content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
In Fig.9.33, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that i) ar (BDE) =1/4 ar (ABC) ii) ar (BDE) = 1/2 ar (BAE) iii) ar (ABC) =
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
In Fig.9.33, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that
i) ar (BDE) =1/4 ar (ABC) ii) ar (BDE) = 1/2 ar (BAE)
iii) ar (ABC) = 2 ar (BEC) iv) ar (BFE) = ar (AFD)
v) ar (BFE) = 2 ar (FED) vi) ar (FED) = 1/8 ar (AFC)
[Hint : Join EC and AD. Show that BE || AC and DE || AB, etc.]
i) Let G and H be the mid-points of side AB and AC
Line segment GH is joining the mid-points and is parallel to the third side. Therefore, GH will be half of the length of BC (mid-point theorem).
∴ GH = 1/2 BC and GH || BD
∴ GH = BD = DC and GH || BD (D is the mid-point of BC)
• GD = HC = HA
• HD = AG = BG
Therefore, clearly, triangle ABC is divided into 4 equal equilateral triangles viz ΔBGD, ΔAGH, ΔDHC, and ΔGHD.
In other words,
ΔBGD = 1/4 ΔABC
Now consider ΔBDG and ΔBDE
BD = BD (Common base)
As both triangles are equilateral triangle, we can say BG = BE
DG = DE
ΔBDG ≅ ΔBDE [By SSS congruency]
The SSS rule states that: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
Thus, area (ΔBDG) = area (ΔBDE)
ar (ΔBDE) = 1/4 ar (ΔABC)
Hence proved.
ii) Join points A and D.
Area (ΔBDE) = Area (ΔAED) (Common base DE and DE || AB)
Area (ΔBDE) - Area (ΔFED) = Area (ΔAED) - Area (ΔFED)
Now Area (ΔBEF) = Area (ΔAFD) ... (1)
Area (ΔABD) = Area (ΔABF) + Area (ΔAFD)
Area (ΔABD) = Area (ΔABF) + Area (ΔBEF) [From equation (1)]
Area (ΔABD) = Area (ΔABE) ...(2)
AD is the median in ΔABC (D is the midpoint of BC)
ar (ΔABD) = 1/2 ar (ΔABC)
ar (ΔBDE) = 1/4 ar (ΔABC) [As proved earlier in (i)]
ar(ΔABD) = 2 ar (ΔBDE) ...(3)
From (2) and (3), we obtain,
2 ar (ΔBDE) = ar (ΔABE)
ar (BDE) = 1/2 ar (BAE)
iii) Join points C and E.
ar (ΔABE) = ar (ΔBEC) (Common base BE and BE || AC)
ar (ΔABF) + ar (ΔBEF) = ar (ΔBEC)
Using equation (1), we obtain,
ar (ΔABF) + ar (ΔAFD) = ar (ΔBEC)
ar (ΔABD) = ar (ΔBEC)
1/2 ar (ΔABC) = ar (ΔBEC)
ar (ΔABC) = 2 ar (ΔBEC)
iv) ΔBDE and ΔAED lie on the same base (DE) and between the parallels DE and AB.
ar (ΔBDE) = ar (ΔAED)
ar (ΔBDE) - ar (ΔFED) = ar (ΔAED) - ar (ΔFED) [Subtracting ar (ΔFED) on both the sides]
ar (ΔBFE) = ar (ΔAFD)
v) Let h be the height of vertex E, corresponding to the side BD in ΔBDE
Let H be the height of vertex A, corresponding to the side BC in ΔABC.
ar (ABC) = 1/2 × BC × H
ar (BDE) = 1/2 × 1/2(BC) × h
In (i), it was shown that ar (BDE) = 1/4 ar (ABC)
Therefore, H = 2h
ar (AFD) = 1/2 × FD × 2h
ar (FED) = 1/2 × (FD) × h
Hence, ar (AFD) = 2 × ar (FED)
In (iv), it was shown that ar (ΔBFE) = ar (ΔAFD).
Therefore, ar (ΔBFE) = ar (ΔAFD) = 2 ar (ΔFED)
Hence proved.
vi) ar (ΔAFC) = ar (ΔAFD) + ar (ΔADC)
= 2 ar (ΔFED) + 1/2 ar (ΔABC) [using (v)]
= 2 ar (ΔFED) + 1/2 [4 × ar (ΔBDE)] [Using result of part (i)]
= 2 ar (ΔFED) + 2 ar (ΔBDE)
= 2 ar (ΔFED) + 2 ar (ΔAED)
[ΔBDE and ΔAED are on the same base and between same parallels]
= 2 ar (ΔFED) + 2 [ar (ΔAFD) + ar (ΔFED)]
= 2 ar (ΔFED) + 2 ar (ΔAFD) + 2 ar (ΔFED)
= 4 ar (ΔFED) + 4 ar (ΔFED) [using result of (v)]
⇒ ar (ΔAFC) = 8 ar (ΔFED)
⇒ ar (ΔFED) = 1/8 ar (ΔAFC)
☛ Check: NCERT Solutions Class 9 Maths Chapter 9
Video Solution:
In Fig.9.33, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that i) ar (BDE) =1/4 ar (ABC) ii) ar (BDE) = 1/2 ar (BAE) iii) ar (ABC) = 2
ar (BEC) iv) ar (BFE) = ar (AFD) v) ar (BFE) = 2 ar (FED) vi) ar (FED) = 1/8 ar (AFC). [Hint : Join EC and AD. Show that BE || AC and DE || AB, etc.]
Maths NCERT Solutions Class 9 Chapter 9 Exercise 9.4 Question 5
If ABC and BDE are two equilateral triangles such that D is the mid-point of BC and AE intersects BC at F, then we can show that ar (BDE) = 1/4 ar (ABC), ar (BDE) = 1/2 ar (BAE), ar (ABC) = 2 ar
ar (BFE) = ar (AFD) , ar (BFE) = 2 ar (FED), and ar (FED) = 1/8 ar (AFC).
☛ Related Questions:
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/ncert-solutions/in-fig-9-33-abc-and-bde-are-two-equilateral-triangles-such-that-d-is-the-mid-point-of-bc-if-ae-intersects-bc-at-f-show-that-i-ar-bde-14-ar-abc-ii-ar-bde-12-ar-bae-iii-ar-abc/","timestamp":"2024-11-09T19:27:22Z","content_type":"text/html","content_length":"253893","record_id":"<urn:uuid:6c2775a8-7103-4ae1-b9db-a036ae2873a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00796.warc.gz"} |
Project Evaluation
Project Evaluation by Paola Cardenas Rodriguez
1. Free Cash Flow (FCF)
1.1. is the
1.1.1. amount by which a business's operating cash flow exceeds its working capital needs and expenditures on fixed assets (known as capital expenditures)
1.1.1.1. Then
1.1.1.1.1. the calculation of the FCF for the shareholder would be
1.1.2. indicator of a company's financial flexibility and is of interest to holders of the company's equity, debt, preferred stock and convertible securities
2. Payback Period (PRI)
2.1. what is
2.1.1. is the amount of time it takes to recover the cost of an investment. Simply put, it is the length of time an investment reaches a breakeven point.
2.1.1.1. You can figure out the payback period
2.1.1.1.1. Payback Period= (Avarage Annual)/(Cash FlowCost of Investment)
2.1.2. This metric is useful before making any decisions, especially when an investor needs to make a snap judgment about an investment venture.
