraw, Dwanda Mcgraw, Erick Mcgraw, Fancy Mcgraw, Felesha Mcgraw, Genette Mcgraw, Graziella Mcgraw, Helaine Mcgraw, Huong Mcgraw, Idania Mcgraw, Ilsa Mcgraw, Iola Mcgraw,
Janifer Mcgraw, Jaynie Mcgraw, Jil Mcgraw, Joyann Mcgraw, Kalina Mcgraw, Kana Mcgraw, Katheleen Mcgraw, Kelie Mcgraw, Kimbly Mcgraw, Ladina Mcgraw, Lamisha Mcgraw, Landra Mcgraw, Laretha Mcgraw,
Lashuna Mcgraw, Latorsha Mcgraw, Lehua Mcgraw, Libbie Mcgraw, Loris Mcgraw, Lovetta Mcgraw, Marilin Mcgraw, Marquis Mcgraw, Medea Mcgraw, Michaella Mcgraw, Milly Mcgraw, Montina Mcgraw, Muna Mcgraw,
Nickcole Mcgraw, Nickia Mcgraw, Quianna Mcgraw, Retina Mcgraw, Rubina Mcgraw, Sallyann Mcgraw, Sangeeta Mcgraw, Shalinda Mcgraw, Shatonya Mcgraw, Tamasha Mcgraw, Tenise Mcgraw, Terrilynn Mcgraw, Teya
Mcgraw, Toma Mcgraw, Trang Mcgraw, Vienna Mcgraw, Zenda Mcgraw, Amisha Mcgraw, Chenell Mcgraw, Cherene Mcgraw, Chimere Mcgraw, Darcell Mcgraw, Desa Mcgraw, Devi Mcgraw, Dove Mcgraw, Eilene Mcgraw,
Jarita Mcgraw, Jenetta Mcgraw, Johnie Mcgraw, Kanani Mcgraw, Kanya Mcgraw, Katiria Mcgraw, Kenyon Mcgraw, Khristi Mcgraw, Kidada Mcgraw, Kristien Mcgraw, Laressa Mcgraw, Lashonta Mcgraw, Lataisha
Mcgraw, Leanda Mcgraw, Leshonda Mcgraw, Letia Mcgraw, Lorenzo Mcgraw, Makenzie Mcgraw, Manya Mcgraw, Maribelle Mcgraw, Maryjean Mcgraw, Moniqua Mcgraw, Myeisha Mcgraw, Mystie Mcgraw, Nakeesha Mcgraw,
Neeley Mcgraw, Neil Mcgraw, Nida Mcgraw, Oletha Mcgraw, Rebel Mcgraw, Rosaland Mcgraw, Satonya Mcgraw, Shanah Mcgraw, Sherly Mcgraw, Sherrilyn Mcgraw, Shonia Mcgraw, Shuna Mcgraw, Shylah Mcgraw,
Tameisha Mcgraw, Telecia Mcgraw, Terrill Mcgraw, Torry Mcgraw, Tracyann Mcgraw, Undrea Mcgraw, Usha Mcgraw, Velicia Mcgraw, Windie Mcgraw, Yehudis Mcgraw, Ameenah Mcgraw, Amika Mcgraw, Anupama
Mcgraw, Bethaney Mcgraw, Brookie Mcgraw, Bryce Mcgraw, Cherika Mcgraw, Courtni Mcgraw, Danyle Mcgraw, Darilyn Mcgraw, Darnita Mcgraw, Eveline Mcgraw, Gilbert Mcgraw, Giovanni Mcgraw, Jacquel Mcgraw,
Jenai Mcgraw, Jocelyne Mcgraw, Jull Mcgraw, Karene Mcgraw, Karisha Mcgraw, Kelvin Mcgraw, Kenzie Mcgraw, Kimblery Mcgraw, Kyleen Mcgraw, Laketta Mcgraw, Larisha Mcgraw, Lashae Mcgraw, Laveta Mcgraw,
Lezli Mcgraw, Marlynn Mcgraw, Melana Mcgraw, Meribeth Mcgraw, Onita Mcgraw, Philicia Mcgraw, Raychel Mcgraw, Roschelle Mcgraw, Rosilyn Mcgraw, Sandhya Mcgraw, Selenia Mcgraw, Seretha Mcgraw, Shadia
Mcgraw, Shalise Mcgraw, Sonnet Mcgraw, Tacey Mcgraw, Taurus Mcgraw, Telly Mcgraw, Tenita Mcgraw, Tiawana Mcgraw, Timica Mcgraw, Warren Mcgraw, Yamile Mcgraw, Yaminah Mcgraw, Adel Mcgraw, Aiyana
Mcgraw, Amybeth Mcgraw, Angee Mcgraw, Angeligue Mcgraw, Areli Mcgraw, Asheley Mcgraw, Bina Mcgraw, Celines Mcgraw, Charlet Mcgraw, Charley Mcgraw, Charonda Mcgraw, Dameka Mcgraw, Danne Mcgraw,
Deloise Mcgraw, Denay Mcgraw, Denene Mcgraw, Donata Mcgraw, Georgianne Mcgraw, Jamelia Mcgraw, Jennetta Mcgraw, Jesusa Mcgraw, Johnda Mcgraw, Juleen Mcgraw, Khaliah Mcgraw, Kimala Mcgraw, Kimble
Mcgraw, Kimmarie Mcgraw, Korene Mcgraw, Lezley Mcgraw, Lindie Mcgraw, Lorilyn Mcgraw, Mayme Mcgraw, Meeka Mcgraw, Natlie Mcgraw, Ranita Mcgraw, Rasheen Mcgraw, Rayetta Mcgraw, Savina Mcgraw,
Scarlette Mcgraw, Shaindy Mcgraw, Sheana Mcgraw, Sheldon Mcgraw, Shoni Mcgraw, Stephan Mcgraw, Taneeka Mcgraw, Tanesia Mcgraw, Tannya Mcgraw, Tionna Mcgraw, Tresia Mcgraw, Tylisha Mcgraw, Vonita
Mcgraw, Vonnetta Mcgraw, Wren Mcgraw, Akiba Mcgraw, Blaire Mcgraw, Buffey Mcgraw, Candita Mcgraw, Charnelle Mcgraw, Cynda Mcgraw, Danesha Mcgraw, Darbie Mcgraw, Deah Mcgraw, Deette Mcgraw, Devita
Mcgraw, Devonia Mcgraw, Dianah Mcgraw, Dustina Mcgraw, Dylan Mcgraw, Elli Mcgraw, Eun Mcgraw, Geneen Mcgraw, Geni Mcgraw, Jenevieve Mcgraw, Julaine Mcgraw, Kaitlyn Mcgraw, Lalania Mcgraw, Latrise
Mcgraw, Mahalia Mcgraw, Manon Mcgraw, Marycatherine Mcgraw, Nyasha Mcgraw, Orlena Mcgraw, Petina Mcgraw, Precilla Mcgraw, Raenell Mcgraw, Ramonica Mcgraw, Reema Mcgraw, Scotti Mcgraw, Shakeya Mcgraw,
Shaylene Mcgraw, Shenique Mcgraw, Shimika Mcgraw, Takiesha Mcgraw, Tambi Mcgraw, Tammatha Mcgraw, Tauheedah Mcgraw, Timiko Mcgraw, Tomisha Mcgraw, Venesa Mcgraw, Venise Mcgraw, Akesha Mcgraw, Anmarie
Mcgraw, Arnette Mcgraw, Brennan Mcgraw, Cayla Mcgraw, Cecil Mcgraw, Coreena Mcgraw, Danessa Mcgraw, Day Mcgraw, Denni Mcgraw, Fernando Mcgraw, Floretta Mcgraw, Fred Mcgraw, Gardenia Mcgraw, Genevie
Mcgraw, Georgann Mcgraw, Grasiela Mcgraw, Jaimy Mcgraw, Jaquelyn Mcgraw, Jerelyn Mcgraw, Jinnifer Mcgraw, Johnelle Mcgraw, Jovonna Mcgraw, Kady Mcgraw, Keala Mcgraw, Kesi Mcgraw, Kyana Mcgraw,
Laketia Mcgraw, Laquanta Mcgraw, Latoy Mcgraw, Laurine Mcgraw, Lynea Mcgraw, Lynee Mcgraw, Maegen Mcgraw, Margarite Mcgraw, Marijane Mcgraw, Mellany Mcgraw, Monia Mcgraw, Monice Mcgraw, Mylene
Mcgraw, Nakiya Mcgraw, Neila Mcgraw, Oleta Mcgraw, Queenie Mcgraw, Quinetta Mcgraw, Rekha Mcgraw, Remi Mcgraw, Rickey Mcgraw, Roselynn Mcgraw, Samanatha Mcgraw, Shakila Mcgraw, Shalandra Mcgraw,
Shamira Mcgraw, Shandel Mcgraw, Shandrea Mcgraw, Shinita Mcgraw, Stayce Mcgraw, Takela Mcgraw, Talibah Mcgraw, Tameria Mcgraw, Taronda Mcgraw, Telia Mcgraw, Ticia Mcgraw, Tovah Mcgraw, Venissa
Mcgraw, Verlinda Mcgraw, Vittoria Mcgraw, Alicyn Mcgraw, Amylynn Mcgraw, Anabell Mcgraw, Anela Mcgraw, Annice Mcgraw, Aubry Mcgraw, Coralie Mcgraw, Coryn Mcgraw, Darlyn Mcgraw, Dejuana Mcgraw, Eleana
Mcgraw, Esmerelda Mcgraw, Georganne Mcgraw, Jacqlyn Mcgraw, Jodine Mcgraw, Joelyn Mcgraw, Josey Mcgraw, Keanna Mcgraw, Kendy Mcgraw, Korine Mcgraw, Kristianna Mcgraw, Laneka Mcgraw, Lannie Mcgraw,
Mathilda Mcgraw, Myiesha Mcgraw, Nessa Mcgraw, Nikiya Mcgraw, Penina Mcgraw, Radha Mcgraw, Ralonda Mcgraw, Shantele Mcgraw, Sharlette Mcgraw, Shelanda Mcgraw, Shelda Mcgraw, Shere Mcgraw, Shuntel
Mcgraw, Stephaney Mcgraw, Tabbitha Mcgraw, Tamkia Mcgraw, Teona Mcgraw, Terrisa Mcgraw, Teryl Mcgraw, Trellis Mcgraw, Vanissa Mcgraw, Abigale Mcgraw, Bonnita Mcgraw, Carren Mcgraw, Clementine Mcgraw,
Coni Mcgraw, Demonica Mcgraw, Denis Mcgraw, Diantha Mcgraw, Edana Mcgraw, Elene Mcgraw, Faithe Mcgraw, Franklin Mcgraw, Haidee Mcgraw, Jaima Mcgraw, Javonda Mcgraw, Jayma Mcgraw, Jeffifer Mcgraw,
Jennel Mcgraw, Jennfer Mcgraw, Katryna Mcgraw, Keta Mcgraw, Kristyne Mcgraw, Krystle Mcgraw, Kymberlee Mcgraw, Kymberli Mcgraw, Lacole Mcgraw, Laranda Mcgraw, Leea Mcgraw, Loryn Mcgraw, Lyndell
Mcgraw, Lyndy Mcgraw, Mande Mcgraw, Marlissa Mcgraw, Marrissa Mcgraw, Meko Mcgraw, Miryam Mcgraw, Neida Mcgraw, Odilia Mcgraw, Pamila Mcgraw, Porscha Mcgraw, Quentina Mcgraw, Riley Mcgraw, Roselia
Mcgraw, Sharifah Mcgraw, Sonyia Mcgraw, Sugar Mcgraw, Tamico Mcgraw, Tanara Mcgraw, Tiombe Mcgraw, Verity Mcgraw, Zoey Mcgraw, Abril Mcgraw, Aine Mcgraw, Aixa Mcgraw, Ange Mcgraw, Annah Mcgraw,
Avelina Mcgraw, Chantil Mcgraw, Charmel Mcgraw, Cheralyn Mcgraw, Chesley Mcgraw, Colin Mcgraw, Cornelius Mcgraw, Cortni Mcgraw, Deserae Mcgraw, Dru Mcgraw, Emeline Mcgraw, Iran Mcgraw, Jesslyn
Mcgraw, Jovon Mcgraw, Kalani Mcgraw, Kaleen Mcgraw, Kamilla Mcgraw, Kemi Mcgraw, Kendrick Mcgraw, Lachele Mcgraw, Letesha Mcgraw, Makita Mcgraw, Maricia Mcgraw, Mariya Mcgraw, Marlise Mcgraw, Nara
Mcgraw, Ngina Mcgraw, Nikkol Mcgraw, Okema Mcgraw, Palma Mcgraw, Priscillia Mcgraw, Ranell Mcgraw, Rashmi Mcgraw, Renessa Mcgraw, Ricarda Mcgraw, Rozlyn Mcgraw, Shamia Mcgraw, Sharissa Mcgraw,
Shianne Mcgraw, Takeesha Mcgraw, Tekia Mcgraw, Tyhesha Mcgraw, Vanassa Mcgraw, Vicenta Mcgraw, Abagail Mcgraw, Addy Mcgraw, Akemi Mcgraw, Analilia Mcgraw, Andreanna Mcgraw, Anh Mcgraw, Annmargaret
Mcgraw, Caitlyn Mcgraw, Carrin Mcgraw, Celestial Mcgraw, Chameka Mcgraw, Chandrika Mcgraw, Charlsie Mcgraw, Cherlynn Mcgraw, Darien Mcgraw, Darlean Mcgraw, Devyn Mcgraw, Dyane Mcgraw, Eman Mcgraw,
Erna Mcgraw, Fontella Mcgraw, Halie Mcgraw, Ilyse Mcgraw, Janece Mcgraw, Jayla Mcgraw, Jeannene Mcgraw, Juwana Mcgraw, Kachina Mcgraw, Keleigh Mcgraw, Kissie Mcgraw, Krisi Mcgraw, Kristyl Mcgraw,
Krystie Mcgraw, Kyesha Mcgraw, Lizandra Mcgraw, Lyda Mcgraw, Majorie Mcgraw, Marnita Mcgraw, Marrisa Mcgraw, Miguelina Mcgraw, Nadege Mcgraw, Nakiesha Mcgraw, Neesha Mcgraw, Raeleen Mcgraw, Rashana
Mcgraw, Renada Mcgraw, Rupa Mcgraw, Shaunette Mcgraw, Sheetal Mcgraw, Sherryann Mcgraw, Tanecia Mcgraw, Tashiba Mcgraw, Tawney Mcgraw, Telma Mcgraw, Tomeeka Mcgraw, Xochil Mcgraw, Yuki Mcgraw, Ahna
Mcgraw, Aneesha Mcgraw, Aphrodite Mcgraw, Areatha Mcgraw, Cam Mcgraw, Chalene Mcgraw, Charnetta Mcgraw, Clarise Mcgraw, Cleta Mcgraw, Danniel Mcgraw, Deshana Mcgraw, Eduardo Mcgraw, Erminia Mcgraw,
Eron Mcgraw, Hettie Mcgraw, Janan Mcgraw, Janda Mcgraw, Jayda Mcgraw, Jeanny Mcgraw, Katanya Mcgraw, Kaylin Mcgraw, Keeva Mcgraw, Kimesha Mcgraw, Kristeena Mcgraw, Kristyna Mcgraw, Lakeyshia Mcgraw,
Latesia Mcgraw, Latoia Mcgraw, Lavona Mcgraw, Lonni Mcgraw, Loukisha Mcgraw, Manika Mcgraw, Marbella Mcgraw, Marcina Mcgraw, Marialuisa Mcgraw, Mariellen Mcgraw, Mariluz Mcgraw, Mikia Mcgraw, Morning
Mcgraw, Myndi Mcgraw, Ronalda Mcgraw, Shanitra Mcgraw, Sherea Mcgraw, Shericka Mcgraw, Sonny Mcgraw, Suzane Mcgraw, Terea Mcgraw, Tersa Mcgraw, Yarnell Mcgraw, Yezenia Mcgraw, Zona Mcgraw, Alissia
Mcgraw, Anjenette Mcgraw, Antwan Mcgraw, Bernette Mcgraw, Cedar Mcgraw, Chandria Mcgraw, Cherina Mcgraw, Chrisandra Mcgraw, Christiann Mcgraw, Cosandra Mcgraw, Danea Mcgraw, Demetress Mcgraw,
Emmeline Mcgraw, Ernesta Mcgraw, Fabiana Mcgraw, Fabienne Mcgraw, Falecia Mcgraw, Georgena Mcgraw, Gerardo Mcgraw, Gwynn Mcgraw, Heavenly Mcgraw, Ivett Mcgraw, Jaya Mcgraw, Jessicia Mcgraw, Jetaun
Mcgraw, Joane Mcgraw, Kalliopi Mcgraw, Kalynn Mcgraw, Karmon Mcgraw, Kischa Mcgraw, Kita Mcgraw, Lanea Mcgraw, Laron Mcgraw, Latonyia Mcgraw, Lavone Mcgraw, Leasha Mcgraw, Liann Mcgraw, Madelyne
Mcgraw, Malissia Mcgraw, Margit Mcgraw, Marlowe Mcgraw, Mashonda Mcgraw, Meggen Mcgraw, Merari Mcgraw, Michille Mcgraw, Missi Mcgraw, Nichoel Mcgraw, Nikie Mcgraw, Nyssa Mcgraw, Pooja Mcgraw, Purvi
Mcgraw, Ryane Mcgraw, Sakeenah Mcgraw, Sam Mcgraw, Shaquan Mcgraw, Sharmel Mcgraw, Shlonda Mcgraw, Tamsen Mcgraw, Tikia Mcgraw, Tomicka Mcgraw, Tremeka Mcgraw, Wenonah Mcgraw, Yavonne Mcgraw, Adraine
Mcgraw, Aiysha Mcgraw, Akila Mcgraw, Almira Mcgraw, Alnita Mcgraw, Ambrosia Mcgraw, Aminta Mcgraw, Anedra Mcgraw, Angeliki Mcgraw, Aymee Mcgraw, Barrett Mcgraw, Batina Mcgraw, Brandey Mcgraw, Brena
Mcgraw, Bryant Mcgraw, Charidy Mcgraw, Davona Mcgraw, Delissa Mcgraw, Denea Mcgraw, Denicia Mcgraw, Deniese Mcgraw, Desta Mcgraw, Dimple Mcgraw, Emelie Mcgraw, Eustacia Mcgraw, Farra Mcgraw, Giulia
Mcgraw, Jannetta Mcgraw, Jen Mcgraw, Jennice Mcgraw, Kareemah Mcgraw, Kaysie Mcgraw, Latice Mcgraw, Letanya Mcgraw, Liticia Mcgraw, Luna Mcgraw, Madge Mcgraw, Marielena Mcgraw, Merrily Mcgraw, Mitsy
Mcgraw, Nachelle Mcgraw, Nolita Mcgraw, Palmira Mcgraw, Parthenia Mcgraw, Ramey Mcgraw, Raymona Mcgraw, Rhodesia Mcgraw, Sanna Mcgraw, Shakema Mcgraw, Shamieka Mcgraw, Shandria Mcgraw, Shaniece
Mcgraw, Sharronda Mcgraw, Sheletha Mcgraw, Shoshannah Mcgraw, Suprina Mcgraw, Tanyia Mcgraw, Tarri Mcgraw, Tequita Mcgraw, Thressa Mcgraw, Twala Mcgraw, Tyneshia Mcgraw, Willena Mcgraw, Aaryn Mcgraw,
Ailene Mcgraw, Alfredo Mcgraw, Alinda Mcgraw, Avia Mcgraw, Brandelyn Mcgraw, Brandolyn Mcgraw, Brieanna Mcgraw, Brittanie Mcgraw, Celest Mcgraw, Charman Mcgraw, Chenise Mcgraw, Colandra Mcgraw,
Damari Mcgraw, Donni Mcgraw, Dorice Mcgraw, Dustine Mcgraw, Elease Mcgraw, Jamara Mcgraw, Jameca Mcgraw, Jessalyn Mcgraw, Joyanna Mcgraw, Kalia Mcgraw, Kennisha Mcgraw, Kewanna Mcgraw, Krystel
Mcgraw, Lashara Mcgraw, Leslieann Mcgraw, Letetia Mcgraw, Letizia Mcgraw, Lisanne Mcgraw, Lovella Mcgraw, Maranatha Mcgraw, Melisia Mcgraw, Meriah Mcgraw, Monaca Mcgraw, Myrtis Mcgraw, Nadean Mcgraw,
Naila Mcgraw, Narissa Mcgraw, Natale Mcgraw, Nealy Mcgraw, November Mcgraw, Petrice Mcgraw, Renie Mcgraw, Robecca Mcgraw, Sarra Mcgraw, Shakeena Mcgraw, Shalunda Mcgraw, Sharin Mcgraw, Shatoya
Mcgraw, Soni Mcgraw, Sulma Mcgraw, Taffany Mcgraw, Tahara Mcgraw, Taralynn Mcgraw, Tarena Mcgraw, Tatyana Mcgraw, Tekeisha Mcgraw, Timothea Mcgraw, Tishana Mcgraw, Toshua Mcgraw, Tresea Mcgraw,
Tykisha Mcgraw, Varonica Mcgraw, Yolinda Mcgraw, Adra Mcgraw, Amia Mcgraw, Aquarius Mcgraw, Baila Mcgraw, Betheny Mcgraw, Cady Mcgraw, Calla Mcgraw, Catrinia Mcgraw, Chanika Mcgraw, Chellie Mcgraw,
Chelsy Mcgraw, Christana Mcgraw, Cicley Mcgraw, Crescent Mcgraw, Deronda Mcgraw, Diamantina Mcgraw, Dyani Mcgraw, Elvina Mcgraw, Emiley Mcgraw, Haneefah Mcgraw, Ivan Mcgraw, Jamesetta Mcgraw, Janele
Mcgraw, Jolean Mcgraw, Jory Mcgraw, Julieanna Mcgraw, Jyoti Mcgraw, Kaela Mcgraw, Katena Mcgraw, Kaycie Mcgraw, Kismet Mcgraw, Kobi Mcgraw, Lakechia Mcgraw, Lakesa Mcgraw, Lalanya Mcgraw, Latiesha
Mcgraw, Lauralyn Mcgraw, Lekita Mcgraw, Loletha Mcgraw, Lonetta Mcgraw, Lukisha Mcgraw, Mele Mcgraw, Nadra Mcgraw, Natia Mcgraw, Ngozi Mcgraw, Nija Mcgraw, October Mcgraw, Olinda Mcgraw, Preeti
Mcgraw, Raine Mcgraw, Ronie Mcgraw, Sharlie Mcgraw, Sharnetta Mcgraw, Shontia Mcgraw, Tamakia Mcgraw, Taquita Mcgraw, Temeca Mcgraw, Tenna Mcgraw, Teresea Mcgraw, Teryn Mcgraw, Tessica Mcgraw, Trayce
Mcgraw, Tujuana Mcgraw, Alys Mcgraw, Aris Mcgraw, Asya Mcgraw, Atina Mcgraw, Ayonna Mcgraw, Cherl Mcgraw, Corretta Mcgraw, Cybill Mcgraw, Gaylynn Mcgraw, Heatherly Mcgraw, Jacquelina Mcgraw, Jeanean
Mcgraw, Jeniece Mcgraw, Johnathan Mcgraw, Krystine Mcgraw, Larua Mcgraw, Lelah Mcgraw, Lilla Mcgraw, Linh Mcgraw, Lisia Mcgraw, Lucilla Mcgraw, Malita Mcgraw, Matisha Mcgraw, Melika Mcgraw, Merci
Mcgraw, Mishell Mcgraw, Monae Mcgraw, Monee Mcgraw, Montrice Mcgraw, Najla Mcgraw, Parrish Mcgraw, Ramsey Mcgraw, Raya Mcgraw, Rejeana Mcgraw, Roanna Mcgraw, Roxan Mcgraw, Shai Mcgraw, Shanen Mcgraw,
Sharalyn Mcgraw, Shuntay Mcgraw, Taheerah Mcgraw, Takeysha Mcgraw, Tamelia Mcgraw, Tanyika Mcgraw, Temperance Mcgraw, Theodosia Mcgraw, Thyra Mcgraw, Tyria Mcgraw, Urania Mcgraw, Yvett Mcgraw,
Angelynn Mcgraw, Anitria Mcgraw, Anny Mcgraw, Arlisha Mcgraw, Aya Mcgraw, Brielle Mcgraw, Careen Mcgraw, Correne Mcgraw, Dagmar Mcgraw, Dalynn Mcgraw, Dannetta Mcgraw, Delight Mcgraw, Denetta Mcgraw,
Desaree Mcgraw, Donyel Mcgraw, Eisha Mcgraw, Eleanora Mcgraw, Felix Mcgraw, Fransisca Mcgraw, Jannel Mcgraw, Jenia Mcgraw, Jettie Mcgraw, Jolina Mcgraw, Jowanna Mcgraw, Junko Mcgraw, Kaaren Mcgraw,
Kathey Mcgraw, Keryn Mcgraw, Ketina Mcgraw, Korrine Mcgraw, Kristalyn Mcgraw, Laini Mcgraw, Lamona Mcgraw, Lanay Mcgraw, Leza Mcgraw, Maddalena Mcgraw, Mellody Mcgraw, Merica Mcgraw, Merita Mcgraw,
Nafeesa Mcgraw, Nicholl Mcgraw, Oneika Mcgraw, Pamelyn Mcgraw, Romonda Mcgraw, Ronique Mcgraw, Rosslyn Mcgraw, Rosy Mcgraw, Sachiko Mcgraw, Sarajane Mcgraw, Satara Mcgraw, Shamra Mcgraw, Tamberly
Mcgraw, Tawona Mcgraw, Tiasha Mcgraw, Tine Mcgraw, Trameka Mcgraw, Tynia Mcgraw, Valentine Mcgraw, Venisha Mcgraw, Veronia Mcgraw, Vinnie Mcgraw, Yuri Mcgraw, Aissa Mcgraw, Aleda Mcgraw, Amenda
Mcgraw, Andromeda Mcgraw, Antigone Mcgraw, Aren Mcgraw, Ariadne Mcgraw, Bambie Mcgraw, Belkis Mcgraw, Bunnie Mcgraw, Claudina Mcgraw, Darcelle Mcgraw, Dulcinea Mcgraw, Enola Mcgraw, Florida Mcgraw,
Francia Mcgraw, Ginni Mcgraw, Grizelda Mcgraw, Israel Mcgraw, Jalynn Mcgraw, Jamil Mcgraw, Jenika Mcgraw, Joeanna Mcgraw, Katesha Mcgraw, Katreena Mcgraw, Keilani Mcgraw, Kiah Mcgraw, Kjersti Mcgraw,
Korrin Mcgraw, Leonda Mcgraw, Liesa Mcgraw, Lyna Mcgraw, Manuelita Mcgraw, Marka Mcgraw, Markia Mcgraw, Maronda Mcgraw, Melondy Mcgraw, Melonee Mcgraw, Micheala Mcgraw, Misa Mcgraw, Nikeisha Mcgraw,
Odalys Mcgraw, Parul Mcgraw, Petula Mcgraw, Rainee Mcgraw, Rainie Mcgraw, Romi Mcgraw, Rubie Mcgraw, Shamone Mcgraw, Shawnia Mcgraw, Sheritha Mcgraw, Shery Mcgraw, Tamula Mcgraw, Tasheena Mcgraw,
Thao Mcgraw, Thereasa Mcgraw, Tjuana Mcgraw, Tomesha Mcgraw, Turquoise Mcgraw, Tyiesha Mcgraw, Vernee Mcgraw, Annastasia Mcgraw, Ardra Mcgraw, Bibi Mcgraw, Blaine Mcgraw, Bridie Mcgraw, Carmelle
Mcgraw, Chalice Mcgraw, Chanie Mcgraw, Chauntelle Mcgraw, Dareth Mcgraw, Demitria Mcgraw, Denetria Mcgraw, Didi Mcgraw, Dimitria Mcgraw, Dwight Mcgraw, Elisabetta Mcgraw, Ellisha Mcgraw, Ileen
Mcgraw, Jamala Mcgraw, Jayleen Mcgraw, Jeny Mcgraw, Jeraldine Mcgraw, Jowanda Mcgraw, Joylene Mcgraw, Kareena Mcgraw, Karrah Mcgraw, Kashonda Mcgraw, Kendel Mcgraw, Ketrina Mcgraw, Lagena Mcgraw,
Lashann Mcgraw, Lasheena Mcgraw, Latish Mcgraw, Leroy Mcgraw, Maryum Mcgraw, Mende Mcgraw, Mieka Mcgraw, Nakea Mcgraw, Niurka Mcgraw, Philippa Mcgraw, Pina Mcgraw, Romanda Mcgraw, Satrina Mcgraw,
Shaletha Mcgraw, Shandrika Mcgraw, Shawonda Mcgraw, Sherae Mcgraw, Shernita Mcgraw, Sheronica Mcgraw, Silena Mcgraw, Sora Mcgraw, Stehanie Mcgraw, Tanikka Mcgraw, Tashi Mcgraw, Tawyna Mcgraw, Tenesia