3. Shareholder Cash Flow (FCA)
3.1. corresponds with
3.1.1. The cash that the company has generated in the year for its participants, understood as both the shareholder and the creditor
3.1.1.1. It is forged under the approach
3.1.1.1.1. that the company has two owners, who are the providers of funds, to whom the company must pay
3.1.1.1.2. represent
3.2. Calculate
3.2.1. to the holders of debt and shares is determined as the sum of the Cash Flow for the shareholder (CFa) and the Cash Flow of the debt (CFd). CCF = FCa + FCd
3.2.1.1. If each component
3.2.1.1.1. replaced by its formulation (seen above) a faster form of calculation is obtained.b
4. Net Present Value (NPV)
4.1. What is
4.1.1. NPV is the difference between the present value of cash inflows and the present value of cash outflows over a period.
4.1.1.1. NPV Important
4.1.1.1.1. o Helps in making investment decisions. o Compares different investment options. o Incorporates the time value of money. o Considers the risk associated with investments.
4.2. It measures
4.2.1. the profitability of an investment.
4.2.2. indicates that the investment is expected to generate more money than it costs.
4.3. Calculate NPV
4.3.1. Formula: NPV = ∑(Ct / (1+r)^t) - C0 o Ct = net cash inflow during the period t o r = discount rate o t = time period o C0 = initial investment
4.4. Decision Making Based on NPV
4.4.1. Positive NPV: Accept the project. Negative NPV: Reject the project. NPV = 0: The project is breakeven.
4.5. Limitations of NPV
4.5.1. o Assumes a constant discount rate. o Does not consider qualitative factors. o May not be accurate for long-term projects.
5. Benefit Cost Ratio (BCR)
5.1. What is BCR?
5.1.1. o is a measure that compares the benefits of a project to its costs. o It is calculated by dividing the present value of benefits by the present value of costs. greater than BCR greater than 1
indicates that the benefits exceed the costs, and the project is profitable.
5.2. Important
5.2.1. o Helps make informed investment decisions. o Allows for comparison of different investment alternatives. o Facilitates communication of financial analysis results. o Is a useful tool for
evaluating both public and private projects.
5.3. Calculate
5.3.1. o Formula: BCR = Present Value of Benefits / Present Value of Costs o Present value is calculated by discounting future cash flows at a discount rate
5.4. Interpreting
5.4.1. o BCR > 1: Benefits exceed costs, the project is profitable. o BCR = 1: Benefits equal costs, the project is marginal. o BCR < 1: Costs exceed benefits, the project is not profitable.
5.5. Limitations
5.5.1. o Does not consider all aspects of a project (e.g., social, environmental impacts). o The choice of discount rate can significantly affect the result. o Quantifying benefits can be difficult
in some cases.
6. Internal Rate of Return (IRR)
6.1. Definition
6.1.1. The discount rate that makes the net present value (NPV) of an investment project equal to zero.
6.2. Objective
6.2.1. Measures the profitability of a project without considering the effect of inflation.
6.3. Calculating
6.3.1. • Trial and error method: Different discount rates are tested until the one that makes NPV zero is found. • Formula: NPV = ∑(Cash Flow / (1 + IRR)^t) - Initial Investment = 0
6.4. Interpreting
6.4.1. • IRR > Required Rate of Return: The project is profitable. • IRR < Required Rate of Return: The project is not profitable. • IRR = Required Rate of Return: The project is indifferent.
6.5. Advantages
6.5.1. • Considers the time value of money. • Is a relative measure of profitability. • Is easy to understand and compare.
6.6. Disadvantages
6.6.1. • Can have multiple solutions. • Not suitable for projects with irregular cash flows. • Can be sensitive to changes in assumptions
6.7. Applications
6.7.1. • Evaluation of investment projects. • Comparison of investment alternatives. • Sensitivity analysis. | {"url":"https://www.mindmeister.com/3473172455/project-evaluation","timestamp":"2024-11-05T02:29:55Z","content_type":"application/xhtml+xml","content_length":"56488","record_id":"<urn:uuid:926302ea-5510-442e-a369-b331a944e3ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00608.warc.gz"} |
How to Insert the Sum Of Two Tables In Oracle?
To insert the sum of two tables in Oracle, you can use a SQL query that combines the data from the two tables using a UNION clause, and then calculates the sum using the SUM function in a subquery.
For example, you can write a query like this:
INSERT INTO new_table (sum_column) SELECT SUM(total_column) FROM ( SELECT total_column FROM table1 UNION ALL SELECT total_column FROM table2 ) as combined_table;
In this query, "table1" and "table2" are the two tables that you want to sum the values from. The SUM function calculates the total sum of the "total_column" from both tables, and then inserts this
sum into a new table called "new_table" in the column "sum_column".
Make sure to adjust the table names, column names, and table structure according to your specific requirements when using this query.
How to optimize the SQL query for inserting the sum of two tables in oracle?
To optimize the SQL query for inserting the sum of two tables in Oracle, you can follow these steps:
1. Use the UNION ALL operator to combine the results of the two tables into a single result set. This will allow you to sum the values from both tables in a single query.
2. Use the SUM() function to calculate the total sum of the values from the combined result set.
3. Use a single INSERT statement to insert the total sum into a new table or update an existing table.
Here is an example of how you can optimize the SQL query for inserting the sum of two tables in Oracle:
1 INSERT INTO total_sum_table (total_sum)
2 SELECT SUM(value_column) AS total_sum
3 FROM (
4 SELECT value_column FROM table1
5 UNION ALL
6 SELECT value_column FROM table2
7 ) combined_tables;
In this example, table1 and table2 are the two tables from which you want to calculate the total sum of the values in the value_column. The total_sum_table is the table where you want to insert the
total sum.
By using the UNION ALL operator, you can combine the results of the two tables into a single result set and then use the SUM() function to calculate the total sum of the values. Finally, the total
sum is inserted into the total_sum_table.
This approach optimizes the query by reducing the number of queries and operations needed to calculate and insert the total sum of the two tables in Oracle.
How to insert the sum of two tables in oracle with different data types?
To insert the sum of two tables with different data types in Oracle, you can use a combination of SQL queries to perform the sum operation and then insert the result into a new table. Here are the
general steps you can follow:
1. Create a new table to store the sum of the two tables. Make sure the data types of the columns in this new table accommodate the sum of the values from the two tables.
1 CREATE TABLE sum_table (
2 id NUMBER,
3 total_amount NUMBER
4 );
1. Use a SELECT statement to retrieve the sum of the values from the two tables, ensuring that any necessary data type conversions are performed.
1 INSERT INTO sum_table (id, total_amount)
2 SELECT
3 t1.id,
4 t1.amount + t2.amount
5 FROM table1 t1
6 JOIN table2 t2 ON t1.id = t2.id;
In this example, assume that table1 has the column "id" and "amount" with numeric data types, and table2 also has the column "id" and "amount" with numeric data types. The SELECT statement joins the
two tables on the "id" column and calculates the sum of the "amount" columns, inserting the result into the sum_table.
1. Execute the INSERT INTO statement to perform the operation and insert the sum of the values from the two tables into the new sum_table.
After completing these steps, you should have successfully inserted the sum of two tables with different data types into a new table in Oracle.
How to insert the sum of two tables in oracle?
To insert the sum of two tables in Oracle, you can use a SQL query with a subquery to calculate the sum and then insert it into a new table or update an existing table. Here is an example of how you
can insert the sum of two tables in Oracle:
1. Create a new table to store the sum:
1 CREATE TABLE total_sum (
2 total NUMBER
3 );
1. Use a SQL query to calculate the sum of two tables and insert it into the new table:
1 INSERT INTO total_sum (total)
2 SELECT SUM(column_name)
3 FROM table1
4 UNION ALL
5 SELECT SUM(column_name)
6 FROM table2;
In this example, replace column_name with the specific column from both table1 and table2 that you want to sum.