Mcgraw, Tikesha Mcgraw, Trenell Mcgraw, Tristina Mcgraw, Vallie Mcgraw, Verla Mcgraw, Vinessa Mcgraw, Walida Mcgraw, Yolandra Mcgraw, Aarti Mcgraw, Aries Mcgraw, Bayyinah Mcgraw, Bertie Mcgraw,
Blakely Mcgraw, Cabrina Mcgraw, Chandrea Mcgraw, Cherrell Mcgraw, Chiquitta Mcgraw, Damarys Mcgraw, Danett Mcgraw, Dedee Mcgraw, Demesha Mcgraw, Denica Mcgraw, Derinda Mcgraw, Deshanda Mcgraw,
Diondra Mcgraw, Edgar Mcgraw, Ewa Mcgraw, Grabiela Mcgraw, Jeanifer Mcgraw, Jemma Mcgraw, Jetta Mcgraw, Jodelle Mcgraw, Joeleen Mcgraw, Johari Mcgraw, Kasaundra Mcgraw, Kewana Mcgraw, Kieran Mcgraw,
Kitina Mcgraw, Kitrina Mcgraw, Lacheryl Mcgraw, Lakeishia Mcgraw, Lasean Mcgraw, Lashawne Mcgraw, Lashica Mcgraw, Latanga Mcgraw, Latreece Mcgraw, Latysha Mcgraw, Leonna Mcgraw, Lilli Mcgraw, Lititia
Mcgraw, Lorette Mcgraw, Makala Mcgraw, Mea Mcgraw, Mendie Mcgraw, Meosha Mcgraw, Milessa Mcgraw, Nathasha Mcgraw, Nefertari Mcgraw, Nieves Mcgraw, Odelia Mcgraw, Phylis Mcgraw, Phyliss Mcgraw, Rami
Mcgraw, Shaena Mcgraw, Shahara Mcgraw, Shajuan Mcgraw, Shalane Mcgraw, Shamaine Mcgraw, Shaquanna Mcgraw, Shauntelle Mcgraw, Sheril Mcgraw, Sherley Mcgraw, Shuronda Mcgraw, Sicily Mcgraw, Sindi
Mcgraw, Sonni Mcgraw, Tamantha Mcgraw, Tejal Mcgraw, Tenley Mcgraw, Tongela Mcgraw, Trenita Mcgraw, Vernica Mcgraw, Ameena Mcgraw, Chancey Mcgraw, Chandi Mcgraw, Charelle Mcgraw, Cherisa Mcgraw,
Corby Mcgraw, Corianne Mcgraw, Crystale Mcgraw, Cyd Mcgraw, Cyndie Mcgraw, Dagny Mcgraw, Daine Mcgraw, Darin Mcgraw, Debralee Mcgraw, Delanda Mcgraw, Donta Mcgraw, Estefana Mcgraw, Falesha Mcgraw,
Fawna Mcgraw, Galen Mcgraw, Geanine Mcgraw, Grazia Mcgraw, Hala Mcgraw, Jadie Mcgraw, Jalyn Mcgraw, Jamilia Mcgraw, Jannice Mcgraw, Jasmina Mcgraw, Jehan Mcgraw, Joannah Mcgraw, Joclyn Mcgraw,
Julietta Mcgraw, Kody Mcgraw, Krisa Mcgraw, Lanaya Mcgraw, Laray Mcgraw, Larra Mcgraw, Larrissa Mcgraw, Leanora Mcgraw, Lelania Mcgraw, Lenetta Mcgraw, Lizett Mcgraw, Lonnette Mcgraw, Lorey Mcgraw,
Lynann Mcgraw, Madalena Mcgraw, Marlita Mcgraw, Marrianne Mcgraw, Maylene Mcgraw, Merida Mcgraw, Midori Mcgraw, Nafeesah Mcgraw, Nekeshia Mcgraw, Parris Mcgraw, Payal Mcgraw, Perlita Mcgraw, Prairie
Mcgraw, Raeanna Mcgraw, Rayette Mcgraw, Sabriya Mcgraw, Sadia Mcgraw, Sequita Mcgraw, Sharine Mcgraw, Sharlena Mcgraw, Spencer Mcgraw, Sugey Mcgraw, Taheera Mcgraw, Taiwan Mcgraw, Tameki Mcgraw,
Tamme Mcgraw, Tawonda Mcgraw, Toscha Mcgraw, Tyronza Mcgraw, Valeska Mcgraw, Vannesa Mcgraw, Vashon Mcgraw, Viktoria Mcgraw, Vonya Mcgraw, Akita Mcgraw, Andraya Mcgraw, Anecia Mcgraw, Annjeanette
Mcgraw, Antonya Mcgraw, Brette Mcgraw, Chase Mcgraw, Chena Mcgraw, Cheris Mcgraw, Chirstine Mcgraw, Claude Mcgraw, Clayton Mcgraw, Clea Mcgraw, Clorissa Mcgraw, Consuello Mcgraw, Cosette Mcgraw,
Darinda Mcgraw, Dawnn Mcgraw, Delene Mcgraw, Deneka Mcgraw, Geisha Mcgraw, Gwendelyn Mcgraw, Hyacinth Mcgraw, Ilia Mcgraw, Jacqulin Mcgraw, Jaynee Mcgraw, Jennifr Mcgraw, Kaysha Mcgraw, Keiana
Mcgraw, Kelsy Mcgraw, Keziah Mcgraw, Lakaisha Mcgraw, Lashawndra Mcgraw, Lennette Mcgraw, Loran Mcgraw, Margarete Mcgraw, Marquise Mcgraw, Marvel Mcgraw, Maurica Mcgraw, Mikell Mcgraw, Nya Mcgraw,
Odetta Mcgraw, Raizy Mcgraw, Samika Mcgraw, Scotty Mcgraw, Senita Mcgraw, Shasha Mcgraw, Shaton Mcgraw, Sheilla Mcgraw, Shital Mcgraw, Shonika Mcgraw, Silver Mcgraw, Tabathia Mcgraw, Tabby Mcgraw,
Taniya Mcgraw, Tawonna Mcgraw, Teddy Mcgraw, Tenicia Mcgraw, Teodora Mcgraw, Tessia Mcgraw, Tinya Mcgraw, Travia Mcgraw, Trivia Mcgraw, Verda Mcgraw, Veta Mcgraw, Waneta Mcgraw, Ysenia Mcgraw, Yumiko
Mcgraw, Zully Mcgraw, Adeana Mcgraw, Adenike Mcgraw, Afton Mcgraw, Angeli Mcgraw, Anicia Mcgraw, Arron Mcgraw, Artie Mcgraw, Ashlei Mcgraw, Ayme Mcgraw, Brieanne Mcgraw, Cira Mcgraw, Cynthea Mcgraw,
Damiana Mcgraw, Darlynn Mcgraw, Dennette Mcgraw, Doreena Mcgraw, Dorris Mcgraw, Dorsey Mcgraw, Drew Mcgraw, Drina Mcgraw, Enrika Mcgraw, Genean Mcgraw, Glendora Mcgraw, Gwendlyn Mcgraw, Hedi Mcgraw,
Honesty Mcgraw, Jackalyn Mcgraw, Jillyn Mcgraw, Jode Mcgraw, Joshlyn Mcgraw, Joylyn Mcgraw, Jullian Mcgraw, Kajuana Mcgraw, Kasee Mcgraw, Kimberla Mcgraw, Kjersten Mcgraw, Kortni Mcgraw, Latronda
Mcgraw, Leonie Mcgraw, Lonette Mcgraw, Lonita Mcgraw, Lynnelle Mcgraw, Marenda Mcgraw, Marybelle Mcgraw, Paisley Mcgraw, Patrena Mcgraw, Rheannon Mcgraw, Rochella Mcgraw, Saleemah Mcgraw, Samala
Mcgraw, Shannie Mcgraw, Shelitha Mcgraw, Shemeika Mcgraw, Shenia Mcgraw, Shinika Mcgraw, Shunte Mcgraw, Sia Mcgraw, Sinead Mcgraw, Tal Mcgraw, Tata Mcgraw, Teela Mcgraw, Tita Mcgraw, Verenice Mcgraw,
Wade Mcgraw, Yesica Mcgraw, Zaira Mcgraw, Zuri Mcgraw, Bobie Mcgraw, Brendalee Mcgraw, Carolanne Mcgraw, Chevette Mcgraw, Debera Mcgraw, Denessa Mcgraw, Desiray Mcgraw, Destinie Mcgraw, Diania
Mcgraw, Edra Mcgraw, Ekaterini Mcgraw, Elane Mcgraw, Eudora Mcgraw, Evelyne Mcgraw, Franci Mcgraw, Gussie Mcgraw, Herbert Mcgraw, Ilse Mcgraw, Jaquay Mcgraw, Jerra Mcgraw, Juliene Mcgraw, Kashia
Mcgraw, Kirstan Mcgraw, Laconya Mcgraw, Lajoyce Mcgraw, Larraine Mcgraw, Lera Mcgraw, Loan Mcgraw, Luwana Mcgraw, Maire Mcgraw, Malene Mcgraw, Marca Mcgraw, Marcene Mcgraw, Mathew Mcgraw, Mayumi
Mcgraw, Nakeysha Mcgraw, Orpha Mcgraw, Quinta Mcgraw, Robie Mcgraw, Sarha Mcgraw, Selah Mcgraw, Sergio Mcgraw, Shalaine Mcgraw, Shazia Mcgraw, Shenae Mcgraw, Shenica Mcgraw, Shineka Mcgraw, Sonnie
Mcgraw, Tashauna Mcgraw, Tashica Mcgraw, Tatjana Mcgraw, Tecia Mcgraw, Therisa Mcgraw, Tracina Mcgraw, Tressia Mcgraw, Vallery Mcgraw, Ximena Mcgraw, Aidee Mcgraw, Alpa Mcgraw, Ambur Mcgraw,
Annaliese Mcgraw, Atasha Mcgraw, Ave Mcgraw, Barbaraann Mcgraw, Bekki Mcgraw, Breena Mcgraw, Catharina Mcgraw, Chandelle Mcgraw, Charmion Mcgraw, Cherilynn Mcgraw, Chesney Mcgraw, Chirsty Mcgraw,
Cynethia Mcgraw, Cyntia Mcgraw, Dane Mcgraw, Dorethea Mcgraw, Elyce Mcgraw, Emery Mcgraw, Garrett Mcgraw, Gittel Mcgraw, Gizelle Mcgraw, Hinda Mcgraw, Jaquelin Mcgraw, Jenaya Mcgraw, Jeneva Mcgraw,
Jenipher Mcgraw, Karessa Mcgraw, Karita Mcgraw, Kelita Mcgraw, Kiyana Mcgraw, Kora Mcgraw, Laquanna Mcgraw, Larkin Mcgraw, Latangela Mcgraw, Linsay Mcgraw, Lisett Mcgraw, Lizzy Mcgraw, Magdalen
Mcgraw, Marcea Mcgraw, Mashawn Mcgraw, Meiko Mcgraw, Mekia Mcgraw, Merisa Mcgraw, Nikcole Mcgraw, Norina Mcgraw, Prima Mcgraw, Rashada Mcgraw, Rasheena Mcgraw, Rozella Mcgraw, Selia Mcgraw, Shaheen
Mcgraw, Shamon Mcgraw, Shaya Mcgraw, Shirly Mcgraw, Shunna Mcgraw, Shyanne Mcgraw, Skylar Mcgraw, Sukari Mcgraw, Sylvana Mcgraw, Tahlia Mcgraw, Taiesha Mcgraw, Tamiki Mcgraw, Tangala Mcgraw, Taniqua
Mcgraw, Tanza Mcgraw, Temesha Mcgraw, Teonna Mcgraw, Tiffane Mcgraw, Torya Mcgraw, Tremaine Mcgraw, Tricha Mcgraw, Vanesha Mcgraw, Yetunde Mcgraw, Zarinah Mcgraw, Amalie Mcgraw, Atoya Mcgraw, Bryony
Mcgraw, Caree Mcgraw, Carron Mcgraw, Celestia Mcgraw, Charma Mcgraw, Chelly Mcgraw, Christon Mcgraw, Chrystle Mcgraw, Conchetta Mcgraw, Dan Mcgraw, Deandria Mcgraw, Donika Mcgraw, Eulanda Mcgraw,
Eyvonne Mcgraw, Gloriana Mcgraw, Hyun Mcgraw, Joslynn Mcgraw, Kewanda Mcgraw, Laetitia Mcgraw, Laketra Mcgraw, Lakshmi Mcgraw, Latawnya Mcgraw, Lawren Mcgraw, Leo Mcgraw, Letonia Mcgraw, Lichelle
Mcgraw, Lue Mcgraw, Lynnmarie Mcgraw, Mauricia Mcgraw, Nakkia Mcgraw, Neka Mcgraw, Parisa Mcgraw, Quentin Mcgraw, Ranada Mcgraw, Rhianon Mcgraw, Roselee Mcgraw, Rozanne Mcgraw, Rozetta Mcgraw, Rozina
Mcgraw, Sada Mcgraw, Sala Mcgraw, Samar Mcgraw, Seandra Mcgraw, Shaela Mcgraw, Sharrell Mcgraw, Shawneen Mcgraw, Shefali Mcgraw, Shekinah Mcgraw, Sherriann Mcgraw, Shuree Mcgraw, Soyini Mcgraw, Teala
Mcgraw, Telitha Mcgraw, Tempie Mcgraw, Tirza Mcgraw, Valori Mcgraw, Veleda Mcgraw, Vivianne Mcgraw, Alexi Mcgraw, Arnetra Mcgraw, Azizi Mcgraw, Belkys Mcgraw, Cailin Mcgraw, Carmell Mcgraw, Charlyne
Mcgraw, Courteney Mcgraw, Danniell Mcgraw, Danyiel Mcgraw, Danylle Mcgraw, Delita Mcgraw, Deseri Mcgraw, Donice Mcgraw, Ebone Mcgraw, Eleshia Mcgraw, Ellison Mcgraw, Giuliana Mcgraw, Harvest Mcgraw,
Hillari Mcgraw, Ijeoma Mcgraw, Janeene Mcgraw, Kem Mcgraw, Kya Mcgraw, Laquandra Mcgraw, Lashane Mcgraw, Latana Mcgraw, Latoi Mcgraw, Lenice Mcgraw, Lethia Mcgraw, Lindsi Mcgraw, Lis Mcgraw, Lorrine
Mcgraw, Marena Mcgraw, Marlayna Mcgraw, Mashanda Mcgraw, Mercedez Mcgraw, Merlene Mcgraw, Najwa Mcgraw, Paraskevi Mcgraw, Princella Mcgraw, Raejean Mcgraw, Raissa Mcgraw, Ravin Mcgraw, Reda Mcgraw,
Reisha Mcgraw, Roland Mcgraw, Sakia Mcgraw, Salma Mcgraw, Shakela Mcgraw, Sharica Mcgraw, Shatisha Mcgraw, Shauntee Mcgraw, Sheriann Mcgraw, Stori Mcgraw, Sueellen Mcgraw, Sylina Mcgraw, Tametra
Mcgraw, Tanieka Mcgraw, Tanikia Mcgraw, Tarisa Mcgraw, Tiffony Mcgraw, Tolanda Mcgraw, Treana Mcgraw, Tumeka Mcgraw, Ursala Mcgraw, Yumi Mcgraw, Zoraya Mcgraw, Alfie Mcgraw, Aliyah Mcgraw, Almeda
Mcgraw, Amarilys Mcgraw, Amberley Mcgraw, Arabella Mcgraw, Ardella Mcgraw, Atonya Mcgraw, Audree Mcgraw, Benedetta Mcgraw, Billiejean Mcgraw, Carlynn Mcgraw, Cherylynn Mcgraw, Christabel Mcgraw,
Clarrisa Mcgraw, Colinda Mcgraw, Cutina Mcgraw, Dalyn Mcgraw, Dametra Mcgraw, Daphna Mcgraw, Emili Mcgraw, Fatema Mcgraw, Glennda Mcgraw, Izetta Mcgraw, Janielle Mcgraw, Jessamy Mcgraw, Jonique
Mcgraw, Karron Mcgraw, Kathia Mcgraw, Katrese Mcgraw, Kendrah Mcgraw, Kevia Mcgraw, Khalia Mcgraw, Kissa Mcgraw, Kizzi Mcgraw, Kutina Mcgraw, Lanetra Mcgraw, Lashonya Mcgraw, Latanja Mcgraw, Latisia
Mcgraw, Latriece Mcgraw, Lenny Mcgraw, Leonarda Mcgraw, Lewanna Mcgraw, Lien Mcgraw, Lyndia Mcgraw, Mallissa Mcgraw, Marlie Mcgraw, Marqueta Mcgraw, Marvetta Mcgraw, Mikayla Mcgraw, Milicent Mcgraw,
Montana Mcgraw, Morgana Mcgraw, Nathania Mcgraw, Neema Mcgraw, Noriko Mcgraw, Ondria Mcgraw, Onna Mcgraw, Patrici Mcgraw, Rachella Mcgraw, Reatha Mcgraw, Rise Mcgraw, Rosabel Mcgraw, Secret Mcgraw,
Shanetha Mcgraw, Sharah Mcgraw, Sharion Mcgraw, Sherilynn Mcgraw, Summar Mcgraw, Tam Mcgraw, Tamanika Mcgraw, Tarhonda Mcgraw, Teddie Mcgraw, Tennelle Mcgraw, Tephanie Mcgraw, Terryl Mcgraw, Tiawanna
Mcgraw, Tischa Mcgraw, Tonesha Mcgraw, Amirah Mcgraw, Amory Mcgraw, Anji Mcgraw, Arasely Mcgraw, Argentina Mcgraw, Aviance Mcgraw, Batsheva Mcgraw, Briar Mcgraw, Cena Mcgraw, Chalon Mcgraw, Chancy
Mcgraw, Chandler Mcgraw, Chivon Mcgraw, Chonita Mcgraw, Chrisa Mcgraw, Darnelle Mcgraw, Deanie Mcgraw, Deeana Mcgraw, Delise Mcgraw, Elayna Mcgraw, Errika Mcgraw, Eurika Mcgraw, Farrell Mcgraw,
Felcia Mcgraw, Isidra Mcgraw, Jai Mcgraw, Jamell Mcgraw, Jamesha Mcgraw, Jere Mcgraw, Jodilyn Mcgraw, Jovonne Mcgraw, Jowana Mcgraw, Juline Mcgraw, Kameshia Mcgraw, Katrisha Mcgraw, Keary Mcgraw,
Keyshia Mcgraw, Kolette Mcgraw, Lakecha Mcgraw, Lakiya Mcgraw, Larrisa Mcgraw, Lasharn Mcgraw, Lasundra Mcgraw, Latica Mcgraw, Latoiya Mcgraw, Laurisa Mcgraw, Lucrezia Mcgraw, Mariateresa Mcgraw,
Maryland Mcgraw, Melicia Mcgraw, Mieko Mcgraw, Molina Mcgraw, Mozella Mcgraw, Nataly Mcgraw, Nayda Mcgraw, Olanda Mcgraw, Onica Mcgraw, Ourania Mcgraw, Quinette Mcgraw, Rayma Mcgraw, Rianne Mcgraw,
Sarika Mcgraw, Shalona Mcgraw, Shawntina Mcgraw, Shellye Mcgraw, Sheva Mcgraw, Shironda Mcgraw, Sonora Mcgraw, Sophronia Mcgraw, Symantha Mcgraw, Tahra Mcgraw, Tearsa Mcgraw, Terrika Mcgraw, Tonyetta
Mcgraw, Treina Mcgraw, Tzipora Mcgraw, Violette Mcgraw, Vivianna Mcgraw, Yanet Mcgraw, Akira Mcgraw, Amanada Mcgraw, Atara Mcgraw, Britten Mcgraw, Camika Mcgraw, Carlo Mcgraw, Celicia Mcgraw, Chanon
Mcgraw, Chanti Mcgraw, Christeena Mcgraw, Consandra Mcgraw, Correen Mcgraw, Delayna Mcgraw, Derenda Mcgraw, Dlynn Mcgraw, Donyetta Mcgraw, Fawne Mcgraw, Feliz Mcgraw, Gisselle Mcgraw, Glen Mcgraw,
Hong Mcgraw, Hyla Mcgraw, Jacques Mcgraw, Jasmyn Mcgraw, Jonel Mcgraw, Khaleelah Mcgraw, Kista Mcgraw, Lachell Mcgraw, Lajoy Mcgraw, Lakethia Mcgraw, Lamia Mcgraw, Lasheba Mcgraw, Laurianne Mcgraw,
Lelani Mcgraw, Lonya Mcgraw, Loucinda Mcgraw, Maressa Mcgraw, Mariesa Mcgraw, Meshia Mcgraw, Messina Mcgraw, Paulett Mcgraw, Rashon Mcgraw, Rayshawn Mcgraw, Rochanda Mcgraw, Ronnica Mcgraw, Rorie
Mcgraw, Saunya Mcgraw, Shakeitha Mcgraw, Shalla Mcgraw, Shalondra Mcgraw, Shamecca Mcgraw, Shavonn Mcgraw, Sherril Mcgraw, Shonell Mcgraw, Sonita Mcgraw, Svetlana Mcgraw, Takila Mcgraw, Tala Mcgraw,
Tamecca Mcgraw, Tiphany Mcgraw, Torria Mcgraw, Vanya Mcgraw, Verdell Mcgraw, Vonna Mcgraw, Zsazsa Mcgraw, Adonia Mcgraw, Alease Mcgraw, Aleena Mcgraw, Alishea Mcgraw, Anetta Mcgraw, Ania Mcgraw,
Anquinette Mcgraw, Antonieta Mcgraw, Archana Mcgraw, Arely Mcgraw, Artavia Mcgraw, Arturo Mcgraw, Batya Mcgraw, Bernardine Mcgraw, Betti Mcgraw, Breck Mcgraw, Camara Mcgraw, Cesar Mcgraw, Chalise
Mcgraw, Charly Mcgraw, Cherrise Mcgraw, Chikita Mcgraw, Cristiana Mcgraw, Dayatra Mcgraw, Ericha Mcgraw, Ermalinda Mcgraw, Eulonda Mcgraw, Fayth Mcgraw, Halina Mcgraw, Hunter Mcgraw, Iveliz Mcgraw,
Ivie Mcgraw, Jahna Mcgraw, Jancy Mcgraw, Janisha Mcgraw, Jeaninne Mcgraw, Jeannemarie Mcgraw, Jessamine Mcgraw, Keeya Mcgraw, Kelisha Mcgraw, Kenyette Mcgraw, Kyley Mcgraw, Kyrie Mcgraw, Ladora
Mcgraw, Laguana Mcgraw, Larri Mcgraw, Leyda Mcgraw, Louie Mcgraw, Marykathryn Mcgraw, Mekesha Mcgraw, Meredeth Mcgraw, Merridith Mcgraw, Mittie Mcgraw, Nakeitha Mcgraw, Narda Mcgraw, Nature Mcgraw,
Neeta Mcgraw, Nitasha Mcgraw, Nuvia Mcgraw, Paticia Mcgraw, Promise Mcgraw, Rabia Mcgraw, Raquelle Mcgraw, Richetta Mcgraw, Shawneequa Mcgraw, Shervon Mcgraw, Siomara Mcgraw, Starlyn Mcgraw, Tahirih
Mcgraw, Taia Mcgraw, Talita Mcgraw, Tamina Mcgraw, Tamya Mcgraw, Tanicka Mcgraw, Tanyanika Mcgraw, Tawan Mcgraw, Tawanya Mcgraw, Tiarra Mcgraw, Timisha Mcgraw, Tinita Mcgraw, Tonnia Mcgraw, Toronda
Mcgraw, Toshika Mcgraw, Triva Mcgraw, Trula Mcgraw, Tykesha Mcgraw, Vaness Mcgraw, Vennessa Mcgraw, Wakeelah Mcgraw, Wilhemina Mcgraw, Yoshiko Mcgraw, Afia Mcgraw, Afiya Mcgraw, Allysa Mcgraw, Alsha
Mcgraw, Amberlyn Mcgraw, Arah Mcgraw, Audri Mcgraw, Bernardette Mcgraw, Bernedette Mcgraw, Binta Mcgraw, Breeann Mcgraw, Britni Mcgraw, Carnetta Mcgraw, Cesilia Mcgraw, Chelli Mcgraw, Dawnna Mcgraw,
Desirie Mcgraw, Fleur Mcgraw, Gianina Mcgraw, Hanh Mcgraw, Ivelis Mcgraw, Jalena Mcgraw, Jenafer Mcgraw, Johann Mcgraw, Joyanne Mcgraw, Kamia Mcgraw, Katerine Mcgraw, Kecha Mcgraw, Keir Mcgraw,
Kennita Mcgraw, Kimra Mcgraw, Kristiann Mcgraw, Levita Mcgraw, Lluvia Mcgraw, Lorian Mcgraw, Lyndsy Mcgraw, Mardell Mcgraw, Maurissa Mcgraw, Mischell Mcgraw, Mistey Mcgraw, Nataya Mcgraw, Nickola
Mcgraw, Obdulia Mcgraw, Preston Mcgraw, Raynetta Mcgraw, Rondell Mcgraw, Shakara Mcgraw, Shalan Mcgraw, Shanya Mcgraw, Shawnika Mcgraw, Shenetha Mcgraw, Shilah Mcgraw, Soo Mcgraw, Tametha Mcgraw,
Tanicia Mcgraw, Tanny Mcgraw, Tennie Mcgraw, Tiffannie Mcgraw, Torra Mcgraw, Trinika Mcgraw, Val Mcgraw, Abeer Mcgraw, Adah Mcgraw, Aleece Mcgraw, Andreya Mcgraw, Annaliza Mcgraw, Arrie Mcgraw,
Ayasha Mcgraw, Berit Mcgraw, Birgit Mcgraw, Bre Mcgraw, Brigetta Mcgraw, Cassundra Mcgraw, Christyl Mcgraw, Cristyn Mcgraw, Danean Mcgraw, Daralyn Mcgraw, Davonne Mcgraw, Deisha Mcgraw, Denina
Mcgraw, Dietrich Mcgraw, Domini Mcgraw, Erinne Mcgraw, Evy Mcgraw, Forrest Mcgraw, Genetta Mcgraw, Gioia Mcgraw, Guenevere Mcgraw, Gwenetta Mcgraw, Gwenette Mcgraw, Hesper Mcgraw, Ignacia Mcgraw,
Irmalinda Mcgraw, Jacquita Mcgraw, Jeanita Mcgraw, Jodiann Mcgraw, Kalin Mcgraw, Kenny Mcgraw, Keon Mcgraw, Kimberlynn Mcgraw, Kimela Mcgraw, Kimm Mcgraw, Kiwanna Mcgraw, Lacreshia Mcgraw, Lakashia
Mcgraw, Latarshia Mcgraw, Latrish Mcgraw, Lauria Mcgraw, Likisha Mcgraw, Lindee Mcgraw, Little Mcgraw, Loreta Mcgraw, Lutisha Mcgraw, Mahasin Mcgraw, Maki Mcgraw, Maxi Mcgraw, Meesha Mcgraw, Melitta
Mcgraw, Mery Mcgraw, Mysty Mcgraw, Quanisha Mcgraw, Ranetta Mcgraw, Rosia Mcgraw, Sadonna Mcgraw, Sangita Mcgraw, Sanora Mcgraw, Shanina Mcgraw, Sharonna Mcgraw, Shavona Mcgraw, Sherrica Mcgraw,
Storm Mcgraw, Tabrina Mcgraw, Tacha Mcgraw, Taleshia Mcgraw, Tamee Mcgraw, Tanetta Mcgraw, Thanh Mcgraw, Twanya Mcgraw, Ulrica Mcgraw, Yaffa Mcgraw, Aarin Mcgraw, Afi Mcgraw, Aliscia Mcgraw, Aneta
Mcgraw, Anetria Mcgraw, Antoinett Mcgraw, Athenia Mcgraw, Atisha Mcgraw, Barb Mcgraw, Brit Mcgraw, Caley Mcgraw, Camala Mcgraw, Carena Mcgraw, Carinne Mcgraw, Carmine Mcgraw, Charlott Mcgraw, Chessa
Mcgraw, Chundra Mcgraw, Cotrina Mcgraw, Drenda Mcgraw, Elanor Mcgraw, Eleftheria Mcgraw, Erricka Mcgraw, Feleshia Mcgraw, Geanna Mcgraw, Harley Mcgraw, Honora Mcgraw, Jakia Mcgraw, Jannah Mcgraw,
Jennene Mcgraw, Jennett Mcgraw, Jobeth Mcgraw, Jordanna Mcgraw, Juanetta Mcgraw, Junie Mcgraw, Kaira Mcgraw, Kashana Mcgraw, Katee Mcgraw, Kateena Mcgraw, Kelleigh Mcgraw, Kenyana Mcgraw, Kerrilyn
Mcgraw, Kessa Mcgraw, Kirk Mcgraw, Lakitha Mcgraw, Laneshia Mcgraw, Lanissa Mcgraw, Lashunta Mcgraw, Leonia Mcgraw, Lester Mcgraw, Letina Mcgraw, Llesenia Mcgraw, Loranda Mcgraw, Lorilynn Mcgraw,
Lotus Mcgraw, Luke Mcgraw, Marisal Mcgraw, Marvis Mcgraw, Maurie Mcgraw, Minnette Mcgraw, Mishawn Mcgraw, Natsha Mcgraw, Ngoc Mcgraw, Nioka Mcgraw, Particia Mcgraw, Phillina Mcgraw, Prisilla Mcgraw,
Raney Mcgraw, Raphaela Mcgraw, Rashaunda Mcgraw, Raynelle Mcgraw, Ross Mcgraw, Rufina Mcgraw, Sarahann Mcgraw, Shaconda Mcgraw, Shanicka Mcgraw, Shaunita Mcgraw, Shekina Mcgraw, Shelie Mcgraw, Sherah
Mcgraw, Shermeka Mcgraw, Shermika Mcgraw, Sonnia Mcgraw, Stephonie Mcgraw, Taffie Mcgraw, Takima Mcgraw, Talea Mcgraw, Tamaya Mcgraw, Taneika Mcgraw, Tanina Mcgraw, Tanisia Mcgraw, Taysha Mcgraw,
Tekeshia Mcgraw, Terie Mcgraw, Therasa Mcgraw, Thomasa Mcgraw, Tifini Mcgraw, Tomie Mcgraw, Tressy Mcgraw, Trissa Mcgraw, Tristy Mcgraw, Tritia Mcgraw, Twania Mcgraw, Tyrene Mcgraw, Uganda Mcgraw,
Wandy Mcgraw, Wileen Mcgraw, Yumeka Mcgraw, Zarina Mcgraw, Adreana Mcgraw, Alandra Mcgraw, Alenda Mcgraw, Amera Mcgraw, Arlicia Mcgraw, Artemis Mcgraw, Avani Mcgraw, Bena Mcgraw, Candia Mcgraw,
Cathyjo Mcgraw, Chala Mcgraw, Charitie Mcgraw, Charlise Mcgraw, Chasiti Mcgraw, Chayla Mcgraw, Chrisanne Mcgraw, Christell Mcgraw, Clarrissa Mcgraw, Cortnee Mcgraw, Dametria Mcgraw, Daneka Mcgraw,
Darlisa Mcgraw, Delorse Mcgraw, Deshone Mcgraw, Dorette Mcgraw, Esperansa Mcgraw, Evagelia Mcgraw, Feliciana Mcgraw, Fikisha Mcgraw, Freeda Mcgraw, Frida Mcgraw, Gaila Mcgraw, Genae Mcgraw, Georgi
Mcgraw, Gitel Mcgraw, Gricel Mcgraw, Hilliary Mcgraw, Ichelle Mcgraw, Ishia Mcgraw, Ivelise Mcgraw, Jacci Mcgraw, Jamesa Mcgraw, Jammy Mcgraw, Jerrilynn Mcgraw, Jillaine Mcgraw, Jillann Mcgraw, Jin
Mcgraw, Joda Mcgraw, Jodeen Mcgraw, Kalisa Mcgraw, Karne Mcgraw, Katura Mcgraw, Kayci Mcgraw, Keegan Mcgraw, Keenan Mcgraw, Khia Mcgraw, Koleen Mcgraw, Krishana Mcgraw, Krislyn Mcgraw, Kurt Mcgraw,
Lafaye Mcgraw, Lakeysa Mcgraw, Lakresha Mcgraw, Lalisa Mcgraw, Lashonne Mcgraw, Leighanna Mcgraw, Lesslie Mcgraw, Lin Mcgraw, Madlyn Mcgraw, Malky Mcgraw, Melenda Mcgraw, Melisssa Mcgraw, Mikita
Mcgraw, Monicia Mcgraw, Monik Mcgraw, Nakima Mcgraw, Naquita Mcgraw, Natash Mcgraw, Raphael Mcgraw, Rashawna Mcgraw, Reannon Mcgraw, Reneta Mcgraw, Reshanda Mcgraw, Riana Mcgraw, Rocky Mcgraw,
Rosamond Mcgraw, Roshan Mcgraw, Saleena Mcgraw, Schuyler Mcgraw, Shaka Mcgraw, Shakesha Mcgraw, Shaleta Mcgraw, Shanekia Mcgraw, Surina Mcgraw, Syrita Mcgraw, Taj Mcgraw, Tajuanna Mcgraw, Tasheen
Mcgraw, Tiffine Mcgraw, Timmie Mcgraw, Tomikia Mcgraw, Trease Mcgraw, Tyrhonda Mcgraw, Wakisha Mcgraw, Yvone Mcgraw, Adalia Mcgraw, Ahuva Mcgraw, Angelicia Mcgraw, Anina Mcgraw, Annitra Mcgraw,
Betsie Mcgraw, Cachet Mcgraw, Callista Mcgraw, Camey Mcgraw, Celita Mcgraw, Cherissa Mcgraw, Collin Mcgraw, Daiana Mcgraw, Equilla Mcgraw, Evetta Mcgraw, Eyvette Mcgraw, Feige Mcgraw, Filicia Mcgraw,
Floyd Mcgraw, Garnett Mcgraw, Geoffrey Mcgraw, Harper Mcgraw, Ione Mcgraw, Iyana Mcgraw, Jametta Mcgraw, Jeanee Mcgraw, Jessalynn Mcgraw, Jonalyn Mcgraw, Juanice Mcgraw, Kaija Mcgraw, Kalene Mcgraw,
Kassidy Mcgraw, Katryn Mcgraw, Kennethia Mcgraw, Keosha Mcgraw, Kresha Mcgraw, Krisandra Mcgraw, Labrina Mcgraw, Lachonda Mcgraw, Lamara Mcgraw, Lashika Mcgraw, Lateka Mcgraw, Laytoya Mcgraw, Lourdez
Mcgraw, Lyndsi Mcgraw, Markeita Mcgraw, Marletta Mcgraw, Marshella Mcgraw, Martiza Mcgraw, Marylouise Mcgraw, Miri Mcgraw, Nani Mcgraw, Payton Mcgraw, Puja Mcgraw, Qianna Mcgraw, Raffaella Mcgraw,
Roanne Mcgraw, Sadiqa Mcgraw, Salvador Mcgraw, Shabnam Mcgraw, Shamar Mcgraw, Shamina Mcgraw, Sharalee Mcgraw, Shavette Mcgraw, Sheilia Mcgraw, Shekia Mcgraw, Shelise Mcgraw, Sherrilynn Mcgraw,
Sheyla Mcgraw, Shy Mcgraw, Suprena Mcgraw, Tahitia Mcgraw, Talley Mcgraw, Tamiya Mcgraw, Tashena Mcgraw, Tashona Mcgraw, Tatisha Mcgraw, Toia Mcgraw, Tonji Mcgraw, Triana Mcgraw, Tymika Mcgraw, Yanna
Mcgraw, Zainab Mcgraw, Amrita Mcgraw, Aprel Mcgraw, Atalie Mcgraw, Aubre Mcgraw, Auria Mcgraw, Barbarita Mcgraw, Bettyann Mcgraw, Boni Mcgraw, Brandace Mcgraw, Bryanne Mcgraw, Calleen Mcgraw, Chakita
Mcgraw, Chamaine Mcgraw, Chamika Mcgraw, Charlett Mcgraw, Chereese Mcgraw, Cherine Mcgraw, Christalyn Mcgraw, Conya Mcgraw, Dalilah Mcgraw, Dela Mcgraw, Delanna Mcgraw, Delinah Mcgraw, Dierdra
Mcgraw, Domitila Mcgraw, Donnis Mcgraw, Dore Mcgraw, Elenor Mcgraw, Elizabethann Mcgraw, Elysha Mcgraw, Erryn Mcgraw, Evalyn Mcgraw, Frannie Mcgraw, Fredda Mcgraw, Ginnifer Mcgraw, Hilaria Mcgraw,
Ilisa Mcgraw, Ivanna Mcgraw, Janeal Mcgraw, Jeena Mcgraw, Jeremie Mcgraw, Jeriann Mcgraw, Julina Mcgraw, Karim Mcgraw, Kenitha Mcgraw, Kery Mcgraw, Korinna Mcgraw, Lachrisha Mcgraw, Lamonda Mcgraw,
Laneisha Mcgraw, Lene Mcgraw, Letita Mcgraw, Lillia Mcgraw, Lindsie Mcgraw, Lissete Mcgraw, Lynsay Mcgraw, Magalie Mcgraw, Makini Mcgraw, Manette Mcgraw, Margareta Mcgraw, Marlette Mcgraw, Marsi
Mcgraw, Maryn Mcgraw, Megon Mcgraw, Melainie Mcgraw, Meria Mcgraw, Merrideth Mcgraw, Moranda Mcgraw, Myah Mcgraw, Myron Mcgraw, Odalis Mcgraw, Rakia Mcgraw, Renesha Mcgraw, Rhanda Mcgraw, Richell
Mcgraw, Salinda Mcgraw, Sammantha Mcgraw, Shamonica Mcgraw, Shaquanda Mcgraw, Sharmain Mcgraw, Shereece Mcgraw, Shron Mcgraw, Tammika Mcgraw, Terence Mcgraw, Teresha Mcgraw, Tiah Mcgraw, Treniece
Mcgraw, Truly Mcgraw, Venicia Mcgraw, Virgil Mcgraw, Winnifred Mcgraw, Yaritza Mcgraw, Yarrow Mcgraw, Zandria Mcgraw, Alacia Mcgraw, Albertha Mcgraw, Andie Mcgraw, Anuradha Mcgraw, Artesia Mcgraw,
Beatrix Mcgraw, Brandin Mcgraw, Brittni Mcgraw, Carrey Mcgraw, Catricia Mcgraw, Chelise Mcgraw, Christien Mcgraw, Dalina Mcgraw, Damia Mcgraw, Deletha Mcgraw, Eldora Mcgraw, Eli Mcgraw, Eufemia
Mcgraw, Fabrienne Mcgraw, Garland Mcgraw, Genessa Mcgraw, Ginnette Mcgraw, Goldy Mcgraw, Grant Mcgraw, Greg Mcgraw, Hasina Mcgraw, Hermila Mcgraw, Ima Mcgraw, Jacquette Mcgraw, Jilian Mcgraw, Jilleen
Mcgraw, Jolita Mcgraw, Jorja Mcgraw, Julita Mcgraw, Juna Mcgraw, Keicha Mcgraw, Keshawn Mcgraw, Ketty Mcgraw, Kimmi Mcgraw, Kizmet Mcgraw, Koni Mcgraw, Kristiane Mcgraw, Krystyn Mcgraw, Laronica
Mcgraw, Lashonia Mcgraw, Lateefa Mcgraw, Latresia Mcgraw, Lequisha Mcgraw, Levi Mcgraw, Lorenia Mcgraw, Lubna Mcgraw, Lucas Mcgraw, Marche Mcgraw, Marlea Mcgraw, Masha Mcgraw, Miah Mcgraw, Michala
Mcgraw, Michelyn Mcgraw, Mistina Mcgraw, Necola Mcgraw, Netra Mcgraw, Niomi Mcgraw, Nira Mcgraw, Nonnie Mcgraw, Phelicia Mcgraw, Quanta Mcgraw, Quita Mcgraw, Raini Mcgraw, Rayanna Mcgraw, Roselind
Mcgraw, Roselinda Mcgraw, Roshana Mcgraw, Sagrario Mcgraw, Sameka Mcgraw, Shaindel Mcgraw, Sharetha Mcgraw, Sharol Mcgraw, Shawnese Mcgraw, Shellia Mcgraw, Sheray Mcgraw, Sherian Mcgraw, Shi Mcgraw,
Stepheni Mcgraw, Subrena Mcgraw, Tabithia Mcgraw, Taleen Mcgraw, Taneha Mcgraw, Tanysha Mcgraw, Tekela Mcgraw, Telesa Mcgraw, Tima Mcgraw, Tonni Mcgraw, Ula Mcgraw, Yitty Mcgraw, Yona Mcgraw, Yonna
Mcgraw, Yuko Mcgraw, Zalika Mcgraw, Aarika Mcgraw, Ajeenah Mcgraw, Alegra Mcgraw, Alichia Mcgraw, Allecia Mcgraw, Amor Mcgraw, Aneatra Mcgraw, Anjannette Mcgraw, Anntionette Mcgraw, Antonetta Mcgraw,
Ardelia Mcgraw, Atlanta Mcgraw, Ayoka Mcgraw, Breeze Mcgraw, Casonya Mcgraw, Cathern Mcgraw, Chanise Mcgraw, Charlina Mcgraw, Charmon Mcgraw, Chemika Mcgraw, Cheresa Mcgraw, Cindra Mcgraw, Claretha
Mcgraw, Coren Mcgraw, Cyndia Mcgraw, Cyntha Mcgraw, Danialle Mcgraw, Danyeal Mcgraw, Davin Mcgraw, Deanda Mcgraw, Deaundra Mcgraw, Dejah Mcgraw, Demica Mcgraw, Demisha Mcgraw, Dinita Mcgraw, Donalyn
Mcgraw, Donesha Mcgraw, Dotty Mcgraw, Fidelia Mcgraw, Golden Mcgraw, Helana Mcgraw, Hidi Mcgraw, Ilena Mcgraw, Jenniferann Mcgraw, Jolleen Mcgraw, Jolynda Mcgraw, Juanna Mcgraw, Karlena Mcgraw,
Karlin Mcgraw, Keandra Mcgraw, Kionna Mcgraw, Konni Mcgraw, Lakeyta Mcgraw, Lechelle Mcgraw, Levon Mcgraw, Lilith Mcgraw, Maliaka Mcgraw, Mallie Mcgraw, Margeaux Mcgraw, Mariaisabel Mcgraw, Marixa
Mcgraw, Memorie Mcgraw, Meshawn Mcgraw, Michole Mcgraw, Myrian Mcgraw, Neela Mcgraw, Nivea Mcgraw, Peggi Mcgraw, Porchia Mcgraw, Racine Mcgraw, Rashaun Mcgraw, Reynalda Mcgraw, Rosaleen Mcgraw,
Sabrine Mcgraw, Samanda Mcgraw, Shanitha Mcgraw, Sharece Mcgraw, Sheretha Mcgraw, Sherwanda Mcgraw, Shonica Mcgraw, Solveig Mcgraw, Soyla Mcgraw, Taka Mcgraw, Taquilla Mcgraw, Tedi Mcgraw, Tema
Mcgraw, Terita Mcgraw, Tippi Mcgraw, Tiziana Mcgraw, Tomieka Mcgraw, Tonga Mcgraw, Trenace Mcgraw, Tyeasha Mcgraw, Tyresha Mcgraw, Valda Mcgraw, Vandy Mcgraw, Victora Mcgraw, Victory Mcgraw, Waverly
Mcgraw, Adair Mcgraw, Agueda Mcgraw, Aiko Mcgraw, Akina Mcgraw, Amaya Mcgraw, Anthonette Mcgraw, Ariann Mcgraw, Averi Mcgraw, Ayde Mcgraw, Belissa Mcgraw, Bernadett Mcgraw, Berniece Mcgraw, Berry
Mcgraw, Bethsaida Mcgraw, Bindu Mcgraw, Bonniejean Mcgraw, Brendy Mcgraw, Cadence Mcgraw, Camy Mcgraw, Canisha Mcgraw, Carilyn Mcgraw, Caronda Mcgraw, Chantrell Mcgraw, Chavela Mcgraw, Chemeka
Mcgraw, Crystie Mcgraw, Dafina Mcgraw, Dangela Mcgraw, Daphnee Mcgraw, Darenda Mcgraw, Darolyn Mcgraw, Davelyn Mcgraw, Desarie Mcgraw, Devone Mcgraw, Dionisia Mcgraw, Donnielle Mcgraw, Ebonique
Mcgraw, English Mcgraw, Eran Mcgraw, Eryka Mcgraw, Falisa Mcgraw, Fareeda Mcgraw, Herman Mcgraw, Inda Mcgraw, Janeice Mcgraw, Janny Mcgraw, Jessicah Mcgraw, Jodean Mcgraw, Julius Mcgraw, Kaisa
Mcgraw, Kanda Mcgraw, Karel Mcgraw, Karia Mcgraw, Kimba Mcgraw, Konya Mcgraw, Kylah Mcgraw, Kyrsten Mcgraw, Laretta Mcgraw, Lasharon Mcgraw, Latascha Mcgraw, Leni Mcgraw, Leontyne Mcgraw, Linell
Mcgraw, Lovena Mcgraw, Maghan Mcgraw, Makeshia Mcgraw, Malorie Mcgraw, Marce Mcgraw, Maryetta Mcgraw, Mechille Mcgraw, Media Mcgraw, Meira Mcgraw, Miosotis Mcgraw, Mirjana Mcgraw, Mlissa Mcgraw,
Nekeya Mcgraw, Nica Mcgraw, Nikkisha Mcgraw, Ortencia Mcgraw, Porche Mcgraw, Raffaela Mcgraw, Raleigh Mcgraw, Reka Mcgraw, Rotunda Mcgraw, Sharline Mcgraw, Sharlonda Mcgraw, Shawnice Mcgraw, Shone
Mcgraw, Shuntell Mcgraw, Sneha Mcgraw, Stephanne Mcgraw, Sundy Mcgraw, Tabita Mcgraw, Takara Mcgraw, Takeia Mcgraw, Talli Mcgraw, Tanganika Mcgraw, Taquana Mcgraw, Thresea Mcgraw, Tila Mcgraw, Tim
Mcgraw, Tyan Mcgraw, Vaishali Mcgraw, Xanthe Mcgraw, Zenja Mcgraw, Zerlina Mcgraw, Aggie Mcgraw, Aleathea Mcgraw, Aleen Mcgraw, Alexcia Mcgraw, Aloma Mcgraw, Angeleque Mcgraw, Araina Mcgraw, Asucena
Mcgraw, Azadeh Mcgraw, Brendan Mcgraw, Candina Mcgraw, Candiss Mcgraw, Cantina Mcgraw, Carlisha Mcgraw, Charra Mcgraw, Chaunta Mcgraw, Dalonda Mcgraw, Damica Mcgraw, Decarla Mcgraw, Deneice Mcgraw,
Derhonda Mcgraw, Derrica Mcgraw, Dorea Mcgraw, Elvera Mcgraw, Estell Mcgraw, Fae Mcgraw, Graceann Mcgraw, Grissel Mcgraw, Hina Mcgraw, Jamielynn Mcgraw, Jesscia Mcgraw, Joley Mcgraw, Junell Mcgraw,
Kalilah Mcgraw, Kathe Mcgraw, Katurah Mcgraw, Kenni Mcgraw, Keysa Mcgraw, Kimie Mcgraw, Lachana Mcgraw, Ladell Mcgraw, Lanore Mcgraw, Laramie Mcgraw, Larrie Mcgraw, Latonna Mcgraw, Latori Mcgraw,
Livier Mcgraw, Loletta Mcgraw, Magdaline Mcgraw, Malaina Mcgraw, Mandalyn Mcgraw, Marshawn Mcgraw, Marticia Mcgraw, Maurisa Mcgraw, Mellonie Mcgraw, Nashira Mcgraw, Orelia Mcgraw, Quanna Mcgraw,
Raisa Mcgraw, Rakisha Mcgraw, Raymonda Mcgraw, Remonia Mcgraw, Robbye Mcgraw, Ronnita Mcgraw, Roshaunda Mcgraw, Sabre Mcgraw, Sabreen Mcgraw, Sharis Mcgraw, Shawnae Mcgraw, Sherrin Mcgraw, Sindia
Mcgraw, Staphanie Mcgraw, Stevi Mcgraw, Tannisha Mcgraw, Tashunda Mcgraw, Tekla Mcgraw, Teralyn Mcgraw, Tomecia Mcgraw, Tonique Mcgraw, Trilby Mcgraw, Unika Mcgraw, Vanisha Mcgraw, Voula Mcgraw,
Waynetta Mcgraw, Yasmina Mcgraw, Amesha Mcgraw, Aniko Mcgraw, Annica Mcgraw, Antione Mcgraw, Astria Mcgraw, Camilia Mcgraw, Caprina Mcgraw, Carah Mcgraw, Catonya Mcgraw, Chalanda Mcgraw, Chasty
Mcgraw, Cherica Mcgraw, Coy Mcgraw, Crystalynn Mcgraw, Dakisha Mcgraw, Davine Mcgraw, Dawnyel Mcgraw, Demetrias Mcgraw, Derica Mcgraw, Dinamarie Mcgraw, Dvora Mcgraw, Elbony Mcgraw, Emanuela Mcgraw,
Gayleen Mcgraw, Genevra Mcgraw, Hadiyah Mcgraw, Harolyn Mcgraw, Hidie Mcgraw, Indy Mcgraw, Jaki Mcgraw, Jalana Mcgraw, Jalila Mcgraw, Janiel Mcgraw, Jennah Mcgraw, Jessia Mcgraw, Jihan Mcgraw, Jinnie
Mcgraw, Jonya Mcgraw, Justa Mcgraw, Justice Mcgraw, Kameela Mcgraw, Kameko Mcgraw, Kaneisha Mcgraw, Karletta Mcgraw, Keneisha Mcgraw, Kimberl Mcgraw, Kimyada Mcgraw, Kirstyn Mcgraw, Krisy Mcgraw,
Kyisha Mcgraw, Labreeska Mcgraw, Lacreasha Mcgraw, Lajeana Mcgraw, Lakiska Mcgraw, Lanya Mcgraw, Laren Mcgraw, Latash Mcgraw, Lataunya Mcgraw, Leiah Mcgraw, Loralyn Mcgraw, Lorre Mcgraw, Malaka
Mcgraw, Malicia Mcgraw, Marga Mcgraw, Margaretann Mcgraw, Markell Mcgraw, Meriam Mcgraw, Micca Mcgraw, Morningstar Mcgraw, Neta Mcgraw, Nigel Mcgraw, Noah Mcgraw, Nuria Mcgraw, Oliver Mcgraw, Osha
Mcgraw, Pierrette Mcgraw, Quinita Mcgraw, Quintessa Mcgraw, Rashan Mcgraw, Ravonda Mcgraw, Raylynn Mcgraw, Reyes Mcgraw, Rhondalyn Mcgraw, Rosey Mcgraw, Roshanna Mcgraw, Ruthy Mcgraw, Sasheen Mcgraw,
Shakeisha Mcgraw, Shakirah Mcgraw, Shaneta Mcgraw, Sharmeka Mcgraw, Sharonica Mcgraw, Shelba Mcgraw, Shelinda Mcgraw, Shell Mcgraw, Sheranda Mcgraw, Stachia Mcgraw, Stefanee Mcgraw, Sydnee Mcgraw,
Tamekka Mcgraw, Tanisa Mcgraw, Telissa Mcgraw, Terissa Mcgraw, Tilda Mcgraw, Tomoko Mcgraw, Trinna Mcgraw, Tulani Mcgraw, Varina Mcgraw, Verlene Mcgraw, Vernisha Mcgraw, Yasha Mcgraw, Yvonnie Mcgraw,
Albertine Mcgraw, Angeletta Mcgraw, Anni Mcgraw, Arrica Mcgraw, Billyjo Mcgraw, Brocha Mcgraw, Calie Mcgraw, Cass Mcgraw, Catie Mcgraw, Chairty Mcgraw, Chakakhan Mcgraw, Chassie Mcgraw, Cherae
Mcgraw, Cheryn Mcgraw, Chisa Mcgraw, Chrystine Mcgraw, Coronda Mcgraw, Curtina Mcgraw, Danuta Mcgraw, Debie Mcgraw, Delona Mcgraw, Demia Mcgraw, Dosha Mcgraw, Enrica Mcgraw, Evone Mcgraw, Fanchon
Mcgraw, Fina Mcgraw, Gaetana Mcgraw, Genevia Mcgraw, Gera Mcgraw, Gracy Mcgraw, Hollyann Mcgraw, Iasia Mcgraw, Iyesha Mcgraw, Jackline Mcgraw, Joanette Mcgraw, Jonnette Mcgraw, Kaamilya Mcgraw,
Karolina Mcgraw, Kathlynn Mcgraw, Kayle Mcgraw, Kayse Mcgraw, Kehaulani Mcgraw, Kella Mcgraw, Kenyatte Mcgraw, Kianga Mcgraw, Kijuana Mcgraw, Kinisha Mcgraw, Kiwanda Mcgraw, Kriss Mcgraw, Lakea
Mcgraw, Lakeithia Mcgraw, Lakeyia Mcgraw, Latusha Mcgraw, Leala Mcgraw, Leina Mcgraw, Lucita Mcgraw, Madia Mcgraw, Mairead Mcgraw, Makiba Mcgraw, Marchella Mcgraw, Marquel Mcgraw, Martita Mcgraw,
Meco Mcgraw, Melissaann Mcgraw, Merilyn Mcgraw, Minyon Mcgraw, Monnica Mcgraw, Monnie Mcgraw, Mora Mcgraw, Netasha Mcgraw, Nkechi Mcgraw, Ocie Mcgraw, Oni Mcgraw, Patricie Mcgraw, Quina Mcgraw,
Quintana Mcgraw, Renu Mcgraw, Rhetta Mcgraw, Rissa Mcgraw, Rozalyn Mcgraw, Salisa Mcgraw, Seena Mcgraw, Shaine Mcgraw, Shandon Mcgraw, Shanisha Mcgraw, Shantai Mcgraw, Sharisa Mcgraw, Shary Mcgraw,
Shaylyn Mcgraw, Sheleta Mcgraw, Shilonda Mcgraw, Sojourner Mcgraw, Syrena Mcgraw, Taleah Mcgraw, Tanekia Mcgraw, Tatianna Mcgraw, Tere Mcgraw, Terez Mcgraw, Teriann Mcgraw, Tewana Mcgraw, Ticey
Mcgraw, Vanette Mcgraw, Vidya Mcgraw, Wonder Mcgraw, Zoie Mcgraw, Aeisha Mcgraw, Alisande Mcgraw, Allan Mcgraw, Amantha Mcgraw, Analee Mcgraw, Ane Mcgraw, Artrice Mcgraw, Audelia Mcgraw, Ayelet
Mcgraw, Benji Mcgraw, Benny Mcgraw, Briann Mcgraw, Brittan Mcgraw, Calvina Mcgraw, Cameka Mcgraw, Cecilie Mcgraw, Cecille Mcgraw, Chanette Mcgraw, Chantia Mcgraw, Charlese Mcgraw, Charolett Mcgraw,
Chenille Mcgraw, Chivonne Mcgraw, Chudney Mcgraw, Clemencia Mcgraw, Courtny Mcgraw, Cristalle Mcgraw, Cynde Mcgraw, Daveda Mcgraw, Derika Mcgraw, Dezarae Mcgraw, Dionicia Mcgraw, Dunia Mcgraw,
Elspeth Mcgraw, Fayette Mcgraw, Glena Mcgraw, Gordon Mcgraw, Gretchin Mcgraw, Halli Mcgraw, Ilissa Mcgraw, Jauna Mcgraw, Johnnette Mcgraw, Joycelynn Mcgraw, Junelle Mcgraw, Kaydee Mcgraw, Kerryn
Mcgraw, Koryn Mcgraw, Kosha Mcgraw, Kyoko Mcgraw, Lala Mcgraw, Lametra Mcgraw, Laresha Mcgraw, Lauraann Mcgraw, Lecretia Mcgraw, Luv Mcgraw, Lyndsie Mcgraw, Malessa Mcgraw, Marcine Mcgraw, Mellissia
Mcgraw, Nakina Mcgraw, Nashonda Mcgraw, Neelam Mcgraw, Nikitia Mcgraw, Orit Mcgraw, Otis Mcgraw, Pasqualina Mcgraw, Porshia Mcgraw, Rayshell Mcgraw, Rico Mcgraw, Rockell Mcgraw, Roshon Mcgraw,
Savanna Mcgraw, Shallan Mcgraw, Shashana Mcgraw, Shawntia Mcgraw, Sherrise Mcgraw, Sofie Mcgraw, Taji Mcgraw, Talanda Mcgraw, Tamella Mcgraw, Tamsyn Mcgraw, Taquisha Mcgraw, Tashea Mcgraw, Terisha
Mcgraw, Tikita Mcgraw, Timmy Mcgraw, Trenese Mcgraw, Valissa Mcgraw, Valli Mcgraw, Vergie Mcgraw, Yancy Mcgraw, Aina Mcgraw, Albina Mcgraw, Aletheia Mcgraw, Alisson Mcgraw, Allisyn Mcgraw, Amorita
Mcgraw, Anginette Mcgraw, Anjana Mcgraw, Anngela Mcgraw, Ariela Mcgraw, Arienne Mcgraw, Arnisha Mcgraw, Arvella Mcgraw, Ayodele Mcgraw, Badia Mcgraw, Brooklynn Mcgraw, Candus Mcgraw, Conswello
Mcgraw, Curtisha Mcgraw, Cythina Mcgraw, Dashia Mcgraw, Dekisha Mcgraw, Denyce Mcgraw, Dilia Mcgraw, Doreatha Mcgraw, Dortha Mcgraw, Ernesto Mcgraw, Fatisha Mcgraw, Fauna Mcgraw, Fifi Mcgraw, Gordana