1. Once you run the above SQL query, the sum of the two tables will be calculated and inserted into the total_sum table. You can then query the total_sum table to retrieve the sum of the two tables.
Remember to adjust the SQL query as per your specific table names, column names, and requirements.
What is the recommended approach for merging data from two tables in oracle in a production environment?
In a production environment, the recommended approach for merging data from two tables in Oracle is to use the "MERGE" statement. The MERGE statement allows you to merge data from one table to
another based on a specified condition, for example matching on a unique key. This statement combines the insert, update, and delete operations into a single statement, making it more efficient and
easier to maintain.
Here is an example of how to use the MERGE statement to merge data from two tables:
1 MERGE INTO target_table t
2 USING source_table s
3 ON (t.id = s.id)
4 WHEN MATCHED THEN
5 UPDATE SET t.column1 = s.column1, t.column2 = s.column2
6 WHEN NOT MATCHED THEN
7 INSERT (id, column1, column2)
8 VALUES (s.id, s.column1, s.column2);
In this example, the MERGE statement is merging data from the source_table into the target_table based on the matching id column. It updates the columns in the target table if a match is found, and
inserts a new row in the target table if there is no match.
Overall, using the MERGE statement is the recommended approach for merging data from two tables in Oracle in a production environment as it provides a concise and efficient way to manage data | {"url":"https://myblogfrog.zoho.to/blog/how-to-insert-the-sum-of-two-tables-in-oracle","timestamp":"2024-11-10T19:16:04Z","content_type":"text/html","content_length":"168860","record_id":"<urn:uuid:fb22187e-f2e4-4109-bf5f-1937e2c0019e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00557.warc.gz"} |
graph theory problems computer science
Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. should be modified to "Graph theory: Solution to '3 utilities problem'
could lead to better computers." "[12] In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. ϕ {\displaystyle \{x,x\}=\{x\}} Algorithms and graph
theory: The major role of graph theory in computer applications is the development of graph algorithms. ( In one more general sense of the term allowing multiple edges,[3][4] a graph is an ordered
triple {\displaystyle x} These algorithms are used to solve the graph theoretical concepts which intern used to solve the corresponding computer science application problems. "Graph theory: Solution
to '3 utilities problem' could lead to better computers." , y In one restricted but very common sense of the term,[1][2] a graph is an ordered pair "While reading our research article, we suddenly
realized that the solution was before our eyes. . The size of a graph is List structures are often preferred for sparse graphs as they have smaller memory requirements. , Under the umbrella of social
networks are many different types of graphs. Let G be a simple directed graph on n nodes.. ( y which is not in Ascertaining relationships among classes (e.g. is called the inverted edge of {\
displaystyle E} Some specific decomposition problems that have been studied include: Many problems involve characterizing the members of various classes of graphs. "We had nearly given up on getting
the last piece and solving the riddle. Specifically, for each edge A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas.[32]. E y
Graph theory is both an area of mathematics and an important tool in computer science. ( Mannheim: Bibliographisches Institut 1969. ) The edges of an undirected simple graph permitting loops y { E
Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem. So to allow loops the definitions must be expanded. . ( y Theoretically one can distinguish between list
and matrix structures but in concrete applications the best structure is often a combination of both. } A GRAPH is a very simple construction used to model things that can be described as objects and
the connections between them. ) Graph Theory has become an important discipline in its own right because of its applications to Computer Science, Communication Networks, and Combinatorial
optimization through the design of efficient algorithms. x The original set cover problem, also called hitting set, can be described as a vertex cover in a hypergraph. , } ∣ . x A veritable brain
teaser. = Two mathematicians from the University of Copenhagen's Department of Computer Science and DTU have now solved a problem that the world's quickest and most clever have been struggling with
since the 1980's. ( In general graphs theory has a wide range of applications in diverse fields. {\displaystyle (x,x)} , ) G , its endpoints A similar approach can be taken to problems in social
media,[9] travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases,[10][11] and many other fields. G Traditionally, syntax and compositional semantics follow
tree-based structures, whose expressive power lies in the principle of compositionality, modeled in a hierarchical graph. Researchers from the University of Copenhagen and the Technical University of
Denmark (DTU) thought that they were five years away from solving a math riddle from the 1980's. { and on , its number of edges. ϕ A subdivision or homeomorphism of a graph is any graph obtained by
subdividing some (or no) edges. y Jacob Holm has been interested in the mathematical conundrum since 1998, but the answer was only revealed while the two researchers were reading through their
already submitted research article. , For undirected multigraphs, the definition of ) {\displaystyle x} E are said to be adjacent to one another, which is denoted { {\displaystyle y} y [30][31] The
proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. Graphs can be used to model many types of relations and
processes in physical, biological, social and information systems. | {"url":"http://tns.com.pe/wiki/page.php?id=graph-theory-problems-computer-science-2b95e5","timestamp":"2024-11-12T15:28:21Z","content_type":"text/html","content_length":"13143","record_id":"<urn:uuid:f27ffac5-9995-4f95-aaa3-938344a2de2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00427.warc.gz"} |
“Less May be More with Math Curriculum”
The books are distributed by an Oregon-based company known as SingaporeMath.com, which counts a private school in Madison as the first of its growing number of clients.
The biggest difference between math instruction in Singapore – a city-state with a population of about 4.4 million – and the United States is a simple premise: Less is more.
Students in Singapore are introduced to roughly half the number of new math topics a year as students in the United States are. Experts and policy analysts say Singapore’s emphasis on depth
over breadth is a formula for success.
The thicker the textbooks and the greater the volume of math topics introduced a year, the less likely American students and teachers are to achieve similar results, says Alan Ginsburg,
director of the policy and program studies service at the U.S. Department of Education.
More on the Connected Math / Singapore Math textbook photos.
Madison Country Day School was the first US school to purchase Singapore Math textbooks, in 1997, according to this article. | {"url":"https://www.schoolinfosystem.org/2006/02/25/less_may_be_mor/","timestamp":"2024-11-07T19:07:25Z","content_type":"text/html","content_length":"6906","record_id":"<urn:uuid:3ff1bd10-8d56-4915-86b2-6be10ccab0f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00380.warc.gz"} |
Advanced Mathematical Concepts: Key Questions and Answers
At the master's level, mathematics delves into sophisticated concepts and methods. In this blog, we explore two advanced mathematical questions and provide detailed answers. For additional support
with such topics, visit mathsassignmenthelp.com for expert assistance. If you're struggling with functional analysis, their solve My Functional Analysis Assignment service can be a valuable resource.
Question 1: Understanding Compactness in Functional Analysis
Describe the concept of compactness in functional analysis. How does compactness of an operator impact the properties of a function space?
In functional analysis, compactness refers to a property of operators and subsets of function spaces that generalizes the concept of compact sets in finite-dimensional spaces. An operator is called
compact if it maps bounded sets to relatively compact sets, meaning the image of any bounded set under the operator has a compact closure.
Compactness has significant implications for function spaces:
• Bounded Operators: A compact operator on a Banach space maps bounded sets to relatively compact sets, which implies that any bounded sequence has a convergent subsequence in the image.
• Spectrum: The spectrum of a compact operator consists of zero and possibly a set of eigenvalues with finite multiplicity. This is different from general operators, where the spectrum can be much
more complex.
• Hilbert Spaces: In Hilbert spaces, compact operators can be understood in terms of eigenvalues and their associated eigenfunctions, which simplifies many problems in analysis and partial
differential equations.