Mcgraw, Indya Mcgraw, Jaina Mcgraw, Janiene Mcgraw, Jarah Mcgraw, Jasmyne Mcgraw, Jeananne Mcgraw, Jeff Mcgraw, Jeronica Mcgraw, Jessaca Mcgraw, Joby Mcgraw, Jodette Mcgraw, Jolisa Mcgraw, Juanda
Mcgraw, Juanette Mcgraw, Kalie Mcgraw, Kaneshia Mcgraw, Karalynn Mcgraw, Kavitha Mcgraw, Kelcy Mcgraw, Kerrilynn Mcgraw, Kierston Mcgraw, Kineta Mcgraw, Kiri Mcgraw, Krisie Mcgraw, Latorria Mcgraw,
Lazara Mcgraw, Lekecia Mcgraw, Lenell Mcgraw, Lesle Mcgraw, Levina Mcgraw, Liat Mcgraw, Lindley Mcgraw, Maiya Mcgraw, Makela Mcgraw, Mariane Mcgraw, Marki Mcgraw, Mashelle Mcgraw, Maud Mcgraw, Medora
Mcgraw, Megumi Mcgraw, Milca Mcgraw, Missey Mcgraw, Mistelle Mcgraw, Nakeeta Mcgraw, Neile Mcgraw, Nicloe Mcgraw, Nykisha Mcgraw, Orly Mcgraw, Patria Mcgraw, Rama Mcgraw, Ramonia Mcgraw, Ranie
Mcgraw, Ranya Mcgraw, Refugio Mcgraw, Rendi Mcgraw, Schwanna Mcgraw, Senia Mcgraw, Shalaunda Mcgraw, Shalia Mcgraw, Shanieka Mcgraw, Shantale Mcgraw, Shatima Mcgraw, Shawon Mcgraw, Shelva Mcgraw,
Shemeca Mcgraw, Shemeeka Mcgraw, Sher Mcgraw, Sinda Mcgraw, Skyler Mcgraw, Stephaie Mcgraw, Suzetta Mcgraw, Suzzane Mcgraw, Synethia Mcgraw, Taesha Mcgraw, Takina Mcgraw, Tanyetta Mcgraw, Taran
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5 Ways to Calculate Binomial Coefficient in Python
Now we are going to see about the binomial coefficient in Python. Here we will learn a lot of methods to calculate the binomial coefficients. In mathematics, binomial helps us to expand some terms
with higher power easily. For example, if we have a number 103 to the power of 7. At that time, binomial is useful to expand this term. A binomial is known as a polynomial of the sum or difference of
two terms.
The binomial coefficient is a positive integer. It means is a positive whole number that is a constant in the binomial theorem. This is useful to expand the highest power. In mathematics, it is one
of the most interesting and beneficial.
What is the formula for the binomial coefficient?
The formula for the binomial coefficient is
In this, the value of n should always be greater than ‘k’.
Let us see how to calculate the binomial coefficient in python in different functions.
Here we are going to calculate the binomial coefficient in various functions they are:
• scipy.special.comb()
• scipy.special.binom()
• math.combo() function
• math.fact() function
• using operator
Method 1: Finding Python Binomial Coefficient Using scipy.special.comb()
What is the scipy module?
Scipy is a python library. It is useful for mathematical and scientific problems. Scipy is open-source. It is a built library in NumPy. scipy has some sub-packages. Now we are going to use the
sub-packages to calculate the binomial coefficient. scipy.binom() and scipy.comb() are the sub-packages we will use.
First, we will see about scipy.comb() function to check the binomial coefficient.
Syntax for scipy.comb()
val1- a value of n (must be greater than k) val2-value of k
binomial coefficient
import scipy.special
First, we are importing a library as scipy. special. This module holds the attribute comb, next to giving 20 and 10 to get the binomial coefficient.
Next using scipy.binom() module to calculate the binomial coefficient.
Method 2: Finding Python Binomial Coefficient Using scipy.special.binom()
Syntax for scipy.binom()
val1- value of n (must be greater than val2) val2-value of k
binomial coefficient
import scipy.special
First, we are importing a library as scipy.special. This module holds the attribute binom, next to giving 20 and 10 to get the binomial coefficient.
Method 3: Finding Python Binomial Coefficient Using math.combo() function
What is math.comb()?
The math module has a comb function that is used to calculate the binomial coefficient.
val1- a value of n (must be greater than k) val2-value of k
binomial coefficient
import math
First, we are importing library math. Next, giving 20 and 10 to calculate the binomial coefficient.
Method 4: Finding Python Binomial Coefficient Using math.fact() function
What is math.fact()?
The math module has the fact() function to calculate the binomial coefficient.
from math import factorial as fact
def binomial(a,b):
return fact(a) // fact(b) // fact(a-b)
First, we are importing the math function—next, declaring a function named binomial. Now giving parameters a and b. And then returning a formula to calculate the binomial coefficient.
Recommended Reading | Python Program for Factorial of a Number
Method 5: Finding Python Binomial Coefficient Using Operator
import math
import operator
from functools import reduce
prod = lambda x,y:reduce(operator.mul, range(x, y+1), 1)
a= 20
b = 10
c=prod(b+1, a) / prod(1, a-b)
First, importing math function and operator. From function tool importing reduce. A lambda function is created to get the product. Next, assigning a value to a and b. And then calculating the
binomial coefficient of the given numbers.
A fast way to calculate binomial coefficient in Python
def binomial(n, k):
if 0 <= k <= n:
a= 1
for t in range(1, min(k, n - k) + 1):
a *= n
b *= t
n -= 1
return a // b
return 0
First, create a function named binomial. The parameters are n and k. Giving if condition to check the range. Next, assign a value for a and b as 1. Now creating for loop to iterate. floor division
method is used to divide a and b. Next, calculating the binomial coefficient.
Finding Binomial Coefficient in Python Using Recursion
def factorial(z):
if z==1:
return 1
return z* factorial(z-1)
def binomial_coefficient(n,k):
a= (factorial(n)) / (factorial(k) * factorial(n-k))
return a
print("The binomial coefficient is:",binomial_coefficient(n,k))
The above code is calculating the binomial coefficient using recursion. First, we are creating a function named factorial. We all know that factorial is one of the best examples of recursion. And
below, we are doing the calculation for factorial.
Next, create another function named binomial_coefficient on the next line using the formula to calculate the binomial coefficient. Giving the value of n and k. And at last, calculating the binomial
But the above code is only useful for small numbers. If we want to go like the greatest numbers, we have to set the recursion limit.
The binomial coefficient is:252.0
Using recursion limit
As we already said, for the greatest numbers, we have to set the recursion limit. What we are going to do now:
import sys
def factorial(z):
if z==1:
return 1
return z* factorial(z-1)
def binomial_coefficient(n,k):
a= (factorial(n)) / (factorial(k) * factorial(n-k))
return a
print("The binomial coefficient is:",binomial_coefficient(n,k))
First, we are going to Import a sys module. Sys module is to set the recursion limit. We are setting the recursion limit as 3000 so that we can calculate to 3000. Creating a function named factorial.
We all know that factorial is one of the best examples of recursion. And below, we are doing the calculation for factorial.
Next, create another function named binomial_coefficient on the next line using the formula to calculate the binomial coefficient. Giving the value of n and k. And at last, calculating the binomial
The binomial coefficient is:2.7028824094543655e+299
Frequently Asked Questions Related to Binomial Coefficient Using Python
1. What are the possible way to calculate the binomial coefficient?
scipy.comb(), scipy.binom(),math.comb(), and math.fact() are the possible ways to calculate binomial coefficient.
2. What is the usage of binomial coefficients?
It is useful for analysis, and also it is a base for the binomial distributions.
3. What is the way of setting the recursion limit?
First importing sys module. And then setting the limit like sys.setrecursionlimit().
Here we have learned a lot about binomial coefficients. It is one of the interesting parts of mathematics. We can use binomial coefficients to expand the difficult term. It is advantageous and easy.
The above-shown methods are the possible ways to calculate the binomial coefficients in Python.
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AP Physics B and C- Equations to be Remembered in Circular Motion and Rotation
In the section ‘Circular motion and rotation’, the sub sections included as common for AP Physics B and C are
(1) Uniform circular motion and
(2) Torque and rotational statics.
The sub sections included for AP Physics C only are
(1) Rotational kinematics and dynamics and
(2) Angular momentum and its conservation.
Circular motion and rotation’ carries 4% of the total points for AP Physics B and 9% of the total points for AP Physics C.
Here are the important equations you need to remember to tackle the AP Physics B Examination:
(1) When an object is in uniform circular motion, the magnitude of its centripetal acceleration (a[c]) is given by
a[c ]= v^2/R where ‘v’ is the speed of the object and R is the radius of the circle. The direction of a[c] is always towards the centre of the circle.
The angular speed ω is related to the speed v by
v = ωR.
Therefore, the centripetal acceleration is a[c] = ω^2R.
If T is the time period of revolution of the object in circular motion and f is its frequency, we have f = 1/T and ω= 2π/T = 2πf.
The centripetal force required to produce circular motion is mv^2/R = mω^2R
(2) Angular acceleration (α) is the time rate of change of angular velocity and is given by
α = dω/dt
In the case of a body in accelerated rotational motion, the angular velocity (ω) after a time t is given by
ω = ω[0] + αt where ω[0 ]is the initial angular velocity and and α is the angular acceleration. [This is similar to the equation, v = v[0] + at in linear motion].
The angular displacement (θ) at the instant t is given by
θ = θ[0] + ω[0]t + (½) αt^2 where θ[0] is the initial displacement (at t =0)
[This is similar to the equation x = x[0] + v[0]t + (½) at^2 in linear motion].
(3) Torque (τ) is the moment of force and is the product of force and the lever arm. By lever arm we mean the perpendicular distance (ON in the figure) of the line of action of the force from the
Therefore, Torque τ produced by the force F acting on the particle at P = ON × F.
Since ON = r sinθ where θ is the angle between r and F,
τ = rF sinθ
[Note that torque (τ) is a vector which is the vector product of the position vector r and the force vector F. Therefore, torque τ = r × F]
(4) A rigid body will be in mechanical equilibrium, if the total force and the total torque on the body are zero. The condition of zero net force will ensure that there is no change in the linear
momentum and the condition of zero net torque will ensure that there is no change in the angular momentum.
For AP Physics C Examination, you will require the following also (in addition to the above):
(5) Consider a particle of mass m and linear momentum p. If the position vector of the particle is r, the angular momentum L of the particle with respect to the origin is given by
L = r × p
L is a vector whose magnitude is rpsinθ where r is the distance of the particle from the origin, p is the magnitude of the linear momentum vector p and θ is the angle between the vectors r and p.
Angular momentum is the moment of the linear momentum and is the product of linear momentum and the lever arm r sinθ. [By lever arm we mean the perpendicular distance of the line of action of the
linear momentum from the origin].
(6) Moment of inertia (I)of a system of particles (as in the case of a rigid body) about an axis (of rotation) is given by
I = ∑mr^2 where m is the mass of a particle at perpendicular distance r from the axis of rotation and the summation is for all the particles.
Radius of gyration (k) is related to moment of inertia (I) and the mass of the body (M) as
I = Mk^2
Note that both I and k depend on the axis of rotation.
Parallel axes theorem states that the moment of inertia (I)of a body about any axis is equal to the sum of the moment of inertia (I[CM] )of the body about a parallel axis through its centre of mass
and the product Ma^2 where M is the mass of the body and a is the distance between the two axes:
I = I[CM] + Ma^2
Perprndicular axes theorem states that the moment of inertia of a lamina about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent
with the perpendicular axis and lying in the plane of the body.
[For example, if the X and Y axes are in the plane of the lamina, the Z-axis is the perpendicular axis and we have I[Z] = I[X] + I[Y] where I[Z], I[X] and I[Y] are the moments of inertia about the Z,
X and Y-axes respectively]
Moments of inertia of some regular bodies are given below:
(i) Thin circular ring (Mass M, Radius R) about its central axis perpendicular to its plane: MR^2
A hollow cylinder (pipe) also has the above value for its moment of inertia about its own axis.
(ii) Thin circular ring (Mass M, Radius R) about any diameter: MR^2/2
(iii) Thin rod (Mass M, Length L) about a perpendicular axis through the mid point: ML^2/12
(iv) Circular disc (Mass M, Radius R) about its central axis perpendicular to its plane: MR^2/2
(v) Circular disc (Mass M, Radius R) about its diameter: MR^2/4
(vi) Solid cylinder (Mass M, Radius R) about the axis of the cylinder: MR^2/2
(vii) Solid sphere (Mass M, Radius R) about its diameter: (2/5)MR^2
(viii) Hollow sphere (Mass M, Radius R) about its diameter: (2/3)MR^2
(7) Angular momentum (L) is given by
L = Iω where I is the moment of inertial about the axis of rotation and ω is the angular velocity.
This is similar to the expression for linear momentum p = mv. In angular motion (rotational motion) I is to be used in place of m and ω is to be used in place of v.
Newton’s 2^nd law in rotational motion is
τ = dL/dt = d(Iω)/dt
If the moment of inertia (I) is constant, as is the case of a rigid body rotating about a fixed axis, we can write
τ = I (dω/dt) = I α where α is the angular acceleration.
The law of conservation of angular momentum states that in the absence of external torque, the angular momentum of a system remains unchanged. This can be expressed as
I[1 ]ω[1] = I[2] ω[2] where I[1] and I[2] are the initial and final moments of inertia and ω[1] and ω[2] are the initial and final angular velocities of a system in the absence of external torques.
(8) Acceleration (a) of a body rolling down an inclined plane of inclination θ is given by
a = gsinθ / [1 + (k^2/R^2)] where R is the radius of the body and k is the radius of gyration about the axis of rolling.
Since Mk^2 = (2/5)MR^2 for a solid sphere, k^2/R^2 = 2/5. This is the least value in the case of regular bodies and hence the acceleration a is maximum in the case of a solid sphere. In contrast, in
the case of a ring (and pipe), the value of k^2/R^2 is 1 and is the maximum in the case of regular bodies and hence the acceleration a is the minimum in the case of a ring.
If differently shaped bodies are allowed to roll down from the top of an inclined plane, the solid sphere will reach the bottom first and the ring (and the pipe) will arrive last. It is interesting
to note that for a given shape, the time of arrival at the bottom is independent of mas and size.
The above equation for acceleration down the plane can also be written as
a = (Mgsinθ) / [M + (I/R^2)] where M is the mass of the body; but it will be better to remember the above form in terms of the radius of gyration, k.
(9) Rotational kinetic energy (K) of a body is given by
K = (½) Iω^2
(10) A rolling body has translational and rotational kinetic energies. The total kinetic energy of a rolling body is therefore given by
K = (½) Mv^2 + (½) Iω^2 where v is the linear velocity of the body
(11) Work (dW) done by a torque τ in producing an angular displacement dθ is given by
dW = τ dθ [By dW we mean the small amount of work done for a small angular displacement dθ]
This is similar to the expression for work, dW = Fds in linear motion.
(12) Power (P) in rotational motion is given by
P = τω
This is similar to the expression for power, P = Fv in linear motion
Different quantities in linear motion and the corresponding quantities in rotational motion (about a fixed axis) are given below:
(a) Displacement x → Angular displacement θ
(b) Velocity v = dx/dt or dr/dt→ Angular velocity ω = dθ/dt
(c) Acceleration a = dv/dt → Angular acceleration α = dω/dt
(d) Mass M→ Moment of inertia I
(e) Linear momentum p = Mv→ Angular momentum L = Iω
(f) Force F = Ma →Torque τ = I α
(g) Work dW = Fds → Work dW = τ dθ
(h) Kinetic energy K = Mv^2/2→ Kinetic energy K = Iω^2/2
(i) Power P = F v→ Power P = τω
In the next post, we will discus typical questions in this section.
3 comments:
1. Allright,I understood the formula of calculating the acceleration of a rolling body down the formula,is there any formula to calculate the velocity of a rolling body down the inclined plane.
2. This comment has been removed by the author.
3. Hello roshanboy,
To obtain the linear velocity down the plane, just substitute the value of the acceleration in the equation of linear motion such as v = u + at or v^2 = u^2 + 2as. You can use the energy
relation, mgh = (1/2) mv^2 +(1/2)Iω2 also to find the velocity. Here ‘h’ is the height of the inclined plane, ‘I’ is the moment of inertia and ‘ω’ is the angular velocity which is v/R where R is
the radius of the rolling body. Don’t think of a ready made formula for ‘v’. | {"url":"http://www.apphysicsresources.com/2008/01/ap-physics-b-and-c-equations-to-be.html","timestamp":"2024-11-08T05:22:10Z","content_type":"application/xhtml+xml","content_length":"124039","record_id":"<urn:uuid:d4a1f10d-8019-43f3-a510-e29a4df0cd01>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00697.warc.gz"} |
Numerical Solution of the Quadratic Formula
Numerical Recipes
Homework Problem 7 in Chapter 11 of Intermediate Physics for Medicine and Biology examines fitting data to a straight line. In that problem, the four data points to be fit are (100, 4004), (101,
4017), (102, 4039), and (103, 4063). The goal is to fit the data to the line y = ax + b. For this data, one must perform the calculation of a and b to high precision or else you get large errors. The
solution manual (available to instructors upon request) says that
This problem illustrates how students can run into numerical problems if they are not careful. With modern calculators that carry many significant figures, this may seem like a moot point. But
the idea is still important and can creep subtly into computer computations and cause unexpected, difficult-to-debug errors.
Are there other examples of numerical errors creeping into calculations? Yes. You can find one discussed in Numerical Recipes that involves the quadratic formula.
We all know the quadratic formula from high school. If you have a quadratic equation of the form
the solution is
For example,
has two solutions
so x = 1 or 2.
Now, suppose the coefficient b is larger,
The solution is
so x = 300 or 0.00667.
This calculation is susceptible to numerical error. For instance, suppose all numerical calculations are performed to only four significant figures. Then when you reach the step
you must subtract 8 from 90,000. You get 89992, which to four significant figures becomes 89990, which has a square root of (again to four significant figures) 300.0. The solutions are therefore x =
300 or 0. The large solution (300) is correct, but the small solution (0 instead of 0.00667) is completely wrong. The main reason is that when using the minus sign for ± you must subtract two numbers
that are almost the same (in this case, 300–299.98667) to get a much smaller number.
You might say “so what! Who uses only four significant figures in their calculations?” Okay, try solving
where I increased b from 300 to 3000. You’ll find that using even six significant figures gives one nonsense solution (try it). As you make b larger and larger, the calculation becomes more and more
difficult. The situation can cause unexpected, difficult-to-debug errors.
What’s the moral to this story? Is it simply that you must use high precision when doing calculations? No. We can do better. Notice that the solution is fine when using the plus sign in the quadratic
equation. We need make no changes. It’s the negative sign that gives the problem,
Let’s try a trick; multiply the expression by a very special form of one:
Simplifying, we get
Voilà! The denominator has the plus sign in front of the square root, so it is not susceptible to numerical error. The numerator is simplicity itself. Try solving x2- 300 x + 2 = 0 using math to four
significant figures,
No error, even with just four sig figs. The problem is fixed!
I should note that the problem is fixed only for negative values of b. If b is positive, you can use an analogous approach to get a slightly different form of the solution (I’ll leave that as an
exercise for the reader).
So, the moral of the story is: if you find that your numerical calculation is susceptible to numerical error, fix it! Look for a trick that eliminates the problem. Often you can find one. | {"url":"https://bradroth.medium.com/numerical-solution-of-the-quadratic-formula-13ea4314d091","timestamp":"2024-11-06T06:05:11Z","content_type":"text/html","content_length":"145730","record_id":"<urn:uuid:8dd04841-a777-4163-b219-06c4a3c38e72>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00000.warc.gz"} |
Divergence and float points issue with Atmospheric Boundary Layers inlet UDF | Ansys Learning Forum
□ November 19, 2023 at 2:54 pm
Hi, I am simulating the wind flowing across a Y-shape building to evaulate the wake effect behind.Â
Mesh was done in Fluent Meshing, with minimum orthogonality of 0.2, and 5 inflation laters added to ground and building surface.
with BOI refinement in volume near the building
To be more accurate towards reality, I want to take the atmospheroc boundry layer for the wind flow inlet condition.
However, divergence and float points popped up on console when initiaizing. (while just using constant input to inlet velocity did not cause such issues), it is earnestly hoped for some
guidance on this and some additional tips on using UDF, thanks!!!!
Error codes:
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0
Divergence detected in AMG solver: ads-0 10 0.000000e+00
Hybrid initialization is done.
Error at host: floating point exception
Error at Node 4: floating point exception
Error at Node 0: floating point exception
Error at Node 1: floating point exception
Error at Node 2: floating point exception
Error at Node 3: floating point exception
Error at Node 5: floating point exception
Error at Node 6: floating point exception
Error at Node 7: floating point exception
Error at Node 8: floating point exception
Error at Node 9: floating point exception
Error at Node 10: floating point exception
Error at Node 11: floating point exception
Error at Node 12: floating point exception
Error at Node 13: floating point exception
Error at Node 14: floating point exception
Error at Node 15: floating point exception
Error at Node 16: floating point exception
Error at Node 17: floating point exception
Error at Node 18: floating point exception
Error at Node 19: floating point exception
===============Message from the Cortex Process================================
Compute processes interrupted. Processing can be resumed.
Error: floating point exception
Error Object: #f
Below is the UDF I taken reference from &t=2395s:
#include "udf.h"
#define UREF 4.491 // reference wind speed at ZREF
#define ZREF 0.16 // height of the building, 2b
#define LREF 0.08 // characteristic length, b
#define CMU 0.09 //
DEFINE_PROFILE(velocity_profile, thread, position)
float x[ND_ND];
float y;
float u;
face_t f;
begin_f_loop(f, thread)
u = UREF*pow(y/ZREF,0.27);
F_PROFILE(f,thread,position) = u;
end_f_loop(f, thread)
/* profile for kinetic energy, k */
DEFINE_PROFILE(k_profile, thread, position)
float x[ND_ND];
face_t f;
float A1 = 0.0042 ;
float A2 = -0.0719 ;
float A3 = 0.2692 ;
float A4 = 0.3671 ;
float k, y, yb;
begin_f_loop(f, thread)
y = x[1];
yb = y/LREF;
k = A1*pow(yb,3)+A2*pow(yb,2)+A3*yb+A4 ;
end_f_loop(f, thread)
/* profile for dissipation rate, epsilon */
DEFINE_PROFILE(dissip_profile, thread, position)
float x[ND_ND];
face_t f;
float A1 = 0.0042 ;
float A2 = -0.0719 ;
float A3 = 0.2692 ;
float A4 = 0.3671 ;
float k, y, yb;
begin_f_loop(f, thread)
yb = y/LREF;
k = A1*pow(yb,3)+A2*pow(yb,2)+A3*yb+A4 ;
[1] Yan MENG and Kazuki HIBI. Turbulent measurements of the flow field around a high-rise building. Journal of wind engineering. No 76, July 1998.
□ November 20, 2023 at 12:31 pm
You may want to check your direction vector for "up". If z is up it's not [1] in the vector definition!Â
☆ January 20, 2024 at 6:56 am
Hi Rob, thank you for the guide.
I have rotated my geometry so that y is pointing upwards to meet the UDF.
Still same errors occur during initialisation. Everything works when I just insert constant velocity for inlet without udf
Sincerely hoped for further advice, Thanks!
□ January 22, 2024 at 11:25 am
I'd have changed the UDF! What is the minimum y value in the domain? Use the Scale tool in Fluent to find out.Â
☆ January 22, 2024 at 1:02 pm
Yes I have altered the ZREF and LREF according to my building geometry with height (z-axis): 150m and cross-sectional width: 64.4m. For the rest of the UDF, I think they describing the
general solving for atmospheric boundary layer in velocity, k and epsilon, so I didnt modify it.Â
#include "udf.h"
#define UREF 4.491 // reference wind speed at ZREF
#define ZREF 150.0 // height of the building, 2b
#define LREF 64.4 // characteristic length, b
#define CMU 0.09 //
Or are you referring to the mini. y+ of my mesh domain?
Thank you for help.
□ January 22, 2024 at 1:09 pm
No, what is the minimum y coordinate on the inlet?Â
Before going too much into y+ how confident are you about the ground position and detail? I see far too many thinking y+ is the only mesh metric, and if you have bushes, trees, curbs etc did
you also resolve all of those?Â
• The topic ‘Divergence and float points issue with Atmospheric Boundary Layers inlet UDF’ is closed to new replies.
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Jeremy Quastel
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Professor Quastel grew up in Vancouver, British Columbia and received his undergraduate degree from McGill. He obtained his Ph.D. from New York University in 1990 under the supervision of S.
Varadhan. His thesis was a tour-de-force in how to derive differential equations from discrete interacting particle systems. After postdoctoral work at the Mathematical Sciences Research Institute,
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Professor Quastel is particularly known for a series of ground-breaking works during his years at
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known as the Kardar-Parisi-Zhang (KPZ) equation. The KPZ equation is a central object in modern probability related to interacting systems from biology, heat flow in inhomogeneous media, polymers,
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Some of these discoveries occurred during his service as chair of the Department of Mathematics from 2017 to 2021. As chair, he reformed the undergraduate teaching program, expanding the hiring of
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For the profound impact of his work, Quastel received the Killam Research Fellowship (2013). He has been recognized as a Fellow of the Royal Society of Canada (2016). He has also received the
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Outputting 'state prediction' time series with Proc Model Switching regression
Hello, I have been working on US recession turning point analysis using LEI data, and I have run into a bit of a snag. One of the models I am working with is a Markov Switching model, seen here:
http://support.sas.com/documentation/cdl/en/etsug/66100/HTML/default/viewer.htm#etsug_model_sect277.... (I believe this is the most famous example of how to do this in SAS by far)
However this code only estimates the parameters of the equation in each state, as well as a series of tests to see if the parameters are significantly different between states. What I am looking for
out of this model is a monthly time series output (I'm using monthly time series data) of which state it believes is active in a given month, so basically a series of regime/state probabilities. Is
there any way to get this working in SAS?
04-24-2017 09:52 PM | {"url":"https://communities.sas.com/t5/SAS-Forecasting-and-Econometrics/Outputting-state-prediction-time-series-with-Proc-Model/td-p/353057","timestamp":"2024-11-13T19:38:14Z","content_type":"text/html","content_length":"186850","record_id":"<urn:uuid:745297b5-8647-4b4f-b51c-58a1b22a90c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00815.warc.gz"} |
Logarithm Calculator with Solution - Tej Calculator
Advanced Logarithm Calculator
Logarithm Calculator: Your Essential Tool for Simplifying Math
If you’ve ever faced the daunting task of calculating logarithms, you know how challenging it can be. Enter the logarithm calculator—your new best friend in navigating the world of logarithmic
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What Are Logarithms?
At its core, a logarithm is a way to express exponents. When you see an expression like log[b](N) ), it translates to: “To what power must the base ( b ) be raised to produce the number ( N )?”
For instance, if we take log[10]100 , we’re asking, “10 raised to what power gives us 100?” The answer? 2, because ( 10^2 = 100 ).
Why Use a Logarithm Calculator?
You might be wondering, “Why can’t I just do these calculations manually?” While you can, a log calculator simplifies the entire process. Here’s why it’s essential:
1. Speed and Efficiency: Whether you’re crunching numbers for a homework assignment or a professional project, a calculator saves you time.
2. Accuracy: Human error can creep in during manual calculations, especially with complex logarithms. A calculator ensures you get it right every time.
3. Versatility: Many calculators offer functions for various types of logarithms, like common logarithms (base 10), natural logarithms (base ( e )), and binary logarithms (base 2).
Types of Logarithms
Understanding the different types of logarithms can help you choose the right calculations for your needs. Here’s a quick overview:
Type of Logarithm Definition Example
Common Logarithm Logarithm with base 10, written as log(N) log(100) = 2
Natural Logarithm Logarithm with base ( e ) (approximately 2.718), written as ln(N) ln(e^2) = 2
Binary Logarithm Logarithm with base 2, written as log[2](N) log[2](8) = 3
How to Use a Logarithm Calculator
Using a logarithmic calculator is a breeze! Here’s a simple guide to get you started:
1. Enter the Number: Input the number ( N ) for which you want to find the logarithm.
2. Select the Base: Choose the base ( b ). If you’re unsure, base 10 and ( e ) are common choices.
3. Hit Calculate: Click the “Calculate” button, and voilà! Your result appears instantly.
Example Calculations
To help illustrate the benefits of using a logarithm calculator, let’s look at some example calculations:
Number (N) Base (b) Logarithm Result ( \log_b(N) )
50 10 1.699
7.389 ( e ) 2
These examples showcase how a logarithm calculator can quickly provide answers that might otherwise take longer to compute manually.
Conclusion: Make Math Simpler with a Logarithm Calculator
Embracing the power of a log calculator can transform how you tackle mathematical challenges. Whether you’re a student trying to get through your math homework or a professional needing quick
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What is the null hypothesis in MyStatLab? | Hire Someone To Do Exam
What is the null hypothesis in MyStatLab? [a 3-D visualization application] This post is for anyone eager to explore a new approach to visualize a relationship in MyStatLab. The MyStatLab is a
graphical visualization application designed to visualize the relationships of relationships and relationships between documents, or relationships and movies. It will allow you to inspect the
relationship, and see how a particular relationship is built or used in the relationship. It’s built on the principles of a natural graph model – a simple node which represents many relationships
made up of two, independent edges, who can be easily or not linked to each other to get the connection. The visualization application will walk you through the relationships of relationships and
understand how each relationship is built and where they can go on their journey in the graphical visualization field. The main purpose of MyStatLab is to observe relationships in visualized
relationships, for those who like to have a “human” relationship or relationships between two people (people you know and not know), who do not know each other, and for those whose relationships and
relationships do not express deep and meaningful relationships, but are always built into other relationships within another world, where there may not be any close or close connections yet! Some
examples of relationships that can be viewed and analyzed in MyStatLab are: I have a party (donor’s part) where I have people I don’t know so I can’t come over often as this makes me feel unhappy. If
I leave the party and there is a missing person on our side, I then meet one of the new people on the other side who is showing me my party. This is my party and I do very well! If I weren’t there, I
would still be having some ill health trouble! but I have discovered I am different and that gives me something to believe and to feel! When I am no longer at one party, I have a personal friend who
falls out of the Continue One thing I have noticed so far, is that there are moments when the couple or people I have met is in our group who has gone to the other side of the street and they notice
before returning. When you give someone a party, you are bringing the home of that person you are closest to! With this site, my feeling is different. When I am coming home, the first thing i do when
I step out of the group are to find my party where a person I don’t know is there. I come back again, this time to find my other partner who you know but is seeing no friends or family. When I come
home, I find myself wondering if the family will meet me at my new place when I come into contact with them and I know I am in that group. I still have this feeling that there are people in other
groups that are not there – particularly because they are not here either! My feeling is that there is someone there who I could easily see in a crowd and hope to have a relationship/relationship
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self-confidence my feeling is that there is someone in the group that could help me find my new person, but will not reach my group until you are there and someone else wants to talk to me before I
leave! This is true of my feeling that there are people of different perspectives in your group that maybe one of you can help me find my true/self-sufficient person! So again we have found that
there are two ways of being helpful for meeting with someone/group and if everyone is there you will be able to find the next someone and he or she will be meeting your new/curious/previous person! I
have chosen the two method of being helpful for meeting with people, but the purpose is to enable your group members in the group to find them and help them to find your next person, they need to be
present and welcoming before you arrive at the next meeting! Below are some examples of how this is explained thoroughly for these visualization applications. As to my feeling but how I do feel with
the three methods of being helpful, this is because I think that all the visualizations in this article make readers feel more confident. What about me? As the image of my friends shows us, this
image describes events by the group members, and it is clearly a very important event in the group. Something like [their] name is there and we have all the information about them. What does it
matter and what makes you feel more confident? We can still feel more confident.
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These three visualization applications are used to understand which relationships and relationships between people need to be more of ongoing and/or ongoing relationshipsWhat is the null hypothesis
in MyStatLab? Let’s give a basic understanding of your problem: If a formula is true, and there is some condition that has a null hypothesis, we may draw a line in the text and find out if we get the
right hypothesis (or fail to get any hypothesis). However, it’s often not clear what lines we should look at when drawing lines when doing this: There are two situations when trying to draw a line
between a boundary and a given point: One rule for checking against null theories of the form \emph{~0}\emph{$\le$}. Therefore, finding such a line is the same as finding a negative line in a text,
but the correct results must also be accepted. For example, if you ran the program for 20 rows, and have 5 tests (this can be done for each row), it’s apparent that line can only exist/break other
lines on the right side of the figure. However, I wrote at least some comments – at least, for anchor while looking at a sample. Apparently, there is a lot when you start out, but you can definitely
get more results to the figure, especially if you are trying to figure out how this looks in real data. Update: Here is my problem! I tried out the second one and it’s exactly where my focus should
be. In \emph{~0};”, we go ahead with each pair of tests (this is what it is). This should show which lines are null if one, or break them if the other (I think part of that code was written to try to
keep my graph in the proper order). To avoid being overcomplicated, in the case above, we got the line \emph{0}\emph{90}. This made my graph clear! A: I managed to solve the problem, but I think
you’re trying to make your data even easier to readWhat is the null hypothesis in MyStatLab? is a statistical test for null hypothesis? If you are using (or you are wondering if) the test which
doesn’t say the null hypothesis here… you can make use of or go can try to find out what the null hypothesis does without having to start using Fermi or the statistical test and try to calculate what
went wrong. You can use the test given in MyStatLab Fermi functions, (as opposed to some distributions) provide an alternative distribution of the given number of bins to the normal distribution.
They have properties of non-negative vector , such as a non-zero vector where , and and . For more information see (based on my familiarity with them and various other functions). Can people use
MyStatLab to find out which of the following are true? The null hypothesis is true, but the test says the null hypothesis is false. MyStatLab has no analytical implementation to prevent negative
features from being found in the data. Can you indicate where these are located? (based on my familiarity with them) Maybe this could be done by my colleagues with something they could do.) Thank
you. 1 2 3 4 5 6 7 8 9 10 Thank you all. The last part will probably appear: Your 6-person team are using myStatsLab (http://statslab.
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com/) to produce estimates and predictions based on which of the following distribution(s) in the data was tested? In (1) (2) (3) (4) have the test chosen randomly for the distribution in the data?
If in such a case the (actual) negative parameters of a distribution are calculated, how can you plot them? These | {"url":"https://paytodoexam.com/what-is-the-null-hypothesis-in-mystatlab-2","timestamp":"2024-11-03T03:28:42Z","content_type":"text/html","content_length":"195605","record_id":"<urn:uuid:9efd7663-10bb-4d01-8428-efacdbe04b03>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00414.warc.gz"} |
cam_align in Axis
cam_align in Axis
13 Mar 2023 23:32
13 Mar 2023 23:38 #266615 by my1987toyota
So for the past week I have been trying to just get the cam_align as a tab to work in Axis.
I copied the files cam_align.ui and cam_align_handler.py from the usr / share / qtvcp/ panel folder.
and added them to my machine's profile folder. In my ini file I added under the display section
EMBEDED_TAB_NAME = cam_align
EMBEDED_TAB_ COMMAND = halcmd loaduser -Wn qtvcp_embed qtvcp -d -c qtvcp-embed -x {XID} cam_align.ui
I also made sure that the .ui and .py files were made executable.
and although I have a tab I don't even have a picture.
I am pretty sure I will need a . hal file for the buttons
but I am lost as to the camera what am I missing?
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14 Mar 2023 01:27 #266625 by cmorley
Please try the sim/axis/qtvcp/qtvcp_tab.ini to confirm all is well.
Also run from a terminal and post the output.
You don't need to add the ui or py files to your config folder unless you want to modify them.
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14 Mar 2023 06:35 #266637 by phillc54
The commands in the ini file should start with EMBED rather than EMBEDED.
Also there are parenthesis missing from the print statement in line 47 of the handler.
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14 Mar 2023 21:26 #266705 by my1987toyota
First off Thank you to phillc54, cmorley, and robertspark. Second I apparently made multiple typing mistakes .
first I typed embeded on my first post not in my . ini file. That didn't matter in reality. I did unfortunately type
loaduser in my ini file instead of loadusr. Next I put a - in one of my qtvcp_embed commands. That's the second
time that one has gotten me.
That said I now have a live picture in a Linuxcnc Tab
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16 Mar 2023 22:00
16 Mar 2023 22:03 #266868 by my1987toyota
how do I edit the cam_view widget. I can do the
-o size=400,400 -o rotincr=0. -o camnumber=0 cam_align
in 2.9 but in 2.8 all I get are errors saying -o is not an option.
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17 Mar 2023 15:38 #266916 by cmorley
Are you using an installed version? If so the problem is your changes could be overwritten when you update (but you still can do it)
If you are planning on doing more customization, you might want to switch to a RIP version of linuxcnc.
In the case of the 'o' option, it also requires changes to Qtvcp.py to accommodate it.
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17 Jan 2024 13:55 #290952 by joneb
Hi There I've been trying to get cam_align working following the instructions in the link below on bookworm and linuxcnc 2.9.2
I've got the camera working but can't work out which files I need to put the code in can anyone point me to where i might find this information
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Formulas for Calculating Time - Page 9
Formulas for Calculating Time
• @Charlene Pons It looks like you may have mistyped a column name (in bold):
=((VALUE(LEFT([End Time]@row, 2) + (VALUE(RIGHT([End Time]@row, 2))/60)) + ([Date]@row - [Start Date]@row) * 24)) - (VALUE(LEFT([Start Time]@row, 2) + (VALUE(RIGHT([Start Time]@row, 2)) / 60)))
• thanks for your posts about using the number stored by Smartsheet to pull the time difference. I had not heard of this before and was enlightened by the idea.
I was able to parse the information and break down how Smartsheet stores the information.
To begin:
=modified@row - Created@row will return the number of days with the decimals for the partial day.
Every day has 86,400 seconds. if you divide 1/86400 you would get 0.0000115740740740741.
However, Smartsheet only works with MAX 5 decimals which is why trying to multiply and divide the number returned by the above formula to get your time would not be accurate because it is
The way I got around this is by multiplying the number to return all the decimals not just the MAX 5 ones that show in the above formula:
=([Time Difference]@row - ROUNDDOWN([Time Difference]@row, 0)) * 100000000000000
Once we have all the full number we can then divide it by 1157407407.40741 to get the qty of seconds into the day.
You can then use the value returned from the above to return the hours, minutes, and seconds in the day:
=INT([Total Seconds]@row / 3600) (to return the hours).
=INT(MOD([Total Seconds]@row, 3600) / 60)
=MOD([Total Seconds]@row, 60)
An example of a fully formatted column is:
="D:" + INT([Time Difference]@row) + " H:" + INT([Total Seconds]@row / 3600) + " M:" + INT(MOD([Total Seconds]@row, 3600) / 60) + " S:" + MOD([Total Seconds]@row, 60)
Please let me know if you found the above to be accurate.
Thank you
• @Leibel Shuchat What did you put in the Time Difference column? Just a basic subtraction of the two cells?
I also wonder how it would react to a rather large gap between the two. I can't remember the exact number but (in addition to the number of digits) there is a "highest number" type limit as well.
What if one of those calculations exceeds that number? I am going to do some digging and see if I can find it again.
• @Leibel Shuchat I found this under the #OVERFLOW error help:
For numeric values the range is -9007199254740992 through 9007199254740992.
I haven't quite wrapped my head around the specifics of your solution, so it could be that I am misinterpreting something or worried about something that just wouldn't be logical (such as
millions of days before the limit is reached).
• Yes, Time Difference column is a subtraction of the two cells.
In regards to the overflow, we are only multiplying the decimals of the difference (the partial day) by 100000000000000. So that should never end up in an #OVERFLOW error.
• @Leibel Shuchat That's what I was thinking but wasn't 100% sure. Thanks for confirming.
@L_123 Check it out. We now have a much easier solution when using system generated date/time stamps.
• @Paul Newcome & @Leibel Shuchat
That's brilliant. I saw the ratio mentioned before and thought it was incorrect, but didn't have the time to check it to be sure. The rounding makes perfect sense. Thank you!
• It also seems that the milliseconds are stored as well. Without getting to complicated if you multiply the results of seconds you can then take that number into excel or the like and calculate
the milliseconds.
I could not find a way to do it in Smartsheet because of decimal and OVERFLOW limitations.
What triggered me looking into this is because I noticed some the second column sometimes showing 60 seconds (Formula: =MOD([Total Seconds]@row, 60). This is because the actual number was
slightly below 60 (hence the milliseconds) but was showing as 60 due to decimal limitations...
• Hi there,
I'm kinda of new to SmartSheet and formulas, there is a project that I really need to find the time spent on each of the attended fault, I wonder if anyone can help.