For a deeper understanding of compactness and its implications, or to get help with complex problems in functional analysis, mathsassignmenthelp.com offers expert assistance. Their Solve My
Functional Analysis Assignment service is tailored to help with advanced topics in this area.
Question 2: Eigenfunction Expansion in Fourier Series
Explain the concept of eigenfunction expansion in the context of Fourier series. How does this expansion help in solving differential equations?
Eigenfunction expansion is a method used to express functions as a series of eigenfunctions of a differential operator. In the context of Fourier series, this involves representing a function as an
infinite sum of sines and cosines, which are eigenfunctions of the second-order differential operator with periodic boundary conditions.
• Fourier Series Expansion: For a given function that is periodic, it can be expanded into a series of sine and cosine functions. Each term in this series is an eigenfunction of the Laplace
operator with periodic boundary conditions, and the coefficients are determined by projecting the original function onto these eigenfunctions.
• Solving Differential Equations: This expansion is particularly useful in solving partial differential equations, such as the heat equation or wave equation. By expressing the solution as a series
of eigenfunctions, the problem can be reduced to solving simpler ordinary differential equations for the coefficients, which are often easier to handle.
Eigenfunction expansion allows for the decomposition of complex problems into simpler components, making it easier to find solutions and analyze the behavior of the system.
For more detailed explanations or assistance with complex problems in functional analysis and related areas, visit mathsassignmenthelp.com. Their Solve My Functional Analysis Assignment service
provides expert help to tackle challenging mathematical issues effectively. | {"url":"https://xn--wo-6ja.com/read-blog/32729_advanced-mathematical-concepts-key-questions-and-answers.html","timestamp":"2024-11-14T08:26:04Z","content_type":"text/html","content_length":"82829","record_id":"<urn:uuid:94317873-e6d3-4dd7-bbdf-8bcab57af773>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00724.warc.gz"} |
Graph Archives - Computing Learner
The graph data structure has many applications. This post will teach you how to use it to solve a real-world problem. To solve the problem I’m showing you here, you can use the implementation for the
undirected simple graph data structure. Graph ADT operations When you want to use a data structure to solve problems, […]
Using the Graph Data Structure to solve real-world problems in C# Read More »
Undirected Simple Graph Data Structure Implementation in C#
The Graph Data Structure is widely used in programming. In this post, you will learn one of the implementations of the TDA and an example of how to use it. To use a graph, you first have to
understand when and why you can use it. As in other topics, you will find in this
Undirected Simple Graph Data Structure Implementation in C# Read More » | {"url":"https://computinglearner.com/category/data-structures/graph/","timestamp":"2024-11-10T20:34:19Z","content_type":"text/html","content_length":"126356","record_id":"<urn:uuid:a949b25d-200c-43a5-9522-c0b461bff536>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00839.warc.gz"} |
Future value of growing annuity excel
Because of the general definition of annuity, an Annuity Calculator might calculate the future value of a savings investment plan (as many online annuity
16 Jul 2019 The Excel future value of a growing annuity calculator, available for download below, is used to compute the future value by entering details 3 Dec 2019 What is Principal? Debt Service
Coverage Ratio (DSCR) Excel Template · Capital Asset Pricing Model (CAPM) Excel Template · Debt Ratio To get the present value of an annuity, you can use the PV function. In the example shown, the
formula in C7 is: 13 Nov 2014 The basic annuity formula in Excel for present value is =PV(RATE,NPER,PMT). Let's break it down: • RATE is the discount rate or interest rate,
9 Dec 2007 This value is referred to as the future value (FV) of an annuity. In plain In practice the FV of an annuity equation is used to calculate the accumulated growth of a series of In Excel the
RATE function is used for this purpose.
FVM Future Value of an annuity allowing for different periodicity of payments per PVEGPerAnn Present Value of an Exponentially Growing PERIODIC Annuity Understanding the calculation of present value
can help you set your retirement so you choose to invest money into an annuity that will make payments each month When using a Microsoft Excel spreadsheet you can use a PV formula to do the Coin in
the bottle and plant growing with savings money put on the wood. 31 Dec 2019 P = The future value of the annuity stream to be paid in the future This value is the amount that a stream of future
payments will grow to, assuming that a certain amount of compounded Excel Formulas and Functions 11 Apr 2010 Present value calculations are the reverse of compound growth $308.39. See
econ422PresentValueProblems.xls for Excel calculations The cash flow for a finite growing annuity pays an amount C, starting next period
9 Dec 2007 This value is referred to as the future value (FV) of an annuity. In plain In practice the FV of an annuity equation is used to calculate the accumulated growth of a series of In Excel the
RATE function is used for this purpose.
To get the present value of an annuity, you can use the PV function. In the example shown, the formula in C7 is: 13 Nov 2014 The basic annuity formula in Excel for present value is =PV
(RATE,NPER,PMT). Let's break it down: • RATE is the discount rate or interest rate,
demonstrates how to value graduated (growing) annuities in Microsoft Excel. calculate the present value or future value of a growing stream of cash flows.
31 Dec 2019 P = The future value of the annuity stream to be paid in the future This value is the amount that a stream of future payments will grow to, assuming that a certain amount of compounded
Excel Formulas and Functions 11 Apr 2010 Present value calculations are the reverse of compound growth $308.39. See econ422PresentValueProblems.xls for Excel calculations The cash flow for a finite
growing annuity pays an amount C, starting next period Since interest rates enable peoples’ money to grow, investors know that In this exampl e, we will use a 6% discount rate to calculate
the present value of FV Excel function to determine the future value of an annuity : =FV(rate,nper C cash flow. FV n future value on date n. PV present value; annuity spreadsheet notation for Why
does the future value of an investment grow faster in later years as shown in. Figure 4.1 ? CALCULATING PRESENT VALUES IN EXCEL .
Understanding the calculation of present value can help you set your retirement so you choose to invest money into an annuity that will make payments each month When using a Microsoft Excel
spreadsheet you can use a PV formula to do the Coin in the bottle and plant growing with savings money put on the wood.
The formula for the future value of a growing annuity is used to calculate the future amount of a series of cash flows, or payments, that grow at a proportionate 16 Jul 2019 The Excel future value
of a growing annuity calculator, available for download below, is used to compute the future value by entering details 3 Dec 2019 What is Principal? Debt Service Coverage Ratio (DSCR) Excel Template
· Capital Asset Pricing Model (CAPM) Excel Template · Debt Ratio To get the present value of an annuity, you can use the PV function. In the example shown, the formula in C7 is: 13 Nov 2014 The
basic annuity formula in Excel for present value is =PV(RATE,NPER,PMT). Let's break it down: • RATE is the discount rate or interest rate, 10 Apr 2019 A growing annuity is a finite stream of equal
cash flows that occur after equal interval of time and grow at a constant rate. It is also called an
Over time, cash flow patterns tend to grow. The following not so well-known formulas will quickly furnish the future value or present value of such growing annuities 20 Mar 2013 Calculate the
present value of a level perpetuity and a growing perpetuity.3. The Future Value of an OrdinaryAnnuity • FVn = FV of annuity at the end of nth Using an Excel Spreadsheet • n = NPER(rate, pmt, pv, fv)
• n | {"url":"https://binaryoptionstxiaq.netlify.app/maryott79972dinu/future-value-of-growing-annuity-excel-125.html","timestamp":"2024-11-12T00:54:51Z","content_type":"text/html","content_length":"33452","record_id":"<urn:uuid:4dc32511-d1b9-480d-950c-7cb353f53388>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00148.warc.gz"} |
how will the knowledge of factoring help in your career?