• @Paul Newcome, I've tried using the formula
=VALUE(MID(Created@row, FIND(" ", Created@row) + 1, FIND(":", Created@row) - (FIND(" ", Created@row) + 1))) + IF(VALUE(MID(Created@row, FIND(" ", Created@row) + 1, FIND(":", Created@row) - (FIND
(" ", Created@row) + 1))) <> 12, IF(FIND("P", Created@row) > 0, 12), IF(FIND("A", Created@row) > 0, -12))
To convert the 12hr format to 24hr from the created column. Once I close the parenthesis on on the value string, it won't let me add the + IF. It just breaks the formula.
Below I'm good when I close the initial string.
As soon as I add the + and type in if, it doesn't allow me to select the IF formula.
anyone that can help me get past this or if there's something better in Dec 2021 please for the sake of my sanity help...
• @Ted Sanborn Try removing one of the closing parenthesis from the end.
• I have a running log in a sheet called SHUTTLE MOVES.
There are 100's of entries added daily by numerous drivers.
The Created field is an automatic date field. (It must be automatic, and cannot be changed to a manually entered date or time field). When the user enters their shuttle move into the smartsheet
form, it is automatically timestamped. (Essentially, it is a unique field, unless 2 drivers just happen to press their submit button at precisely the same moment, which hasn't happened yet).
The driver also enters his/her name on each entry.
In my screenshot, I used a filter to see only today's entries for all drivers. However, in the sheet, it is a running log with thousands of entries for all dates/times. (I have no clue how many
entries there will be in a day, so I cannot refer to a specific cell address in a function. It must be a flexible reference).
I would like to tell smartsheet to look at the Created column, determine the first entry for today, determine the last entry for today, and calculate the number of hours between the first and
last entry per driver.
I'm not really sure how to do this with both a date and time captured inside of the same cell.
Maybe I need to convert the contents of the cell to a number and find the difference? I'm really not sure.
Thanks for the assistance.
• @Paul Newcome I have been reading through multiple threads and still having some trouble with my time calculations. Can you take a look at my sheet here to see if there is anything you can help
me with?
Since this is strictly a calculation sheet I am ok with adding helper columns in to do the calcs.
• @GarrettDyer Are you able to provide more details? How is the sheet supposed to work? What are the formulas you are currently using? How is the sheet supposed to work?
• @Paul Newcome did the link to the sheet work? I have details in the sheet along with the formula that explains. I thought that might be easier than typing it out in the comment. Let me know if
not and I will try to republish the link. | {"url":"https://community.smartsheet.com/discussion/comment/311258","timestamp":"2024-11-03T22:14:31Z","content_type":"text/html","content_length":"432383","record_id":"<urn:uuid:3a90bc69-c350-43f7-88a7-05d00debf721>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00720.warc.gz"} |
Theory of Combinatorial Algorithms
Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)
Mittagsseminar Talk Information
Date and Time: Thursday, November 17, 2011, 12:15 pm
Duration: 30 minutes
Location: OAT S15/S16/S17
Speaker: Thomas Rast
Pancake flipping is NP-hard
Assume a stack of pancakes can only be sorted (by size) by inserting a spatula between two pancakes, and flipping the entire upper part of the stack. This is formally known as sorting by prefix
reversal (SBPR). The question for the number of reversals needed has been open since 1975.
There is a "natural" efficient sequence of flips in which some SBPR instances can be solved, giving rise to a lower bound. In a very recent arXiv paper, Bulteau, Fertin and Rusu show that deciding
whether a given instance attains this lower bound is NP-hard. Thus more generally, determining the number of flips needed is NP-hard.
I will give an overview of the problem and known results, and then show some details of the (rather surprising at least for me) reduction from 3-SAT.
Upcoming talks | All previous talks | Talks by speaker | Upcoming talks in iCal format (beta version!)
Previous talks by year: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996
Information for students and suggested topics for student talks
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CSE 333/533 Assignment 2: Transformation, Viewing and Projection
The given program renders a cube in orthographic projection. Study the program and
understand how transformation, viewing and projection matrices are applied in the
OpenGL programmable pipeline.
Different parts of the assignment are divided into various files (transformations.cpp,
viewing.cpp, and torus.cpp) under StudentCode folder. Look for TODO comments to know
what to implement.
1. Generate screenshots after applying the following transformations (not successively) to
the cube
a. Rotate about the z-axis by 15 degrees [1 mark]
b. Scale along the y-axis by 2.0 [1 mark]
c. Translate by (20, 10) [1 mark]
2. Compose the above transformations in the order and . Generate
screenshots. Give reasons why the result is identical or different in the two cases [2
3. GLM provides a lookAt function to construct the viewing matrix.
a. Write your own version of the same as myLookAt(). [3 marks]
b. Verify correctness of your routine by comparing it with the GLM function used in the
provided code (i.e., by generating screenshots for comparison).
In case of any differences, explain. [2 marks]
4. Perform appropriate projection transformations and generate screenshots for the
following views of the cube:
a. Orthographic: top, front elevation, side elevation, and isometric. [2.5 marks]
b. Perspective: one-point, two-point, three-point (bird’s eye view), three-point (rat’s
eye view). [2.5 marks]
5. A torus is given by this equation
where , is the radius of the center of the hole to the center of the torus
tube, and is the radius of the tube. Generate a torus object and display it. [5 marks]
x = (c + a cosv) cos u
y = (c + a cosv)sin u
z = a sin v
u,v ∈ [0, 2π) c
1 of 2
• C/C++ code (make sure to upload full Cmake Project).
• 2~3 page PDF Report written with Latex/MS Word. Use the acmlarge option (single
column) (see sample-acmlarge.tex if writing with Latex).
Total marks for this assignment: 20 marks
Bonus (bonus marks to a maximum of 5 will be awarded for the following features. This
part is completely optional)
• Implement a trackball to rotate the cube with mouse. Doing so, you will have to
appropriately convert screen (x, y) coordinates into camera (x, y, z) position, and rotation.
[5 marks]
You may want to understand Trackball for the same:
Note: Your code should be written by you and be easy to read. You are NOT permitted to
use any code that is not written by you. (Any code provided by the TA can be used with
proper credits within your program)
2 of 2 | {"url":"https://codingprolab.com/answer/cse-333-533-assignment-2-transformation-viewing-and-projection/","timestamp":"2024-11-02T14:36:43Z","content_type":"text/html","content_length":"108863","record_id":"<urn:uuid:976798d5-b638-434b-89e2-7722dce49f27>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00276.warc.gz"} |
BuildIts in Progress
I was playing with my
thermal test motor
, and noticed that at high speeds, some pretty significant harmonics showed up in the phase currents. Here's an example with an obvious 5th harmonic - I'm commanding 20A, peak, and the harmonics add
a couple amps of ripple on top of that:
This ripple shows up because there's a 5th harmonic in the motor's flux linkage (and therefore back-emf). From the perspective of the field-oriented control, these harmonics at the phases show up as
AC disturbances on the D/Q voltages. The two current loops will try to squash these disturbances, but it's ability to do so depends on the current loop bandwidth and the magnitude of the
disturbances. Since both the magnitude and frequency go up with motor speed, even really high-bandwidth current loops have trouble keeping up. For example, in the scope shot at the top, the current
loop had around 2 kHz closed-loop bandwidth. The motor's electrical frequency was 300 Hz, so the harmonic (in D/Q) was at 1.8 kHz, so the controller couldn't quite reject it.
So how exactly do harmonics in the flux linkage at each phase show up when transformed into D and Q? In the sinusoidal flux linkage case with no position-varying inductance, the motor's dynamics in
the D/Q frame is exactly the classic voltage equations:
$$V_{d} = Ri_{d} + L\frac{di_{d}}{dt} - \omega Li_{q}$$
$$V_{q} = Ri_{q} + L\frac{di_{q}}{dt} + \omega Li_{d} + \omega\lambda$$
Where \(L\) is the synchronous inductance, \(R\) is the synchronous resistance, \(\omega\) is the (electrical) angular velocity, and \(\lambda\) is the peak flux linkage of one phase.
In the equations above, the flux linkage of a single phase \(\lambda_{p}\) is assumed to be of the form \(\lambda_{p} = \lambda \cos(\theta)\), where \(\theta\) is the rotor angle. Other phases will
have an offset of \(\pm \frac{2\pi}{3}\).
If the flux linkage has a fifth-harmonic in it, for a single phase it looks like \(\lambda \cos(\theta) + \lambda_{2}\cos(5 \theta) \). With the fifth harmonic in flux linkage, the voltage equations
in D and Q become:
$$V_{d} = Ri_{d} + L\frac{di_{d}}{dt} - \omega Li_{q} + 5\omega\lambda_{2}\sin(6\theta)$$
$$V_{q} = Ri_{q} + L\frac{di_{q}}{dt} + \omega Li_{d} + \omega\lambda - 5\omega\lambda_{2}\cos(6\theta)$$
(I have no interest in doing tedious trig and algebra by hand, so I wrote a
Mathematica script
to do the transforms analytically for me.)
That's interesting - a fifth harmonic at the phases shows up as a 6th harmonic through the transforms. It turns out, a 7th harmonic at the phases
shows up as a 6th harmonic through the transforms - just with both positive sine and cosine components, rather than positive sine negative cosine. 11 and 13 show up as 12, and so on. Also (although
this may be more intuitively obvious), odd multiples of 3 disappear through the transforms.
Just to confirm, here's the output when plugging a fifth harmonic into my
motor simulator
, using the the other motor parameters from the U12 and same controller gains as the hardware.
Phase currents:
D and Q axis currents, as well as rotor angle. You can count the 6 periods in the AC component of the current per electrical cycle.
So what if you want to control out those harmonics? Or add in harmonics of your own, for some reason? If you know how the voltages of the harmonics appear on the D and Q axes, theoretically you could
feed-forward them out, but there's a more interesting way to closed-loop deal with them, without knowing the specific amplitude of phasing of the harmonic, just it's frequency.
Adaptive Feedforward Cancellation
. This is a weird technique which was briefly introduced at the very end of
, which has the seemingly magical property of giving your controller infinite gain at a specific frequency - i.e. it will perfectly track a signal at that frequency, or perfectly reject a disturbance
at that frequency. AFC is probably most well-known for its use in things like hard-disk read heads, which have to follow very fine tracks of data on the hard drive platters. Since the platters have
some runout as they spin, the read-head has to track in the presence of periodic disturbances. Interestingly, AFC is a lot like field oriented control: FOC gives your controller infinite gain at the
fundamental electrical frequency of your motor, letting you track sinusoidal currents of much higher frequency than the bandwidth of your current controllers.
Here's a block diagram of how AFC works, taken from the paper "
Adaptive feedforward cancellation viewed from an oscillator amplitude control perspective
". \(r(t)\) is the reference (current, in my case), and \(d(t)\) is a disturbance (back-emf harmonics, in my case), and \(y(t)\) is the control effort (D or Q voltage in my case).
And here's how the AFC(s) are integrated into the control loop. The red box is your normal feedback controller (in my case the PI current controllers for D and Q current).
The AFC works kind of like a pair of integrators on the sine and cosine components of the periodic part of the error. If there's zero error \(e(t)\), the AFC outputs the sum of \(\cos(\omega t)\cdot
(\text{cos integral})\) and \(\sin(\omega t)\cdot (\text{sin integral})\), which is some sinusoidal thing. If there is some error, the portion of it at the same frequency as \(\omega\) doesn't
average to zero when multiplied by the \(\cos(\omega t)\) or \(\sin(\omega t)\)'s before the integrators. The integrators then integrate up to whatever value drives the \(\omega\) frequency component
of the error to zero.
The slight modification to get this to work for motor currents is changing the \(\omega t\) to be \(\theta\), the electrical angle of the motor. The periodic disturbances are not a fixed frequency in
, they are a fixed frequency in
This basically just works. Implementing a 6th harmonic AFC on both the D and Q current loops completely eliminates the 5th harmonic in phase currents. Here's the simulation result:
Phase currents:
D/Q currents:
And here's what the integrators in the AFC do:
There's one simple addition to the controller which makes it behave a little better. The magnitude of the harmonic voltage is proportional to how fast the motor is spinning - so if the motor is
accelerating, the magnitude is increasing, and the AFC integrators are constantly playing catch-up. This means that at a constant acceleration, there will be a constant error in the AFC:
The AFC still does a pretty good job even at high acceleration, but how well it does depends on how large the AFC gain is. Below you can see the integrators ramping up with speed:
The simple fix for this is to divide the error by angular velocity before integrating, and then multiply by angular velocity at the end of the AFC calculations. This way, the AFC integrators
integrate up to the harmonic in the flux linkage, which is constant, rather than the back-emf, which is proportional to speed. Honestly, this change doesn't make a huge difference, because the time
constant of the integrator can be quite fast compared to how fast the motor can accelerate, but it's kind of cool none the less:
Now the integrators just converge to constants, even as the motor accelerates:
Pseudo-code implementation:
1 h = 6.0f; // Harmonic number
2 k_afc = 100; // AFC Gain - larger gain converges faster
4 s = sin(h*theta);
5 c = cos(h*theta);
7 i_error = i_ref - i;
8 if(abs(thetadot) > 1) //avoid dividing by zero
9 {
10 afc_cos_int += dt*i_error*k_afc*c/thetadot;
11 afc_sin_int += dt*i_error*k_afc*s/thetadot;
12 }
13 afc_out = thetadot*(s*afc_sin_int + c*afc_cos_int);
15 i_error = i_ref - i + afc_out; //Now use this error in the PI current loop.
And here it is after implementing on the hardware:
The AFC behaves a little strangely when the voltage saturates (causing higher harmonics to show up), and I haven't put any effort into fixing that, but for most operating points it works quite well.
Here's an FFT of the phase current at the beginning of the post. There are significant components at 5, 7 and 11 times the electrical frequency:
And here's the FFT after adding an AFC at 6 times the electrical frequency:
In addition to canceling out harmonics, you can use this technique to add in harmonics of your own, or track periodic current references.
For example, a typical way to cancel out cogging torque is to add in an AC component to your Q-axis current reference which tries to cancel out the cogging torque. The controller's ability to track
this reference depends on how fast the motor is spinning, and its bandwidth. But if you add in an AFC at the cogging frequencies, the AFC just takes care of tracking the cogging current.
Or if you want to add certain harmonics at the phases, you can do so by taking the dq0 transform of the harmonics you want, then adding those to your D/Q current references with an AFC to track them.
Say, for example, you wanted your phase currents to be more trapezoidal, you could take the transform of the trapezoidal waveform, use that as your D/Q current references, and add AFCs at the first
few Fourier coefficients in order to track the reference at high speed. | {"url":"https://build-its-inprogress.blogspot.com/2018/09/","timestamp":"2024-11-14T03:31:11Z","content_type":"application/xhtml+xml","content_length":"105584","record_id":"<urn:uuid:475f14ff-614e-4742-9189-a0f8ce03654d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00728.warc.gz"} |
Choosing dt for time evolve
I am doing a periodic boundary Ising model with N sites (say 40 sites) and I am time evolving with the following code:
Python: Select all
#Time evolution parameters
evol_params = {
'compression_method': 'SVD', ### same results for 'SVD' and 'variational'
'trunc_params': {"chi_max": chi,"svd_min": 1e-10,},
evol = mpo_evolution.ExpMPOEvolution(psi, M, evol_params)
for t in range(t_max):
psi_t = evol.run()
It turns out that the quantities of my interest are converging to different values for different choices of dt.
What should I keep in mind to choose dt for N (say 40) sites? What choice of dt ensures that my results converge to the same value?
Re: Choosing dt for time evolve
There's no simple, generic rule how large/small dt has to be to ensure convergence,
I'm somewhat alarmed by the "periodic boundary conditions". By default, in TeNPy this would just give you one (or ~Ly, if you have a 2D torus) very long-range couplings between the ends of the MPS.
Both TDVP and the ExpMPOEvolution might not correctly include those long-range terms into the time evolution - TDVP focuses on the MPS tangent space, which might project it out (if the coupling is
not diagonal), and the W_I/W_II approximations to U=exp(-iHdt) in the ExpMPOEvolution is also just correct for "non-overlapping" terms. In both cases, a tiny dt should eventually get it correct, but
tiny might be really tiny, blowing up the evolution cost - and eventually also the truncation error, since you need to truncate after each step!
If you really need the periodic bc, you should hence use the lattice with a order="folded" parameter, see here, such that you don't have a single long-range coupling, but many couplings of twice the
range compared to open bc. | {"url":"https://tenpy.johannes-hauschild.de/viewtopic.php?p=1421&sid=67e7138472c7ae804bcd92ea99e1f041","timestamp":"2024-11-07T03:16:46Z","content_type":"text/html","content_length":"25528","record_id":"<urn:uuid:454f8395-95fb-4d39-9577-45a29d04f26a>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00502.warc.gz"} |
[ANSWERED] A nonlinear system is given along with the graphs of both - Kunduz
A nonlinear system is given along with the graphs of both
Last updated: 11/21/2023
A nonlinear system is given along with the graphs of both equations in the system Verify that the points of intersection specified on the graph are solutions of the system by substituting directly
into both equations Substitute the coordinates of the point 2 1 into the left side of the first equation x y 0 6 5 4 3 2 1 4 3 3 6 5 5 x y 5 116 | {"url":"https://kunduz.com/questions-and-answers/a-nonlinear-system-is-given-along-with-the-graphs-of-both-equations-in-the-system-verify-that-the-points-of-intersection-specified-on-the-graph-are-solutions-of-the-system-by-substituting-directly-319584/","timestamp":"2024-11-03T22:24:16Z","content_type":"text/html","content_length":"209435","record_id":"<urn:uuid:c06718db-1d4f-421a-9659-b0cb091cece7>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00042.warc.gz"} |
In recent years, model free reinforcement learning has had great success in controlling physically simulated environments. This is largely thanks to the development of powerful algorithms such as
proximal policy optimisation (PPO) [Sch17P] and soft actor critic methods (SAC) [Haa18aS]. Nevertheless, these algorithms often remain prohibitively expensive due to their high sample complexity:
even simple control problems may require millions of training steps.
Most 3D physics engines treat objects (such as robots) as “trees” of rigid bodies, with each branch having 6 degrees of freedom (position and rotation), and with positions and velocities constrained,
e.g. to prevent the bodies from penetrating each other. This “linear complementarity problem” (LCP) has been studied extensively over the course of many decades [Fea08R], with the most recent studies
being on numerical stability and speed. Different simulators, like Mujoco, PhysX or Bullet treat contacts and friction in different ways, with higher accuracy often coming at the cost of throughput.
One line of recent development involves the differentiability of the parameters of such engines. Similarly to how automatic differentiation libraries, like Tensorflow or PyTorch, enable efficient
differentiation of complex functions, and thus can train neural networks’ parameters to minimise a loss function, the parameters of physics engines could also be optimised to find the optimal
friction, body structure, initial position, joint torque, etc. for a robot to solve a certain task. In the simplest case, minimising a loss through gradient descent on the simulation parameters is
enough to solve the control problem, without the need for a neural network. This constitutes a fully model-based approach to robotic control.
Nevertheless, model-free approaches still hold the highest potential for efficiency and accuracy in highly complex environments. Differentiable environments will greatly help in reducing the drop in
performance (the so called sim-to-real gap) that models trained in a virtual environment witness when deployed in the real world. In particular, it will be possible to fine tune the friction and
contact forces of custom-made environments directly from video input, as done in a recent paper [Jat21G].
Brax [Fre21B] is one of the newest differentiable rigid body simulators. Written in JAX, it puts the engine and RL optimizer together on the same GPU/TPU chip, obtaining speed-ups of RL training of
up to 100-1000x. Such speed comes at a slight cost in accuracy, as contacts and friction are computed within the simplifying framework of “spring joints”. Other simulators, like Mujoco, rely on more
accurate methods, but are much slower or not fully differentiable. The choice of one simulator over another depends on the problem at hand: typically, planning and navigation do not require complex
contact modelling, while manipulation and grasping do.
If you feel like giving Brax a try, you can run it for free in this colab notebook. You can try it out on the OpenAI gym environments and even train some PyTorch models. Another example is related to
analytic policy gradient, i.e. the possibility to train a neural network directly using the gradients returned by the simulator. It is interesting to notice how we can leverage the differentiability
of the simulator for all sorts of hybrid approaches between classic control and neural networks. Finally, if you want to dig deeper, you can build a brand-new environment and run it on your local
machine: Brax is open-source! | {"url":"https://transferlab.ai/pills/2022/brax/","timestamp":"2024-11-13T16:31:44Z","content_type":"text/html","content_length":"36379","record_id":"<urn:uuid:5b857dc7-901e-4b8a-8353-00da7ba37038>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00387.warc.gz"} |
OpenStax College Physics, Chapter 28, Problem 32 (Problems & Exercises)
Prove that for any relative velocity $v$ between two observers, a beam of light sent from one to the other will approach at speed $c$ (provided that $v$ is less than $c$, of course).
Question by
is licensed under
CC BY 4.0
Final Answer
Please see the solution video.
Solution video
OpenStax College Physics, Chapter 28, Problem 32 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. We are going to show that when two observers are approaching each other with a relative velocity of v that when light is emitted from one observer
with a speed of c of course, we are going to show using this formula that the other observer will also report a velocity for this light as c. So suppose this first observer is labeled with the
velocity v that means the velocity for the light that they report is labeled u prime and we are told u prime is c so the question is what velocity does the second observer report for this light? So
the velocity that the second observer reports is going to be v plus u prime over 1 plus v times u prime over c squared. So u prime is c so we plug in c, wherever we see u prime here and this vc over
c squared becomes just v over c and then we can multiply top and bottom by c and on the top, let's just leave it outside of some brackets and on the bottom, the c gets distributed into this binomial
and so we have c plus c times v over c, which is just v and this numerator and denominator are the same so this fraction is 1 and this ends up being c and so we have shown that this second observer
will report the same thing as the first observer when they are talking about the speed of light. | {"url":"https://collegephysicsanswers.com/openstax-solutions/prove-any-relative-velocity-v-between-two-observers-beam-light-sent-one-other","timestamp":"2024-11-08T19:00:16Z","content_type":"text/html","content_length":"159787","record_id":"<urn:uuid:ad419ae3-1726-4582-8b03-e8b352c62e95>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00850.warc.gz"} |
conversion mesure liquide
We just eased conversions from dry measurements to liquid measurements. Then click the Convert Me button. Milliliters to Liters, = 65 mg” gram: milligram: 1 g = 1000 mg: A x 1000 = B: B/1000 = A :
gram: grain: 1 g = 15.4 gr. We also refer to it as the “capacity” or the “volume” of the vessel. Full disclaimer.Do not use calculations for anything where loss of life, … Online Conversion is a
resource for weights, measures, calculators, converters. Also, explore many other unit converters or learn more about volume unit conversions. The different tools used for liquid measurement are: 1.
To switch the unit simply find the one you want on the page and click it. Volume measureme Please Note: The figures mentioned in the charts above contain the approximate values. It is generally not a
good idea to scale a recipe up or down by more than 3 or 4 times. Cubic Feet to Cubic Meters, tank size). A dry ounce is 28.4g, while a liquid ounce is 29.6g (or rounded to 30mL). instructions for
how to enable JavaScript. Full disclaimer. Please help me spread the word by sharing this with friends or on your website/blog. In the past, many systems of measurement were define… Pour se faciliter
la vie, voici un récapitulatif des mesures et des équivalences les plus courantes et des tables de conversion … Male / 60 years old level or over / A retired person / Very /. Using definite measures
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for Emergency Substitutions, Liquid Volumes, Dry Weights, Temperature Conversions, Grain … Conversion Charts For Cooking and Baking The Ultimate No-Bake Banoffee Pie Recipe 20 mins Ratings.
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Converting B to A: Notes: grain: milligram: 1 gr. Dry: Milliliters: Fluid Ounces: 1 Pint: 551 mL: 19 fl. To help with productivity, we now set a cookie to store the last units you have converted from
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What is the Difference Between Nominal, Effective and APR Interest Rates? fluid ounces to milliliters, see the oz to mL question thread. Liquid Measure). Thank you. Liquid Measurement The liquid
measurement is the amount of liquid a vessel contains and its measurement in standard units. can you include British imperial measurements? Many different liquid and dry volume conversions. Your
value gets instantly converted to all other units on the … This is a conversion chart for gallon (U.S. This is the perfect time to use a printable liquid conversion chart for your reference. Liters
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Utilisez notre outil de conversion How many ml in a teaspoon? For instance, 4.93 mL is rounded off to 5 mL, and so on. Barrels (oil) (bbl (oil)), Barrels (UK) (bbl), Barrels (US), Board Feet, Bushels
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Tablespoons (Imperial), Tablespoons (US), Teaspoons (Metric), Teaspoons (Imperial), Teaspoons (US). There are several ways to measure liquids including teaspoons, tablespoons, quarts, fluid ounces,
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as a liquid volume calculator with output in … oz. Infant milk bottle with measurement markings in milliliter and juice bottle with 1 liter contents. 2: Enter the value you want to convert (quart).
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British imperial measurements? Use this liquid volume converter to convert instantly between barrels, cubic feet, gallons, liters, pints, tablespoons and other metric and imperial liquid volume
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determine measurements in different units. Un tableau de conversion de liquide s’utilise pour passer d’une unité de liquide vers une autre en toute facilité. Your dry measure table incorrectly refers
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Which relationships is commonly also known as the range rule getting practical deviation - Loonbedrijf Ekelmans
Which relationships is commonly also known as the range rule getting practical deviation
The standard departure and you will variety are both strategies of one’s bequeath of a data lay. For each number confides in us in individual ways how spaced out the data is, because they’re one
another a way of measuring adaptation. Even though there is not a specific relationship between the range and you will fundamental departure, there is certainly a guideline and this can be beneficial
to connect these two statistics.
The range rule tells us that important departure regarding a great attempt is approximately comparable to one-next of one’s variety of the information and knowledge. Simply put s = (Restrict –
Minimum)/cuatro. This might be an incredibly simple algorithm to use, and ought to just be used as a highly rough estimate from the quality deviation.
An illustration
Observe a good example of the way the range code works, we’ll glance at the adopting the example. Imagine i begin by the details philosophy from twelve, twelve, 14, fifteen, 16, 18, 18, 20, 20,
twenty five. This type of opinions keeps a hateful out of 17 and you can a standard deviation of approximately cuatro.step 1. In the event the as an alternative i first assess the variety of our very
own analysis as the 25 – 12 = thirteen after which separate this matter of the four we have our imagine of your own important deviation since thirteen/4 = 3.twenty five. That it count is relatively
close to the real important deviation and you can best for a harsh estimate.
Why does It Works?
You may be thinking including the assortment signal is a bit unusual. How come they really works? Doesn’t it search entirely arbitrary to just separate the number by the five? As to the reasons would
not we divide of the a different sort of amount? There is in fact particular analytical excuse going on behind-the-scenes.
Recall the characteristics of your bell curve therefore the likelihood of a standard regular delivery. You to definitely function has to pure do with the level of investigation that drops inside a
specific amount of simple deviations:
• Around 68% of the information is in one standard departure (higher or down) on the mean.
• Everything 95% of your own info is within this a couple practical deviations (highest otherwise all the way down) regarding mean.
• Everything 99% is actually about three standard deviations (higher otherwise all the way down) on the suggest.
The number that individuals use is due to 95%. We are able to declare that 95% out-of a few fundamental deviations underneath the mean so you’re able to a couple of practical deviations over the
imply, we have 95% of our research. Hence the majority of all of our typical shipments perform loosen up more than a column segment that’s all in all, four practical deviations long.
Not all information is generally speaking distributed and bell contour designed. But most data is really-behaved adequate you to definitely going several standard deviations off the imply grabs
nearly all of the information. I estimate and claim that five standard deviations was everything the fresh new size of the range, therefore the assortment split up of the five are a crude
approximation of one’s simple departure.
Purposes for the range Laws
The number code is helpful in a few configurations. First, it’s a highly small imagine of your own important departure. The quality deviation needs us to first select the imply, upcoming deduct this
mean away from for every investigation section, rectangular the differences, create this type of, split because of the you to less than how many investigation facts, next (finally) make square-root.
At exactly the same time, the number laws merely requires you to subtraction and another division.
Other places where assortment laws is helpful is when we keeps incomplete guidance. Algorithms like that to decide take to dimensions wanted around three pieces of data: the desired margin out-of
error, the amount of trust therefore the practical deviation of one’s people we are investigating. Several times it is impossible to understand what the population important deviation is. On
diversity rule, we can guess it fact, then know the way high we want to create our very own decide to try.
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How to Analyze and Resolve Conflicts with the Help of Games?
Can game theory help in negotiations and resolution of interstate conflicts? How has regulation based on the Prisoner's Dilemma contributed to the fight against cartels in Europe? Why is it important
to take into consideration future interactions for the game’s outcome? These and other questions were discussed in a recent episode of the "Economics Out Loud" podcast with NES Professor Sergei
Izmalkov. GURU shares a summary of the episode.
The interaction of people, companies, and states can be represented as a type of game with participants simultaneously trying to achieve their goals by choosing the optimal strategy. Therefore, we
can say that game theory deals with fundamental life issues.
People have been thinking for centuries about ways to formalize a person's choice in various situations. In the 1920s, mathematician Emile Borel was one of the first to describe the concept of
strategy as a plan of action for arbitrary situations. Any situation can be viewed as a model, i.e. a game where each player has a set of strategies and each pair of strategies can bring players
certain rewards.
In 1944, mathematician John von Neumann and economist Oskar Morgenstern published the book Theory of Games and Economic Behavior, which became the cornerstone of formal analysis in game theory.
Neumann and Morgenstern showed that for any zero-sum game, it is mathematically possible to find a solution – an equilibrium in which both players can adhere to an optimal strategy for themselves.
In zero-sum games, one player's gain is equivalent to another’s loss. Zero-sum games were actively used to analyze military conflicts, and there appeared a large number of papers on this topic.
However, military conflicts are in fact games with a non-zero sum. Losses are most often borne by both sides, while in zero-sum games, the loss of one side must be equal to the gain of the other. A
nuclear confrontation between the United States and the USSR would also be a non-zero-sum game, because when using nuclear weapons, both sides were threatened with mutual destruction.
In games with a non-zero sum, it is also always possible to come to an equilibrium. This was proved by the famous mathematician and a Nobel Prize winner in Economics John Nash, whose biography became
the basis of the Beautiful Mind film. The Nash equilibrium is a situation where it is disadvantageous for each of the players to deviate from their strategy while the behavior of the other remains
Game theory remains the only way to analyze strategic interactions and is actively applied in practice. For example, European and American antitrust authorities used to fight against cartels
employing a method based on the famous Prisoner's Dilemma, in which prisoners choose between a cooperative strategy and an egoistic one. Firms involved in a cartel were offered exemption from fines
if they were the first to tell the authorities about the cartel. The rest of the participants in the collusion were to be punished. When there was no guarantee of exemption from fines, it was easier
for firms to maintain a collusion, but the introduction of exemptions for the first company to disclose the cartel changed the whole nature of the interaction of cartel participants.
Many interactions in the real world can be described with the Prisoner's Dilemma, where the optimal strategy is non-cooperative, i.e. betrayal. But that is true only for a static game, one in which a
single decision is made by each player. If the game is dynamic, one in which players move sequentially or repeatedly, then the nature of interaction may change: cooperation becomes the optimal
strategy if players value future interaction.
In economies with inefficient institutions, people and firms often do not value future interactions due to high uncertainty. And vice versa: the lower the uncertainty, the more valuable the future
interaction is and the easier it is to maintain cooperative rather than selfish, individual strategies. This is very important for investments – both corporate and personal, for example, in human
capital. Reducing uncertainty is a task for the state, which must guarantee the protection of property rights and an effective judicial system. | {"url":"https://guru.nes.ru/en/en/kak-analizirovat-i-razreshat-konfliktyi-pri-pomoshhi-igr.html","timestamp":"2024-11-02T14:07:17Z","content_type":"text/html","content_length":"44800","record_id":"<urn:uuid:a4e3da22-77d0-4337-8dc4-b34511624882>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00017.warc.gz"} |
Force on a
Force on a current-carrying conductor in a magnetic field
We note also that the speed
If we therefore substitute equations 2 and 3 into 1, we have:
The direction of the force is determined in a similar way we determine the direction of motion of charged particles in a magnetic field, i.e., with the help of the right-hand screw rule. The curled
fingers represent the magnetic field
Another way to determine the direction of the force is using the left-hand rule: the magnetic lines go into the palm of the left hand, the outstretched fingers are the direction of the current and
the outstretched thumb is the direction of the magnetic force.
Determining the direction of the magnetic force on a current-carrying conductor using the left-hand rule
Note: A circle with a cross means the direction of the vector into the monitor (away from the observer), and a circle with a dot would mean the direction of the vector out of the monitor (towards the
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Initially the system shown in figure is in equilibrium. At the ... | Filo
Initially the system shown in figure is in equilibrium. At the moment, the string is cut the downward acceleration of blocks and are respectively and . The magnitudes of and are
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As soon as string is cut (on ) suddenly becomes zero. Therefore a force of acting on upward direction on suddenly becomes zero.
So net force on it will become downwards.
Spring force does not become instantly zero. So acceleration of will not change abruptly.
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Question Text Initially the system shown in figure is in equilibrium. At the moment, the string is cut the downward acceleration of blocks and are respectively and . The magnitudes of and are
Updated On Nov 5, 2022
Topic Work, Energy and Power
Subject Physics
Class Class 11
Answer Type Text solution:1 Video solution: 2
Upvotes 229
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Directed Reading Program
The mathematics department at UNC-CH leads a directed reading program where graduate students act as mentors for interested undergraduate math majors. These motivated undergraduates apply and express
interest in one or two fields or projects. Graduate student volunteers are subsequently matched with applicants and the two decide on an appropriate topic, guiding text, and goal/motivation. Mentees
present their semester’s work in an end of semester capstone talk given to other DRP mentees and mentors. For more information, visit the DRP homepage.
• In Spring 2021 I supervised an early math major in a reading of Quantum Calculus, by Pokman Cheung and Victor Kac.
• In Spring 2022 I supervised a project geared towards an overview of algebraic geometry and its applications with guiding text Ideals, Varieties, and Algorithms, by David A. Cox, John Little, and
Donal O’Shea. | {"url":"https://tarheels.live/josephcompton/mentoring/","timestamp":"2024-11-06T05:35:57Z","content_type":"text/html","content_length":"56715","record_id":"<urn:uuid:a2974fa5-e171-4d1c-9ed1-3d411b8a04c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00531.warc.gz"} |
Online scheduling of two job types on a set of multipurpose machines with unit processing times
We study a problem of scheduling a set of n jobs with unit processing times on a set of m multipurpose machines in which the objective is to minimize the makespan. It is assumed that there are two
different job types, where each job type can be processed on a unique subset of machines. We provide an optimal offline algorithm to solve the problem in constant time and an online algorithm with a
competitive ratio that equals the lower bound. We show that the worst competitive ratio is obtained for an inclusive job-machine structure in which the first job type can be processed on any of the m
machines while the second job type can be processed only on a subset of m/2 machines. Moreover, we show that our online algorithm is 1-competitive if the machines are not flexible, i.e., each machine
can process only a single job type.
Conference 28th European Conference on Operational Research
Country/Territory Poland
City Poznan,
Period 3/07/16 → 6/07/16
Dive into the research topics of 'Online scheduling of two job types on a set of multipurpose machines with unit processing times'. Together they form a unique fingerprint.
• 1 Organizing a conference, workshop, ...
• Karhi, S. (Participation - Conference participant)
3 Jul 2016 → 6 Jul 2016
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Crude oil pip calculator
Calculate margin values and profit and loss to plan your trades accordingly. 100oz Palladium, 1000oz Silver, 100 Barrel Oil. Number of Pips: Calculate. 0.00.
Divide the day's crude oil price by 42. One barrel of crude contains 42 gallons. This will tell you the dollar amount per gallon of refined gasoline attributed to crude. For example, if crude oil is
$100 per barrel, then about $2.38 of the price of a gallon of gas comes from the crude price. Pressure drop in oil pipes - viscosities ranging 100 - 600 Saybolt Universal Seconds Oil pipe pressure
drops in pounds per square inch (psi) per 100 ft pipe are indicated in the diagrams below. Oil viscosity is ranging 100 - 600 Saybolt Universal Seconds. NPS 1/2" Scientific Online Calculator . How to
calculate position size for oil? 1 lot of oil = 1 contract at HotForex. It doesn’t have units like currencies and metals. If I want to open an oil trade and want to risk $1000 and have stop loss of
200 pips how many lots should I buy or sell? I also don’t know how to open a position for indices. If I want to risk $1000 on CAC40 and have stop loss of 200 pips how many lots should I hi, just need
to check if my calculation for pip movement is correct: USOIL 82.00 (1 lot) (.01 / 82.00) x 100,000 = $12.20 per pip XAU_USD 1195.91 (1 lot) every lot of crude oil consists of 1000 barrelsthe price
quoted is per barrel.. for example, if you buy a lot of crude oil @82.50
Oil prices go up or down in “pips” (like forex), the abbreviation for “percentage in “WTI Crude Oil”, Western Texas Intermediate crude oil is easier to refine and
Oil Pipe Diameter A guide to the appropriate sizing of Hydraulic pipes can be carried out here. It should be noted that the purpose of the pipe must be known, is it Suction Pipe, Return Pipe or
Pressure Pipe? A guide to “normal” fluid speeds within these types of pipes are shown below. Fuel Pump & Pipe Sizing Calculator. This pump and pipe sizing tool is applicable for diesel generator and
boiler fuel systems. It is based on the use of positive displacement pumps that are sensitive to suction line conditions, so the first couple of screens ask for information regarding the layout of
the main tank(s) relative to the location of the pumps. Divide the day's crude oil price by 42. One barrel of crude contains 42 gallons. This will tell you the dollar amount per gallon of refined
gasoline attributed to crude. For example, if crude oil is $100 per barrel, then about $2.38 of the price of a gallon of gas comes from the crude price. Pressure drop in oil pipes - viscosities
ranging 100 - 600 Saybolt Universal Seconds Oil pipe pressure drops in pounds per square inch (psi) per 100 ft pipe are indicated in the diagrams below. Oil viscosity is ranging 100 - 600 Saybolt
Universal Seconds. NPS 1/2" Scientific Online Calculator . How to calculate position size for oil? 1 lot of oil = 1 contract at HotForex. It doesn’t have units like currencies and metals. If I want
to open an oil trade and want to risk $1000 and have stop loss of 200 pips how many lots should I buy or sell? I also don’t know how to open a position for indices. If I want to risk $1000 on CAC40
and have stop loss of 200 pips how many lots should I hi, just need to check if my calculation for pip movement is correct: USOIL 82.00 (1 lot) (.01 / 82.00) x 100,000 = $12.20 per pip XAU_USD
1195.91 (1 lot) every lot of crude oil consists of 1000 barrelsthe price quoted is per barrel.. for example, if you buy a lot of crude oil @82.50
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Oil Pipe Diameter A guide to the appropriate sizing of Hydraulic pipes can be carried out here. It should be noted that the purpose of the pipe must be known, is it Suction Pipe, Return Pipe or
Pressure Pipe? A guide to “normal” fluid speeds within these types of pipes are shown below. Fuel Pump & Pipe Sizing Calculator. This pump and pipe sizing tool is applicable for diesel generator and
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In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can
be seen as a:
• Group with a partial function replacing the binary operation;
• Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory.^[1] A
groupoid where there is only one object is a usual group.
In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to
another, and form a dependent family of types, thus morphisms might be typed ${\displaystyle g:A\rightarrow B}$, ${\displaystyle h:B\rightarrow C}$, say. Composition is then a total function: ${\
displaystyle \circ :(B\rightarrow C)\rightarrow (A\rightarrow B)\rightarrow A\rightarrow C}$, so that ${\displaystyle h\circ g:A\rightarrow C}$.
Special cases include:
Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups.^[2]
A groupoid can be viewed as an algebraic structure consisting of a set with a binary partial function. Precisely, it is a non-empty set ${\displaystyle G}$ with a unary operation ${\displaystyle {}^
{-1}:G\to G,}$ and a partial function ${\displaystyle *:G\times G\rightharpoonup G}$ . Here * is not a binary operation because it is not necessarily defined for all pairs of elements of ${\
displaystyle G}$ . The precise conditions under which ${\displaystyle *}$ is defined are not articulated here and vary by situation.
The operations ${\displaystyle \ast }$ and ^−1 have the following axiomatic properties: For all ${\displaystyle a}$ , ${\displaystyle b}$ , and ${\displaystyle c}$ in ${\displaystyle G}$ ,
1. Associativity: If ${\displaystyle a*b}$ and ${\displaystyle b*c}$ are defined, then ${\displaystyle (a*b)*c}$ and ${\displaystyle a*(b*c)}$ are defined and are equal. Conversely, if one of ${\
displaystyle (a*b)*c}$ or ${\displaystyle a*(b*c)}$ is defined, then they are both defined (and they are equal to each other), and ${\displaystyle a*b}$ and ${\displaystyle b*c}$ are also
2. Inverse: ${\displaystyle a^{-1}*a}$ and ${\displaystyle a*{a^{-1}}}$ are always defined.
3. Identity: If ${\displaystyle a*b}$ is defined, then ${\displaystyle a*b*{b^{-1}}=a}$ , and ${\displaystyle {a^{-1}}*a*b=b}$ . (The previous two axioms already show that these expressions are
defined and unambiguous.)
Two easy and convenient properties follow from these axioms:
• ${\displaystyle (a^{-1})^{-1}=a}$ ,
• If ${\displaystyle a*b}$ is defined, then ${\displaystyle (a*b)^{-1}=b^{-1}*a^{-1}}$ .^[3]
Category theoretic
A groupoid is a small category in which every morphism is an isomorphism, i.e., invertible.^[1] More explicitly, a groupoid G is a set G[0] of objects with
• for each pair of objects x and y a (possibly empty) set G(x,y) of morphisms (or arrows) from x to y; we write f : x → y to indicate that f is an element of G(x,y);
• for every object x a designated element ${\displaystyle \mathrm {id} _{x}}$ of G(x,x);
• for each triple of objects x, y, and z a function ${\displaystyle \mathrm {comp} _{x,y,z}:G(y,z)\times G(x,y)\rightarrow G(x,z):(g,f)\mapsto gf}$ ;
• for each pair of objects x, y a function ${\displaystyle \mathrm {inv} :G(x,y)\rightarrow G(y,x):f\mapsto f^{-1}}$ satisfying, for any f : x → y, g : y → z, and h : z → w:
□ ${\displaystyle f\ \mathrm {id} _{x}=f}$ and ${\displaystyle \mathrm {id} _{y}\ f=f}$ ;
□ ${\displaystyle (hg)f=h(gf)}$ ;
□ ${\displaystyle ff^{-1}=\mathrm {id} _{y}}$ and ${\displaystyle f^{-1}f=\mathrm {id} _{x}}$ .
If f is an element of G(x,y) then x is called the source of f, written s(f), and y is called the target of f, written t(f).
A groupoid G is sometimes denoted as ${\displaystyle G_{1}\rightrightarrows G_{0}}$ , where ${\displaystyle G_{1}}$ is the set of all morphisms, and the two arrows ${\displaystyle G_{1}\to G_{0}}$
represent the source and the target.
More generally, one can consider a groupoid object in an arbitrary category admitting finite fiber products.
Comparing the definitions
The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets
of morphisms from x to y). Then ${\displaystyle \mathrm {comp} }$ and ${\displaystyle \mathrm {inv} }$ become partial operations on G, and ${\displaystyle \mathrm {inv} }$ will in fact be defined
everywhere. We define ∗ to be ${\displaystyle \mathrm {comp} }$ and ^−1 to be ${\displaystyle \mathrm {inv} }$ , which gives a groupoid in the algebraic sense. Explicit reference to G[0] (and hence
to ${\displaystyle \mathrm {id} }$ ) can be dropped.
Conversely, given a groupoid G in the algebraic sense, define an equivalence relation ${\displaystyle \sim }$ on its elements by ${\displaystyle a\sim b}$ iff a ∗ a^−1 = b ∗ b^−1. Let G[0] be the set
of equivalence classes of ${\displaystyle \sim }$ , i.e. ${\displaystyle G_{0}:=G/\!\!\sim }$ . Denote a ∗ a^−1 by ${\displaystyle 1_{x}}$ if ${\displaystyle a\in G}$ with ${\displaystyle x\in G_{0}}
$ .
Now define ${\displaystyle G(x,y)}$ as the set of all elements f such that ${\displaystyle 1_{x}*f*1_{y}}$ exists. Given ${\displaystyle f\in G(x,y)}$ and ${\displaystyle g\in G(y,z),}$ their
composite is defined as ${\displaystyle gf:=f*g\in G(x,z)}$ . To see that this is well defined, observe that since ${\displaystyle (1_{x}*f)*1_{y}}$ and ${\displaystyle 1_{y}*(g*1_{z})}$ exist, so
does ${\displaystyle (1_{x}*f*1_{y})*(g*1_{z})=f*g}$ . The identity morphism on x is then ${\displaystyle 1_{x}}$ , and the category-theoretic inverse of f is f^−1.
Sets in the definitions above may be replaced with classes, as is generally the case in category theory.
Vertex groups and orbits
Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G(x,x), where x is any object of G. It follows easily from the axioms above that these are
indeed groups, as every pair of elements is composable and inverses are in the same vertex group.
The orbit of a groupoid G at a point ${\displaystyle x\in X}$ is given by the set ${\displaystyle s(t^{-1}(x))\subseteq X}$ containing every point that can be joined to x by a morphism in G. If two
points ${\displaystyle x}$ and ${\displaystyle y}$ are in the same orbits, their vertex groups ${\displaystyle G(x)}$ and ${\displaystyle G(y)}$ are isomorphic: if ${\displaystyle f}$ is any morphism
from ${\displaystyle x}$ to ${\displaystyle y}$ , then the isomorphism is given by the mapping ${\displaystyle g\to fgf^{-1}}$ .
Orbits form a partition of the set X, and a groupoid is called transitive if it has only one orbit (equivalently, if it is connected as a category). In that case, all the vertex groups are isomorphic
(on the other hand, this is not a sufficient condition for transitivity; see the section below for counterexamples).
Subgroupoids and morphisms
A subgroupoid of ${\displaystyle G\rightrightarrows X}$ is a subcategory ${\displaystyle H\rightrightarrows Y}$ that is itself a groupoid. It is called wide or full if it is wide or full as a
subcategory, i.e., respectively, if ${\displaystyle X=Y}$ or ${\displaystyle G(x,y)=H(x,y)}$ for every ${\displaystyle x,y\in Y}$ .
A groupoid morphism is simply a functor between two (category-theoretic) groupoids.
Particular kinds of morphisms of groupoids are of interest. A morphism ${\displaystyle p:E\to B}$ of groupoids is called a fibration if for each object ${\displaystyle x}$ of ${\displaystyle E}$ and
each morphism ${\displaystyle b}$ of ${\displaystyle B}$ starting at ${\displaystyle p(x)}$ there is a morphism ${\displaystyle e}$ of ${\displaystyle E}$ starting at ${\displaystyle x}$ such that $
{\displaystyle p(e)=b}$ . A fibration is called a covering morphism or covering of groupoids if further such an ${\displaystyle e}$ is unique. The covering morphisms of groupoids are especially
useful because they can be used to model covering maps of spaces.^[4]
It is also true that the category of covering morphisms of a given groupoid ${\displaystyle B}$ is equivalent to the category of actions of the groupoid ${\displaystyle B}$ on sets.
Given a topological space ${\displaystyle X}$ , let ${\displaystyle G_{0}}$ be the set ${\displaystyle X}$ . The morphisms from the point ${\displaystyle p}$ to the point ${\displaystyle q}$ are
equivalence classes of continuous paths from ${\displaystyle p}$ to ${\displaystyle q}$ , with two paths being equivalent if they are homotopic. Two such morphisms are composed by first following the
first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of ${\displaystyle X}$ , denoted ${\
displaystyle \pi _{1}(X)}$ (or sometimes, ${\displaystyle \Pi _{1}(X)}$ ).^[5] The usual fundamental group ${\displaystyle \pi _{1}(X,x)}$ is then the vertex group for the point ${\displaystyle x}$ .
The orbits of the fundamental groupoid ${\displaystyle \pi _{1}(X)}$ are the path-connected components of ${\displaystyle X}$ . Accordingly, the fundamental groupoid of a path-connected space is
transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are equivalent as
categories (see the section below for the general theory).
An important extension of this idea is to consider the fundamental groupoid ${\displaystyle \pi _{1}(X,A)}$ where ${\displaystyle A\subset X}$ is a chosen set of "base points". Here ${\displaystyle \
pi _{1}(X,A)}$ is a (wide) subgroupoid of ${\displaystyle \pi _{1}(X)}$ , where one considers only paths whose endpoints belong to ${\displaystyle A}$ . The set ${\displaystyle A}$ may be chosen
according to the geometry of the situation at hand.
Equivalence relation
If ${\displaystyle X}$ is a setoid, i.e. a set with an equivalence relation ${\displaystyle \sim }$ , then a groupoid "representing" this equivalence relation can be formed as follows:
• The objects of the groupoid are the elements of ${\displaystyle X}$ ;
• For any two elements ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X}$ , there is a single morphism from ${\displaystyle x}$ to ${\displaystyle y}$ (denote by ${\displaystyle
(y,x)}$ ) if and only if ${\displaystyle x\sim y}$ ;
• The composition of ${\displaystyle (z,y)}$ and ${\displaystyle (y,x)}$ is ${\displaystyle (z,x)}$ .
The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples:
• If every element of ${\displaystyle X}$ is in relation with every other element of ${\displaystyle X}$ , we obtain the pair groupoid of ${\displaystyle X}$ , which has the entire ${\displaystyle
X\times X}$ as set of arrows, and which is transitive.
• If every element of ${\displaystyle X}$ is only in relation with itself, one obtains the unit groupoid, which has ${\displaystyle X}$ as set of arrows, ${\displaystyle s=t=id_{X}}$ , and which is
completely intransitive (every singleton ${\displaystyle \{x\}}$ is an orbit).
• If ${\displaystyle f:X_{0}\to Y}$ is a smooth surjective submersion of smooth manifolds, then ${\displaystyle X_{0}\times _{Y}X_{0}\subset X_{0}\times X_{0}}$ is an equivalence relation^[6] since
${\displaystyle Y}$ has a topology isomorphic to the quotient topology of ${\displaystyle X_{0}}$ under the surjective map of topological spaces. If we write, ${\displaystyle X_{1}=X_{0}\times _
{Y}X_{0}}$ then we get a groupoid
${\displaystyle X_{1}\rightrightarrows X_{0}}$
which is sometimes called the banal groupoid of a surjective submersion of smooth manifolds.
• If we relax the reflexivity requirement and consider partial equivalence relations, then it becomes possible to consider semidecidable notions of equivalence on computable realisers for sets.
This allows groupoids to be used as a computable approximation to set theory, called PER models. Considered as a category, PER models are a cartesian closed category with natural numbers object
and subobject classifier, giving rise to the effective topos introduced by Martin Hyland.
Čech groupoid
A Čech groupoid^[6]^p. 5 is a special kind of groupoid associated to an equivalence relation given by an open cover ${\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}$ of some manifold ${\
displaystyle X}$ . Its objects are given by the disjoint union
${\displaystyle {\mathcal {G}}_{0}=\coprod U_{i}}$ ,
and its arrows are the intersections
${\displaystyle {\mathcal {G}}_{1}=\coprod U_{ij}}$ .