how will the knowledge of factoring help in your career? Related topics: maths quiz questions and answers printable for 5th 6th standards
how to solve math application questions
math 1710 precalculus i
algebra free worksheet
ti 84 online trial
prentice hall math oklahoma pre algerbra answers
exponents grade 10
beginner trigonometry worksheet middle school
Author Message
Ornee25 Posted: Monday 10th of May 09:03
I have a difficulty with my math that requires urgent solution. The problem is with how will the knowledge of factoring help in your career? . I have been on the look out for somebody
who can prepare me without delay as my exam is fast approaching . But it's difficult to find somebody fast enough besides it being expensive . Can somebody direct me? It will be a huge
Back to top
Vofj Timidrov Posted: Monday 10th of May 15:30
What exactly don't you understand about how will the knowledge of factoring help in your career? ? I remember having problem with the same thing in Intermediate algebra, so I might be
able to give you some suggestions on how to approach such problems. However if you want help with algebra on a long term basis, then you should check out Algebrator, that's what I did
in my College Algebra, and I have to say it's so cool! It's less costly than a private teacher and you can use it anytime you want . It's not hard to use it, even if you never ever had
a similar software . I would advise you to get it as soon as you can and forget about getting a algebra teacher. You won't regret it!
From: Bulgaria
Back to top
Vnode Posted: Tuesday 11th of May 10:30
I agree. Algebrator not only gets your homework done faster, it actually improves your understanding of the subject by providing very useful tips on how to solve similar questions. It
is a very popular online help tool among students so you should try it out.
From: Germany
Back to top
susdnguru® Posted: Thursday 13th of May 08:26
Friends , Thanks a lot for the replies that you have given . I just had a look at the Algebrator available at: https://softmath.com/ordering-algebra.html. The best part that I liked
was the money back guarantee that they are offering there. I went ahead and bought Algebrator. It is really user friendly and turns up to be a remarkable tool for College Algebra.
Over the
Back to top
cufBlui Posted: Saturday 15th of May 07:40
Don’t worry my friend . As what I said, it shows the solution for the problem so you won’t really have to copy the answer only but it makes you understand how did the software came up
with the answer. Just go to this page https://softmath.com/ordering-algebra.html and prepare to learn and solve faster .
From: Scotland
Back to top
TihBoasten Posted: Saturday 15th of May 13:59
mixed numbers, difference of squares and trigonometric functions were a nightmare for me until I found Algebrator, which is really the best math program that I have come across. I have
used it frequently through several algebra classes – Pre Algebra, Algebra 2 and Remedial Algebra. Simply typing in the algebra problem and clicking on Solve, Algebrator generates
step-by-step solution to the problem, and my algebra homework would be ready. I highly recommend the program.
Back to top | {"url":"https://www.softmath.com/algebra-software-3/how-will-the-knowledge-of.html","timestamp":"2024-11-14T07:01:57Z","content_type":"text/html","content_length":"43372","record_id":"<urn:uuid:a4ed6a5a-d6b4-4fb0-ba89-e63447985c93>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00755.warc.gz"} |
Why does Mathematica insist that a non-convergent series has limit?
4867 Views
2 Replies
1 Total Likes
Why does Mathematica insist that a non-convergent series has limit?
Now, I don't claim to be a Mathematics-overlord (I'm still a freshman ;P ), but I really don't understand the output Mathematica produces. I really think it is simply false. No I don't think I found
a bug; I'm pretty positive this is a design choice; but I really don't understand it.
Here we go:
In[1]:= Clear[q, k]
In[2]:= Sum[q^k, {k, Infinity}]
Out[2]= -(q/(-1 + q))
In[3]:= Reduce[
ForAll[q, Element[q, Reals],
Sum[q^k, {k, Infinity}] == -q/(-1 + q)], q, Reals]
Out[3]= True
In[4]:= SumConvergence[q^k, k]
Out[4]= Abs[q] < 1
I'm sorry, but while I can understand the output of the Sum function alone, all of this simply turns false if put under the universal quantifier. For some values of q the series is not convergent. It
is especially weird for q=1, as we have division by zero then. Hence it is not true that for all real values of q this sum equals -q/(-1+q).
Could somebody be kind enough and explain this to me? Am I failing to understand something, or am I doing something wrong?
Thanks in advance, Marcin G.
2 Replies
If the conditions for which the sum converges are needed, then the GenerateConditions option is handy:
Sum[q^k, {k, Infinity}, GenerateConditions -> True]
resulting in the output
ConditionalExpression[-(q/(-1 + q)), Abs[q] < 1]
It's slightly subtle. To get a sense of why your In[3] returns True you need to see how the expression is evaluated. The short answer is that the expression inside of it,
Sum[q^k, {k, Infinity}] == -q/(-1 + q)
is evaluated before the other things are evaluated. So it first is turned into
-q/(-1 + q) == -q/(-1 + q)
and then the rest of your quantifiers are subsequently being applied to the simple expression which is just the symbol
which of course is always True. To see this unfold make use of TracePrint to see some the evaluations that Mathematica is doing
ForAll[q, Element[q, Reals],
Sum[q^k, {k, Infinity}] == -q/(-1 + q)] // TracePrint
which gives (it displays in a notebook differently but I have changed it to InputForm for readability in this forum):
HoldForm[ForAll[q, Element[q, Reals], Sum[q^k, {k, Infinity}] == -q/(-1 + q)]]
HoldForm[ForAll[q, Element[q, Reals], True]]
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/399943?sortMsg=Recent","timestamp":"2024-11-07T06:34:59Z","content_type":"text/html","content_length":"100808","record_id":"<urn:uuid:2ee79ba0-ba5e-4cbb-a650-d5dc2cba9ea0>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00150.warc.gz"} |
Average trees and maximum clade credibility trees
I have posted a couple of times on so-called average trees. This is the practice of identifying the phylogeny with the minimum sum of squares distances to the trees in a set. This is something I got
into primarily because I was interested in summarizing distributions of trees from comparative methods such as stochastic character mapping.
Though I stumbled into it by accident, I'm discovering that there is a literature on average trees - although in phylogenetic biology, at least, it doesn't seem to be particularly large (compared to
that on the different task of computing consensus trees). One significant contribution is an article dating to 1997 by Lapointe & Cucumel entitled “The Average Consensus Procedure: Combination of
Weighted Trees Containing Identical or Overlapping Sets of Taxa.” This paper describes a procedure that I independently discovered (20 years later) and have implemented in the phytools function
According to this consensus method, the 1 through n for n trees patristic distance matrices, D[i] are computed and averaged to produce D, then a tree & branch lengths are identified in which the sum
of squares difference between D and the patristic distance matrix of the consensus tree, D[consensus] is minimized.
Here is a quick demo using the sample of really bad primate trees from before:
## Best Q = 0.0052509716725781
## Solution found after 1 set of nearest neighbor interchanges.
Now, average trees have been criticized - not least because (evidently) they are not guaranteed to have the Pareto property, meaning that it is not guaranteed that every bipartition present in the
consensus can be found in at least one of the input trees!
However, in this simple case I thought it might be interesting to see how our average tree by this method compared to, say, a maximum clade credibility tree from the same input set. Note that this is
not a maximum clade credibility from a Bayesian posterior distribution - however these trees do come from a probability distribution with expectation equal to the true tree and broad variance, so
this could be equivalent to a sample from the posterior distribution under some circumstances, such as when there is a lot of uncertainty regarding phylogenetic relationships in the data.
We can compute the maximum clade credibility tree using phangorn - however to do this one will have to install the latest phangorn version from GitHub.
Topologically this tree is a pretty good tree, but it is not the correct tree.
(To see how it is wrong, we can do this:
## Rotating nodes to optimize matching...
## Done.
Cool, right?)
Something that we usually cannot do is compute the clade credibility of a tree not in the set that we have drawn from our (posterior) probability distribution. However, here we have a tree who's
credibility is of interest, and so we can compute its clade credibility* simply by tacking it on to the end of our "multiPhylo" object. (*Note that this is not precisely the clade credibility because
we have included the clades from our reference tree in the calculation of the clade frequencies - but it is pretty close.)
xlab="log(clade credibility)")
"clade credibility of \"average\" tree",adj=c(0,0.3),srt=90)
So that means the average tree has a higher probability (credibility) than any tree in the set. Interesting.
One could reasonably point out that if we'd merely sampled more trees we would have surely sampled the tree with highest credibility, and then we would have found it to have the same credibility as
our LS consensus tree. That's probably true in this case - but, remember, the number of trees for many taxa is vast and so it may not be the case - depending on the shape of our posterior density -
that we sample the highest probability topology in our MCMC. (I'm not saying that we don't - just that it cannot be assumed to be guaranteed.)
5 comments:
1. I'm following this series of posts with great interest, thanks for sharing! I assume you are already familiar with this paper, but anyway since it is on my list on favorites:
They talk about Bayesian estimators that minimize the risk under several distances.
1. Thanks Leonardo!
2. This comment has been removed by the author.
3. Hi Liam. I used this function to calculate an MCCT but the tree I got does not have support values from the branches. Any ideas to fix this? Gracias!!
4. This is very informative and fascinating stuff. Thanks!
Note: due to the very large amount of spam, all comments are now automatically submitted for moderation. | {"url":"http://blog.phytools.org/2016/04/average-trees-and-maximum-clade.html","timestamp":"2024-11-12T10:58:16Z","content_type":"text/html","content_length":"185444","record_id":"<urn:uuid:1e7d16c8-488a-4d6c-a121-1834f8a3f3b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00774.warc.gz"} |
What Is 3.25 As A Fraction?
What Is 3.25 As A Fraction?
The decimal 3.25 would be written a couple of different ways as a fraction:
• 3.25 = 3 1/4 as a mixed number
• 3.25 = 13/4 as an improper fraction
Place Value
When you’re working with decimals, it’s important to remember place value. Each of the places to the right of the decimal point represent amounts that are less than one. The first place is the
tenths, then hundredths, thousandths, and ten-thousandths.
Leave a Comment | {"url":"https://thestudyish.com/what-is-3-25-as-a-fraction/","timestamp":"2024-11-13T06:27:58Z","content_type":"text/html","content_length":"52327","record_id":"<urn:uuid:99bc6a11-3fee-48ec-ac3b-5f6cb9942dfb>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00134.warc.gz"} |
Distance from Biloela to Boggabri
Distance between Biloela and Boggabri
The distance from Biloela to Boggabri is 835 kilometers by road. Road takes approximately 17 hours and 16 minutes and goes through Taroom, Wandoan, Miles, Goondiwindi, Boggabilla, Moree and Narrabri.
Shortest distance by air 703 km ✈️
Car route length 835 km 🚗
Driving time 17 h 16 min
Fuel amount 66.8 L
Fuel cost 108.9 AUD
Compare this route in other services:
Point Distance Time Fuel
Biloela 0 km 00 min 0.0 L
60 45 km, 35 min
Banana 45 km 35 min 3.3 L
60 A5 141 km, 2 h 41 min
Taroom 186 km 3 h 16 min 12.9 L
A5 64 km, 1 h 55 min
Wandoan 250 km 5 h 12 min 17.8 L
A5 80 km, 2 h 10 min
Miles 330 km 7 h 22 min 25.2 L
A5 213 km, 5 h 00 min
Goondiwindi 543 km 12 h 22 min 42.8 L
A39 38 km, 35 min
Boggabilla 581 km 12 h 58 min 44.5 L
A39 73 km, 1 h 11 min
Moree 654 km 14 h 10 min 48.9 L
A39 136 km, 2 h 33 min
Narrabri 789 km 16 h 43 min 61.1 L
B51 46 km, 33 min
Boggabri 835 km 17 h 16 min 66.7 L
Hotels of Boggabri
Frequently Asked Questions
How much does it cost to drive from Biloela to Boggabri?
Fuel cost: 108.9 AUD
This fuel cost is calculated as: (Route length 835 km / 100 km) * (Fuel consumption 8 L/100 km) * (Fuel price 1.63 AUD / L)
You can adjust fuel consumption and fuel price here.
How long is a car ride from Biloela to Boggabri?
Driving time: 17 h 16 min
This time is calculated for driving at the maximum permitted speed, taking into account traffic rules restrictions.
• 475 km with a maximum speed 90 km/h = 5 h 16 min
• 149 km with a maximum speed 80 km/h = 1 h 51 min
• 8 km with a maximum speed 60 km/h = 7 min
• 5 km with a maximum speed 50 km/h = 6 min
• 1 km with a maximum speed 40 km/h = 1 min
• 198 km with a maximum speed 20 km/h = 9 h 52 min
The calculated driving time does not take into account intermediate stops and traffic jams.
How far is Biloela to Boggabri by land?
The distance between Biloela and Boggabri is 835 km by road.
Precise satellite coordinates of highways were used for this calculation. The start and finish points are the centers of Biloela and Boggabri respectively.
How far is Biloela to Boggabri by plane?
The shortest distance (air line, as the crow flies) between Biloela and Boggabri is 703 km.
This distance is calculated using the Haversine formula as a great-circle distance between two points on the surface of a sphere. The start and finish points are the centers of Biloela and Boggabri
respectively. Actual distance between airports may be different.
How many hours is Biloela from Boggabri by plane?
Boeing 737 airliner needs 52 min to cover the distance of 703 km at a cruising speed of 800 km/h.
Small plane "Cessna 172" needs 3 h 11 min to flight this distance at average speed of 220 km/h.
This time is approximate and do not take into account takeoff and landing times, airport location and other real world factors.
How long is a helicopter ride from Biloela to Boggabri?
Fast helicopter "Eurocopter AS350" or "Hughes OH-6 Cayuse" need 2 h 55 min to cover the distance of 703 km at a cruising speed of 240 km/h.
Popular "Robinson R44" needs 3 h 20 min to flight this distance at average speed of 210 km/h.
This time is approximate and do not take into account takeoff and landing times, aerodrome location and other real world factors.
What city is halfway between Biloela and Boggabri?
The halfway point between Biloela and Boggabri is Moonie. It is located about 38 km from the exact midpoint by road.
The distance from Moonie to Biloela is 456 km and driving will take about 10 h 21 min. The road between Moonie and Boggabri has length 380 km and will take approximately 6 h 55 min.
The other cities located close to halfway point:
• Condamine is in 376 km from Biloela and 459 km from Boggabri
• Miles is in 330 km from Biloela and 505 km from Boggabri
• Goondiwindi is in 543 km from Biloela and 293 km from Boggabri
Where is Biloela in relation to Boggabri?
Biloela is located 703 km north of Boggabri.
Biloela has geographic coordinates: latitude -24.39977, longitude 150.51419.
Boggabri has geographic coordinates: latitude -30.70454, longitude 150.04435.
Which highway goes from Biloela to Boggabri?
The route from Biloela to Boggabri follows A5, A39.