The source and target maps are then given by the induced maps
{\displaystyle {\begin{aligned}s=\phi _{j}:U_{ij}\to U_{j}\\t=\phi _{i}:U_{ij}\to U_{i}\end{aligned}}}
and the inclusion map
${\displaystyle \varepsilon :U_{i}\to U_{ii}}$
giving the structure of a groupoid. In fact, this can be further extended by setting
${\displaystyle {\mathcal {G}}_{n}={\mathcal {G}}_{1}\times _{{\mathcal {G}}_{0}}\cdots \times _{{\mathcal {G}}_{0}}{\mathcal {G}}_{1}}$
as the ${\displaystyle n}$ -iterated fiber product where the ${\displaystyle {\mathcal {G}}_{n}}$ represents ${\displaystyle n}$ -tuples of composable arrows. The structure map of the fiber product
is implicitly the target map, since
${\displaystyle {\begin{matrix}U_{ijk}&\to &U_{ij}\\\downarrow &&\downarrow \\U_{ik}&\to &U_{i}\end{matrix}}}$
is a cartesian diagram where the maps to ${\displaystyle U_{i}}$ are the target maps. This construction can be seen as a model for some ∞-groupoids. Also, another artifact of this construction is
${\displaystyle [\sigma ]\in {\check {H}}^{k}({\mathcal {U}},{\underline {A}})}$
for some constant sheaf of abelian groups can be represented as a function
${\displaystyle \sigma :\coprod U_{i_{1}\cdots i_{k}}\to A}$
giving an explicit representation of cohomology classes.
Group action
If the group ${\displaystyle G}$ acts on the set ${\displaystyle X}$ , then we can form the action groupoid (or transformation groupoid) representing this group action as follows:
• The objects are the elements of ${\displaystyle X}$ ;
• For any two elements ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X}$ , the morphisms from ${\displaystyle x}$ to ${\displaystyle y}$ correspond to the elements ${\displaystyle
g}$ of ${\displaystyle G}$ such that ${\displaystyle gx=y}$ ;
• Composition of morphisms interprets the binary operation of ${\displaystyle G}$ .
More explicitly, the action groupoid is a small category with ${\displaystyle \mathrm {ob} (C)=X}$ and ${\displaystyle \mathrm {hom} (C)=G\times X}$ and with source and target maps ${\displaystyle s
(g,x)=x}$ and ${\displaystyle t(g,x)=gx}$ . It is often denoted ${\displaystyle G\ltimes X}$ (or ${\displaystyle X\rtimes G}$ for a right action). Multiplication (or composition) in the groupoid is
then ${\displaystyle (h,y)(g,x)=(hg,x)}$ which is defined provided ${\displaystyle y=gx}$ .
For ${\displaystyle x}$ in ${\displaystyle X}$ , the vertex group consists of those ${\displaystyle (g,x)}$ with ${\displaystyle gx=x}$ , which is just the isotropy subgroup at ${\displaystyle x}$
for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the orbit of the group action, and the groupoid is transitive if
and only if the group action is transitive.
Another way to describe ${\displaystyle G}$ -sets is the functor category ${\displaystyle [\mathrm {Gr} ,\mathrm {Set} ]}$ , where ${\displaystyle \mathrm {Gr} }$ is the groupoid (category) with one
element and isomorphic to the group ${\displaystyle G}$ . Indeed, every functor ${\displaystyle F}$ of this category defines a set ${\displaystyle X=F(\mathrm {Gr} )}$ and for every ${\displaystyle
g}$ in ${\displaystyle G}$ (i.e. for every morphism in ${\displaystyle \mathrm {Gr} }$ ) induces a bijection ${\displaystyle F_{g}}$ : ${\displaystyle X\to X}$ . The categorical structure of the
functor ${\displaystyle F}$ assures us that ${\displaystyle F}$ defines a ${\displaystyle G}$ -action on the set ${\displaystyle G}$ . The (unique) representable functor ${\displaystyle F}$ : ${\
displaystyle \mathrm {Gr} \to \mathrm {Set} }$ is the Cayley representation of ${\displaystyle G}$ . In fact, this functor is isomorphic to ${\displaystyle \mathrm {Hom} (\mathrm {Gr} ,-)}$ and so
sends ${\displaystyle \mathrm {ob} (\mathrm {Gr} )}$ to the set ${\displaystyle \mathrm {Hom} (\mathrm {Gr} ,\mathrm {Gr} )}$ which is by definition the "set" ${\displaystyle G}$ and the morphism ${\
displaystyle g}$ of ${\displaystyle \mathrm {Gr} }$ (i.e. the element ${\displaystyle g}$ of ${\displaystyle G}$ ) to the permutation ${\displaystyle F_{g}}$ of the set ${\displaystyle G}$ . We
deduce from the Yoneda embedding that the group ${\displaystyle G}$ is isomorphic to the group ${\displaystyle \{F_{g}\mid g\in G\}}$ , a subgroup of the group of permutations of ${\displaystyle G}$
Finite set
Consider the group action of ${\displaystyle \mathbb {Z} /2}$ on the finite set ${\displaystyle X=\{-2,-1,0,1,2\}}$ which takes each number to its negative, so ${\displaystyle -2\mapsto 2}$ and ${\
displaystyle 1\mapsto -1}$ . The quotient groupoid ${\displaystyle [X/G]}$ is the set of equivalence classes from this group action ${\displaystyle \{[0],[1],[2]\}}$ , and ${\displaystyle [0]}$ has a
group action of ${\displaystyle \mathbb {Z} /2}$ on it.
Quotient variety
Any finite group ${\displaystyle G}$ that maps to ${\displaystyle GL(n)}$ gives a group action on the affine space ${\displaystyle \mathbb {A} ^{n}}$ (since this is the group of automorphisms). Then,
a quotient groupoid can be of the form ${\displaystyle [\mathbb {A} ^{n}/G]}$ , which has one point with stabilizer ${\displaystyle G}$ at the origin. Examples like these form the basis for the
theory of orbifolds. Another commonly studied family of orbifolds are weighted projective spaces ${\displaystyle \mathbb {P} (n_{1},\ldots ,n_{k})}$ and subspaces of them, such as Calabi–Yau
Fiber product of groupoids
Given a diagram of groupoids with groupoid morphisms
{\displaystyle {\begin{aligned}&&X\\&&\downarrow \\Y&\rightarrow &Z\end{aligned}}}
where ${\displaystyle f:X\to Z}$ and ${\displaystyle g:Y\to Z}$ , we can form the groupoid ${\displaystyle X\times _{Z}Y}$ whose objects are triples ${\displaystyle (x,\phi ,y)}$ , where ${\
displaystyle x\in {\text{Ob}}(X)}$ , ${\displaystyle y\in {\text{Ob}}(Y)}$ , and ${\displaystyle \phi :f(x)\to g(y)}$ in ${\displaystyle Z}$ . Morphisms can be defined as a pair of morphisms ${\
displaystyle (\alpha ,\beta )}$ where ${\displaystyle \alpha :x\to x'}$ and ${\displaystyle \beta :y\to y'}$ such that for triples ${\displaystyle (x,\phi ,y),(x',\phi ',y')}$ , there is a
commutative diagram in ${\displaystyle Z}$ of ${\displaystyle f(\alpha ):f(x)\to f(x')}$ , ${\displaystyle g(\beta ):g(y)\to g(y')}$ and the ${\displaystyle \phi ,\phi '}$ .^[7]
Homological algebra
A two term complex
${\displaystyle C_{1}{\overset {d}{\rightarrow }}C_{0}}$
of objects in a concrete Abelian category can be used to form a groupoid. It has as objects the set ${\displaystyle C_{0}}$ and as arrows the set ${\displaystyle C_{1}\oplus C_{0}}$ ; the source
morphism is just the projection onto ${\displaystyle C_{0}}$ while the target morphism is the addition of projection onto ${\displaystyle C_{1}}$ composed with ${\displaystyle d}$ and projection onto
${\displaystyle C_{0}}$ . That is, given ${\displaystyle c_{1}+c_{0}\in C_{1}\oplus C_{0}}$ , we have
${\displaystyle t(c_{1}+c_{0})=d(c_{1})+c_{0}.}$
Of course, if the abelian category is the category of coherent sheaves on a scheme, then this construction can be used to form a presheaf of groupoids.
While puzzles such as the Rubik's Cube can be modeled using group theory (see Rubik's Cube group), certain puzzles are better modeled as groupoids.^[8]
The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed).^[9]^[10]^[11] This groupoid acts on configurations.
Mathieu groupoid
The Mathieu groupoid is a groupoid introduced by John Horton Conway acting on 13 points such that the elements fixing a point form a copy of the Mathieu group M[12].
Relation to groups
Group-like structures
Total Associative Identity Cancellation Commutative
Partial magma Unneeded Unneeded Unneeded Unneeded Unneeded
Semigroupoid Unneeded Required Unneeded Unneeded Unneeded
Small category Unneeded Required Required Unneeded Unneeded
Groupoid Unneeded Required Required Required Unneeded
Commutative Groupoid Unneeded Required Required Required Required
Magma Required Unneeded Unneeded Unneeded Unneeded
Commutative magma Required Unneeded Unneeded Unneeded Required
Quasigroup Required Unneeded Unneeded Required Unneeded
Commutative quasigroup Required Unneeded Unneeded Required Required
Unital magma Required Unneeded Required Unneeded Unneeded
Commutative unital magma Required Unneeded Required Unneeded Required
Loop Required Unneeded Required Required Unneeded
Commutative loop Required Unneeded Required Required Required
Semigroup Required Required Unneeded Unneeded Unneeded
Commutative semigroup Required Required Unneeded Unneeded Required
Associative quasigroup Required Required Unneeded Required Unneeded
Commutative-and-associative quasigroup Required Required Unneeded Required Required
Monoid Required Required Required Unneeded Unneeded
Commutative monoid Required Required Required Unneeded Required
Group Required Required Required Required Unneeded
Abelian group Required Required Required Required Required
If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group.^[12] Many concepts of group theory
generalize to groupoids, with the notion of functor replacing that of group homomorphism.
Every transitive/connected groupoid - that is, as explained above, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above) ${\
displaystyle (G,X)}$ . By transitivity, there will only be one orbit under the action.
Note that the isomorphism just mentioned is not unique, and there is no natural choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object ${\
displaystyle x_{0}}$ , a group isomorphism ${\displaystyle h}$ from ${\displaystyle G(x_{0})}$ to ${\displaystyle G}$ , and for each ${\displaystyle x}$ other than ${\displaystyle x_{0}}$ , a
morphism in ${\displaystyle G}$ from ${\displaystyle x_{0}}$ to ${\displaystyle x}$ .
If a groupoid is not transitive, then it is isomorphic to a disjoint union of groupoids of the above type, also called its connected components (possibly with different groups ${\displaystyle G}$ and
sets ${\displaystyle X}$ for each connected component).
In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to
a multiset of unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets ${\displaystyle X}$ , but only the groups ${\displaystyle G.}$ For
• The fundamental groupoid of ${\displaystyle X}$ is equivalent to the collection of the fundamental groups of each path-connected component of ${\displaystyle X}$ , but an isomorphism requires
specifying the set of points in each component;
• The set ${\displaystyle X}$ with the equivalence relation ${\displaystyle \sim }$ is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but an isomorphism
requires specifying what each equivalence class is:
• The set ${\displaystyle X}$ equipped with an action of the group ${\displaystyle G}$ is equivalent (as a groupoid) to one copy of ${\displaystyle G}$ for each orbit of the action, but an
isomorphism requires specifying what set each orbit is.
The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural. Thus when groupoids arise in terms of
other structures, as in the above examples, it can be helpful to maintain the entire groupoid. Otherwise, one must choose a way to view each ${\displaystyle G(x)}$ in terms of a single group, and
this choice can be arbitrary. In the example from topology, one would have to make a coherent choice of paths (or equivalence classes of paths) from each point ${\displaystyle p}$ to each point ${\
displaystyle q}$ in the same path-connected component.
As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification
of vector spaces with one endomorphism is nontrivial.
Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup ${\displaystyle H}$
of a group ${\displaystyle G}$ yields an action of ${\displaystyle G}$ on the set of cosets of ${\displaystyle H}$ in ${\displaystyle G}$ and hence a covering morphism ${\displaystyle p}$ from, say,
${\displaystyle K}$ to ${\displaystyle G}$ , where ${\displaystyle K}$ is a groupoid with vertex groups isomorphic to ${\displaystyle H}$ . In this way, presentations of the group ${\displaystyle G}$
can be "lifted" to presentations of the groupoid ${\displaystyle K}$ , and this is a useful way of obtaining information about presentations of the subgroup ${\displaystyle H}$ . For further
information, see the books by Higgins and by Brown in the References.
Category of groupoids
The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, and is denoted by Grpd.
The category Grpd is, like the category of small categories, Cartesian closed: for any groupoids ${\displaystyle H,K}$ we can construct a groupoid ${\displaystyle \operatorname {GPD} (H,K)}$ whose
objects are the morphisms ${\displaystyle H\to K}$ and whose arrows are the natural equivalences of morphisms. Thus if ${\displaystyle H,K}$ are just groups, then such arrows are the conjugacies of
morphisms. The main result is that for any groupoids ${\displaystyle G,H,K}$ there is a natural bijection
${\displaystyle \operatorname {Grpd} (G\times H,K)\cong \operatorname {Grpd} (G,\operatorname {GPD} (H,K)).}$
This result is of interest even if all the groupoids ${\displaystyle G,H,K}$ are just groups.
Another important property of Grpd is that it is both complete and cocomplete.
Relation to Cat
The inclusion ${\displaystyle i:\mathbf {Grpd} \to \mathbf {Cat} }$ has both a left and a right adjoint:
${\displaystyle \hom _{\mathbf {Grpd} }(C[C^{-1}],G)\cong \hom _{\mathbf {Cat} }(C,i(G))}$
${\displaystyle \hom _{\mathbf {Cat} }(i(G),C)\cong \hom _{\mathbf {Grpd} }(G,\mathrm {Core} (C))}$
Here, ${\displaystyle C[C^{-1}]}$ denotes the localization of a category that inverts every morphism, and ${\displaystyle \mathrm {Core} (C)}$ denotes the subcategory of all isomorphisms.
The nerve functor ${\displaystyle N:\mathbf {Grpd} \to \mathbf {sSet} }$ embeds Grpd as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always a Kan complex.
The nerve has a left adjoint
${\displaystyle \hom _{\mathbf {Grpd} }(\pi _{1}(X),G)\cong \hom _{\mathbf {sSet} }(X,N(G))}$
Here, ${\displaystyle \pi _{1}(X)}$ denotes the fundamental groupoid of the simplicial set X.
Groupoids in Grpd
There is an additional structure which can be derived from groupoids internal to the category of groupoids, double-groupoids.^[13]^[14] Because Grpd is a 2-category, these objects form a 2-category
instead of a 1-category since there is extra structure. Essentially, these are groupoids ${\displaystyle {\mathcal {G}}_{1},{\mathcal {G}}_{0}}$ with functors
${\displaystyle s,t:{\mathcal {G}}_{1}\to {\mathcal {G}}_{0}}$
and an embedding given by an identity functor
${\displaystyle i:{\mathcal {G}}_{0}\to {\mathcal {G}}_{1}}$
One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares
${\displaystyle {\begin{matrix}\bullet &\to &\bullet \\\downarrow &&\downarrow \\\bullet &\xrightarrow {a} &\bullet \end{matrix}}}$ and ${\displaystyle {\begin{matrix}\bullet &\xrightarrow {a} &\
bullet \\\downarrow &&\downarrow \\\bullet &\to &\bullet \end{matrix}}}$
with ${\displaystyle a}$ the same morphism, they can be vertically conjoined giving a diagram
${\displaystyle {\begin{matrix}\bullet &\to &\bullet \\\downarrow &&\downarrow \\\bullet &\xrightarrow {a} &\bullet \\\downarrow &&\downarrow \\\bullet &\to &\bullet \end{matrix}}}$
which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.
Groupoids with geometric structures
When studying geometrical objects, the arising groupoids often carry a topology, turning them into topological groupoids, or even some differentiable structure, turning them into Lie groupoids. These
last objects can be also studied in terms of their associated Lie algebroids, in analogy to the relation between Lie groups and Lie algebras.
Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, in Poisson geometry one has the notion of a symplectic groupoid, which
is a Lie groupoid endowed with a compatible symplectic form. Similarly, one can have groupoids with a compatible Riemannian metric, or complex structure, etc.
See also
1. ^ ^a ^b Dicks & Ventura (1996). The Group Fixed by a Family of Injective Endomorphisms of a Free Group. p. 6.
2. ^ "Brandt semi-group", Encyclopedia of Mathematics, EMS Press, 2001 [1994], ISBN 1-4020-0609-8
3. ^ Proof of first property: from 2. and 3. we obtain a^−1 = a^−1 * a * a^−1 and (a^−1)^−1 = (a^−1)^−1 * a^−1 * (a^−1)^−1. Substituting the first into the second and applying 3. two more times
yields (a^−1)^−1 = (a^−1)^−1 * a^−1 * a * a^−1 * (a^−1)^−1 = (a^−1)^−1 * a^−1 * a = a. ✓
Proof of second property: since a * b is defined, so is (a * b)^−1 * a * b. Therefore (a * b)^−1 * a * b * b^−1 = (a * b)^−1 * a is also defined. Moreover since a * b is defined, so is a * b * b^
−1 = a. Therefore a * b * b^−1 * a^−1 is also defined. From 3. we obtain (a * b)^−1 = (a * b)^−1 * a * a^−1 = (a * b)^−1 * a * b * b^−1 * a^−1 = b^−1 * a^−1. ✓
4. ^ J.P. May, A Concise Course in Algebraic Topology, 1999, The University of Chicago Press ISBN 0-226-51183-9 (see chapter 2)
5. ^ "fundamental groupoid in nLab". ncatlab.org. Retrieved 2017-09-17.
6. ^ ^a ^b Block, Jonathan; Daenzer, Calder (2009-01-09). "Mukai duality for gerbes with connection". arXiv:0803.1529 [math.QA].
7. ^ "Localization and Gromov-Witten Invariants" (PDF). p. 9. Archived (PDF) from the original on February 12, 2020.
8. ^ An Introduction to Groups, Groupoids and Their Representations: An Introduction; Alberto Ibort, Miguel A. Rodriguez; CRC Press, 2019.
9. ^ Jim Belk (2008) Puzzles, Groups, and Groupoids, The Everything Seminar
10. ^ The 15-puzzle groupoid (1) Archived 2015-12-25 at the Wayback Machine, Never Ending Books
11. ^ The 15-puzzle groupoid (2) Archived 2015-12-25 at the Wayback Machine, Never Ending Books
12. ^ Mapping a group to the corresponding groupoid with one object is sometimes called delooping, especially in the context of homotopy theory, see "delooping in nLab". ncatlab.org. Retrieved
13. ^ Cegarra, Antonio M.; Heredia, Benjamín A.; Remedios, Josué (2010-03-19). "Double groupoids and homotopy 2-types". arXiv:1003.3820 [math.AT].
14. ^ Ehresmann, Charles (1964). "Catégories et structures : extraits". Séminaire Ehresmann. Topologie et géométrie différentielle. 6: 1–31.
• Brandt, H (1927), "Über eine Verallgemeinerung des Gruppenbegriffes", Mathematische Annalen, 96 (1): 360–366, doi:10.1007/BF01209171, S2CID 119597988
• Brown, Ronald, 1987, "From groups to groupoids: a brief survey," Bull. London Math. Soc. 19: 113–34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic
forms. The downloadable version updates the many references.
• —, 2006. Topology and groupoids. Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
• —, Higher dimensional group theory. Explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications in homotopy theory and in group cohomology. Many
• Dicks, Warren; Ventura, Enric (1996), The group fixed by a family of injective endomorphisms of a free group, Mathematical Surveys and Monographs, vol. 195, AMS Bookstore, ISBN 978-0-8218-0564-0
• Dokuchaev, M.; Exel, R.; Piccione, P. (2000). "Partial Representations and Partial Group Algebras". Journal of Algebra. 226. Elsevier: 505–532. arXiv:math/9903129. doi:10.1006/jabr.1999.8204.
ISSN 0021-8693. S2CID 14622598.
• F. Borceux, G. Janelidze, 2001, Galois theories. Cambridge Univ. Press. Shows how generalisations of Galois theory lead to Galois groupoids.
• Cannas da Silva, A., and A. Weinstein, Geometric Models for Noncommutative Algebras. Especially Part VI.
• Golubitsky, M., Ian Stewart, 2006, "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. 43: 305–64
• "Groupoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Higgins, P. J., "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) 13 (1976) 145–149.
• Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an orbit space", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer,
Berlin (1982), 115–122.
• Higgins, P. J., 1971. Categories and groupoids. Van Nostrand Notes in Mathematics. Republished in Reprints in Theory and Applications of Categories, No. 7 (2005) pp. 1–195; freely downloadable.
Substantial introduction to category theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in
topology, e.g. fundamental groupoid.
• Mackenzie, K. C. H., 2005. General theory of Lie groupoids and Lie algebroids. Cambridge Univ. Press.
• Weinstein, Alan, "Groupoids: unifying internal and external symmetry — A tour through some examples." Also available in Postscript., Notices of the AMS, July 1996, pp. 744–752.
• Weinstein, Alan, "The Geometry of Momentum" (2002)
• R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". In Algebraic and geometric combinatorics, volume 423 of Contemp. Math., 305–324. Amer. Math. Soc.,
Providence, RI (2006)
• fundamental groupoid at the nLab
• core at the nLab | {"url":"https://www.knowpia.com/knowpedia/Groupoid","timestamp":"2024-11-14T07:45:24Z","content_type":"text/html","content_length":"535597","record_id":"<urn:uuid:801574b7-7e08-4cca-9740-1e8beaff030a>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00056.warc.gz"} |
ands to Inches
Hands to Inches Converter
β Switch toInches to Hands Converter
How to use this Hands to Inches Converter π €
Follow these steps to convert given length from the units of Hands to the units of Inches.
1. Enter the input Hands value in the text field.
2. The calculator converts the given Hands into Inches in realtime β using the conversion formula, and displays under the Inches label. You do not need to click any button. If the input changes,
Inches value is re-calculated, just like that.
3. You may copy the resulting Inches value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Hands to Inches?
The formula to convert given length from Hands to Inches is:
Length[(Inches)] = Length[(Hands)] × 4
Substitute the given value of length in hands, i.e., Length[(Hands)] in the above formula and simplify the right-hand side value. The resulting value is the length in inches, i.e., Length[(Inches)].
Calculation will be done after you enter a valid input.
Consider that a horse is measured to be 16 hands tall.
Convert this height from hands to Inches.
The length in hands is:
Length[(Hands)] = 16
The formula to convert length from hands to inches is:
Length[(Inches)] = Length[(Hands)] × 4
Substitute given weight Length[(Hands)] = 16 in the above formula.
Length[(Inches)] = 16 × 4
Length[(Inches)] = 64
Final Answer:
Therefore, 16 hand is equal to 64 in.
The length is 64 in, in inches.
Consider that a racehorse stands at 15.5 hands.
Convert this measurement from hands to Inches.
The length in hands is:
Length[(Hands)] = 15.5
The formula to convert length from hands to inches is:
Length[(Inches)] = Length[(Hands)] × 4
Substitute given weight Length[(Hands)] = 15.5 in the above formula.
Length[(Inches)] = 15.5 × 4
Length[(Inches)] = 62
Final Answer:
Therefore, 15.5 hand is equal to 62 in.
The length is 62 in, in inches.
Hands to Inches Conversion Table
The following table gives some of the most used conversions from Hands to Inches.
Hands (hand) Inches (in)
0 hand 0 in
1 hand 4 in
2 hand 8 in
3 hand 12 in
4 hand 16 in
5 hand 20 in
6 hand 24 in
7 hand 28 in
8 hand 32 in
9 hand 36 in
10 hand 40 in
20 hand 80 in
50 hand 200 in
100 hand 400 in
1000 hand 4000 in
10000 hand 40000 in
100000 hand 400000 in
A hand is a unit of length used primarily to measure the height of horses. One hand is equivalent to 4 inches or approximately 0.1016 meters.
The hand is defined as 4 inches, providing a standardized measurement for assessing horse height, ensuring consistency across various contexts and practices.
Hands are used in the equestrian industry to measure the height of horses, from the ground to the highest point of the withers. The unit offers a convenient and traditional method for expressing
horse height and remains in use in equestrian competitions and breed standards.
An inch (symbol: in) is a unit of length used mainly in the United States, the United Kingdom, and Canada. One inch is equal to 2.54 centimeters.
The inch has origins in ancient times, originally based on the width of a human thumb. Its current definition, established in 1959, is exactly 2.54 centimeters.
Inches are commonly used to measure smaller lengths and distances, such as screen sizes and fabric lengths. Despite the widespread adoption of the metric system, the inch remains in use in these
Frequently Asked Questions (FAQs)
1. What is the formula for converting Hands to Inches in Length?
The formula to convert Hands to Inches in Length is:
Hands * 4
2. Is this tool free or paid?
This Length conversion tool, which converts Hands to Inches, is completely free to use.
3. How do I convert Length from Hands to Inches?
To convert Length from Hands to Inches, you can use the following formula:
Hands * 4
For example, if you have a value in Hands, you substitute that value in place of Hands in the above formula, and solve the mathematical expression to get the equivalent value in Inches. | {"url":"https://convertonline.org/unit/?convert=hands-inches","timestamp":"2024-11-12T04:03:19Z","content_type":"text/html","content_length":"89476","record_id":"<urn:uuid:c00998f9-c0ea-480c-bf0c-dac733516c9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00610.warc.gz"} |