Other minor sections pass along the road:
• 60: 45 km
• B51: 36 km
• A2: 1 km
The distance between Biloela and Boggabri is ranked
in the ranking popularity. | {"url":"https://au.drivebestway.com/distance/biloela/boggabri/","timestamp":"2024-11-04T08:46:50Z","content_type":"text/html","content_length":"113149","record_id":"<urn:uuid:45af2636-ced0-4137-a3f1-71fd0b82a6ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00822.warc.gz"} |
[Solved] Find the equation of the straight line passing through... | Filo
Find the equation of the straight line passing through and perpendicular to the line
Not the question you're searching for?
+ Ask your question
Let be the slope of the line whose equation is to be found out which is perpendicular to the line .
The slope of the given line is -1 .
Therefore, using the condition of perpendicularity of lines, we have or
Hence, the required equation of the line is or
Was this solution helpful?
Video solutions (3)
Learn from their 1-to-1 discussion with Filo tutors.
3 mins
Uploaded on: 9/23/2023
Was this solution helpful?
6 mins
Uploaded on: 7/19/2023
Was this solution helpful?
Found 5 tutors discussing this question
Discuss this question LIVE for FREE
12 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions from Mathematics Exemplar (NCERT Exemplar)
Practice questions from Mathematics Exemplar (NCERT Exemplar)
View more
Practice more questions from Straight Lines
Practice questions on similar concepts asked by Filo students
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text Find the equation of the straight line passing through and perpendicular to the line
Updated On Oct 26, 2023
Topic Straight Lines
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 3
Upvotes 384
Avg. Video Duration 4 min | {"url":"https://askfilo.com/math-question-answers/find-the-equation-of-the-straight-line-passing-through-12-and-perpendicular-to","timestamp":"2024-11-03T04:09:48Z","content_type":"text/html","content_length":"456815","record_id":"<urn:uuid:3b02cfc2-95c8-4426-ab35-5578b6c5f57e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00046.warc.gz"} |
Elbow GreaseThe Gibbons Style Guide for writing assignmentsGetting Help with Proofs at the QSR Center and the Writing CenterA day of work in honor of MLKjr's legacyBook Club!Wrapping up the remote semester - a scratchpad of thoughtsMentoring Undergraduate ResearchWorking at odd hoursHamilton's Spring 2020 Grading PoliciesWorkflows: "live" Office HoursAdaptationsDoing a bad job and being okay with itExponential Growth - activity for kiddosRemote Semester OrientationWomp womp!Principles in conflict?Learning curvesChallenges and silver liningsSetting up a home workspaceClass PreferencesNothing ever happens on MarsLaTeX in Blogger? Q.E.Done! \(\square\)And Now for Something Completely Different!How to use Office HoursHow to Study for ExamsA Golden (Formal) Power Series
tag:blogger.com,1999:blog-8551680375687404762024-09-01T05:49:35.101-04:00Thoughts, confessions, ideas, and advice from a working mathematician.Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-855168037568740476.post-75898478157967423492021-11-01T16:26:00.008-04:002021-11-01T16:26:43.915-04:00<p>If you've taken
proofs classes from different professors, you know that we share some basic beliefs about writing, but we also have our own quirks and preferences. (If you haven't yet taken proofs classes from
different professors... get ready! We share some basic beliefs, but we also have our own quirks and preferences). What you'll see below are the guidelines that I consider the standards for submitting
to the Gibbons Journal of Mathematics (by which I mean, sending in a writing assignment to be assessed in a proofs class that you're taking with me).<br /><br />To me, mathematical writing breaks
into three (entangled) pieces: Logic, Mathematics, and Style.</p><p><span style="font-family: CMBX12; font-size: 17pt;">Logic.</span></p> <div class="page" title="Page 1"> <div class="layoutArea">
<div class="column"> <p><span style="font-family: 'CMR12'; font-size: 12.000000pt;">The authors </span></p> <p></p><ul style="text-align: left;"><li><span style="font-size: 12pt;"><span style=
"font-family: CMSY10;">s</span></span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">tate claims (propositions, theorems, lemmas, etc) as logical statements;<br /> </span></li><li><span
style="font-family: 'CMR12'; font-size: 12.000000pt;">state claims appropriate to the assigned problem;</span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">write proofs with
first and last sentences that indicate the proof technique they are using;<br /></span></li><ul><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">this criterion relaxes a bit after the
first proofs course; if you're proving a theorem directly, you can just get to the proof without repeating the hypotheses at the beginning of the proof. But if you're in your first proofs course,
please include the first and last sentences that lay out your hypotheses and conclusions!</span></li></ul><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">define notation and symbols
before using them;<br /> </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">employ a valid proof technique for the statement as written;<br /> </span></li><li><span style=
"font-family: 'CMR12'; font-size: 12.000000pt;">employ a valid proof technique for the statement appropriate to the problem;<br /> </span></li><li><span style="font-family: 'CMR12'; font-size:
12.000000pt;">include all necessary cases in the proof;<br /> </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">write examples with a clear topic sentence;<br /> </span></
li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">write prose using evidence to support claims;<br /> </span></li><li><span style="font-size: 12pt;"><span style="font-family: CMSY10;
">e</span></span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">mploy sound logic even in mathematical prose. </span></li></ul><p></p> <p><span style="font-family: 'CMBX12'; font-size:
17.000000pt;">Mathematics.</span></p> <p><span style="font-family: 'CMR12'; font-size: 12.000000pt;">The authors </span></p> <p></p><ul style="text-align: left;"><li><span style="font-family:
'CMR12'; font-size: 12.000000pt;">use definitions, theorems, and other results from the appropriate chapter(s); </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">cite
definitions, theorems, and other results from the appropriate chapter(s); </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">cite named definitions, theorems, and
results;<br /> </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">employ correct calculations; </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">do
not re-prove prior results. </span></li></ul><p></p> <p><span style="font-family: 'CMBX12'; font-size: 17.000000pt;">Style.</span></p> <p><span style="font-family: 'CMR12'; font-size:
12.000000pt;">The authors</span></p><p></p><ul style="text-align: left;"><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">use the appropriate LaTeX template for the course;</span></li>
<li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">adhere to mathematical style conventions, such as: </span></li><ul><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">
using math mode for math symbols, </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">writing sentences that do not start in math mode, </span></li><li><span style=
"font-family: 'CMR12'; font-size: 12.000000pt;"><b>not</b> writing “two-column” proofs (more here: </span><span style="color: rgb(100.000000%, 0.000000%, 100.000000%); font-family: 'CMR12';
font-size: 12.000000pt;">Paragraph Proofs</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">), </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">using
aligned equations when a string of equations is too long for a page,</span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">using inline equations for minor mathematical
manipulations;</span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">using appropriate LaTeX environments (</span><span style="font-family: 'CMTT12'; font-size: 12.000000pt;">
proof</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">, </span><span style="font-family: 'CMTT12'; font-size: 12.000000pt;">prop*</span><span style="font-family: 'CMR12'; font-size:
12.000000pt;">, example*, etc);</span></li></ul><li><span style="font-family: CMR12;">discerning when to write an example instead of a proposition and proof;</span></li><li><span style="font-family:
'CMR12'; font-size: 12.000000pt;">adhere to the conventions of grammar;<br /> </span></li><li><span style="font-family: 'CMR12'; font-size: 12.000000pt;">write clearly and concisely in a way that
does not obscure the logic or mathematics. </span></li></ul><div><span style="font-family: CMR12;">Here's a note about the difference between a proposition in math (versus formal logic). </
span><span style="font-family: CMR12;">In math, we understand propositions (and theorems, lemmas, corollaries, etc) to be statements that are <b>true</b> (and proved/provable) so that we can
cite them when we need to use them. You can see the confusion that would ensue if a paper contained "Proposition 1: All even integers are prime" under that assumption, even if followed immediately by
a counterexample! So, if you are showing that a statement is not true as part of a writing assignment, here's what that might look like:<br /><br /><b>Example.</b> It is not the case that every
even integer is prime. Notice that the integer 8 is even. Although 8 divides 24 = (2)(</span>12)<span style="font-family: CMR12;">, it divides neither 2 nor 12. Therefore, by (definition of prime/
proposition about prime numbers/whatever it is), 8 is not prime.</span></div><div><span style="font-family: CMR12;"><br /></span></div><div><span style="font-family: CMR12;">To break that down into
more of a template:<br /><br /></span></div><div><span style="font-family: CMR12;"><b>Example.</b> [Topic sentence: what's the point of this example?] [Supporting mathematical evidence, less
formal than a proof, but still following the stylistic conventions of good mathematical writing!] [Concluding sentence.]</span></div><p></p> </div> </div> </div> Courtney R Gibbonshttp://
For help with the math:</b></p><p>The tutors at the QSR are great, and they will happily help you think through the math that goes into a proof. In order to make sure that you are getting the best
possible math help, here are some expectations about the kind of help you'll get at the QSR when it comes to your proofs assignments. We (the math faculty) ask the tutors to help you with the math,
<i>but not the writing</i>, part of the proofs.</p><p>Here are some guidelines for working on proofs at the QSR:</p><p></p><ul style="text-align: left;"><li>No working out the math in Overleaf at the
QSR! (It's your job to typeset the math nicely after you have figured it out.)</li><li>As a corollary: you may only work things out using on paper (or whiteboard, or otherwise "by hand") with a
tutor at the QSR. Why? Usually, thinking through a problem and understanding what's happening is necessary before you can write it up, and when you write it up, it may look a lot different than what
you worked out on paper first.</li><li>Figuring out what properties you are using in a proof is a math problem; figuring out what order to write equations or how to break up the math and exposition
is a writing challenge. Make sure you are asking about math, not writing.</li></ul><p></p><p>As long as you don't abuse the goodwill of the QSR tutors and director, you can also ask simple
LaTeX questions if you can't find an answer on the internet or elsewhere. (Many of the tutors are also LaTeX gurus, and they can help troubleshoot your code, but don't forget that Google is a great
debugger, too!)<br /><br /><b>Conversely... for help with the writing:</b></p><p>If you've figured out the math but you're having trouble writing it up, make an appointment to talk to a Writing
Center peer counselor! There are usually a couple tutors at the Writing Center that have been recommended by math faculty, and we ask that they help you with the writing, <i>but not the math</i>,
part of the proofs.<br /><br /><b>The Upshot:</b> You can use both centers for an assignment to get help with the math and the writing! But, you can't use one center for both.</p>Courtney R
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-53895741369376391322021-01-19T22:17:00.003-05:002021-01-19T22:18:00.613-05:00<p>You know I’m an academic
because my first instinct was block out space for an abstract for this blog post.</p> <p><strong>Abstract:</strong> On MLK day, I, a white mathematics professor, consciously and intentionally
dedicated myself to working <em>only</em> in honor of Martin Luther King Jr.'s legacy. That is, all of my professional activities were focused on social equity, especially along the axes of race,
ethnicity, and socioeconomic status. I’m writing this post for accountability, transparency, invitations for collaboration on any of these ideas, and for constructive feedback for those feeling
generous. The content? A break down of my day, starting at 9am.</p> <span></span><span><a name='more'></a></span><h3 id="active-bystander-workshop-9am">Active Bystander Workshop (9am)</h3> <p>Here
are some highlights from this workshop (run by <a href="https://twitter.com/Rhodes_Perry">Rhodes Perry</a> and focused on interventions on a college campus).</p> <ul> <li> <p>What an active bystander
is.</p> </li> <li> <p>Why we don’t intervene even when we know we should: we don’t know how (so we have to practice!), we’re afraid of repurcussions (so we have to understand the difference between
being uncomfortable and being unsafe), we’re afraid of escalating the situation (so we have to learn to embrace mistakes, because we’re definitely going to make some mistakes).</p> </li> <li> <p>A 4
Step Approach to being an Active Bystander (“microinterventions”)</p> <ol> <li>Assess your safety (<strong>comfort is not safety</strong>)</li> <li>Make the invisible visible (raise consciousness of
the commenter)</li> <li>Educate the person commenting (what matters is the impact of their comment, not their intention – decenter their point of view)</li> <li>Have patience and expect progress
(redirect and educate about harm)</li> </ol> </li> <li> <p>What if I mess up?</p> <ul> <li>Embrace mistakes and don’t quit</li> <li>Keep trying to build stamina and moral courage</li> <li>If you’re
exhausted, you’re probably on the right track</li> <li>Super important to model this on campus where students are watching!</li> <li>Lean into the “active” part of being an active bystander</li> </
ul> </li> </ul> <p>Further actions for me following this workshop:</p> <ul> <li class="task-list-item"><input checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Fill out
personal reflection worksheet</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Set up a place/time for colleagues to practice bystander
intervention together. <ul> <li class="task-list-item"><input checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Send some emails to follow up with interested parties</li>
<li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Zoom with colleague who is interested in coordinating this effort to fill in the details <ul> <li
class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Determine a platform</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled=""
type="checkbox" /> Develop some starter scenarios (“Your professional society just announced they will award a [sic] <em>Fellowship for a Black Mathematician</em> to the dismay of potential
recipients imagining putting that on their CV…”)</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Develop process for sharing intervention ideas</
li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Develop process for giving feedback on intervention ideas</li> <li class="task-list-item"><input
class="task-list-item-checkbox" disabled="" type="checkbox" /> Develop accountability system</li> </ul> </li> </ul> </li> </ul> <h3 id="syllabus-review-modern-algebra-10am">Syllabus Review, Modern
Algebra (10am)</h3> <p>I initially intended to grab my notes from a webinar on cultivating a liberatory classroom, but I ended up working with<a href="https://www.unco.edu/nhs/
stem-inclusive-excellence-collective/pdf/syllabus-review-protocol.pdf">Syllabus Review for Equity-Minded Practice</a> (h/t @wrinkle_nancy on the bird app for this awesome resource!), put out by <a
href="https://twitter.com/Center4UrbanEd">The Center for Urban Education in the School of Education at USC Rossier</a>. (I’m not abandoning the original plan, just triaging it for now.)</p> <p>From
the guide, here are a few introductory ideas.</p> <blockquote> <p><strong>What is syllabus review?</strong><br /> Syllabus review is an inquiry tool for promoting racial/ethnic equity and
equity-minded practice. To achieve this goal, the syllabus review process promotes faculty inquiry into teaching approaches and practices, especially how they affect African American, Latinx, Native
American, Pacific Islander, and other racially/ ethnically minoritized students; facilitates a self-assessment of these teaching approaches and practices from a racial/ethnic equity lens; and allows
faculty to consider changes that result in more equitable teaching approaches and practice.</p> <p><strong>What is in the guide?</strong><br /> The Syllabus Review Guide is comprised of six parts
that provide the conceptual knowledge and practical know-how to conduct equity-minded self-reflection on an essential document in academic life: the syllabus. Throughout the Guide are examples that
illustrate the ideas motivating syllabus review, as well opportunities to practice inquiry and to reflect on how to change your syllabi—and your teaching more generally—so are more equity-minded.</p>
</blockquote> <p>What I learned from this resource so far:</p> <ul> <li>You can write stuff in your syllabus specifically to support and encourage students. It doesn’t have to be a boring contract;
it can communicate a lot more about the class, including the environment (eg, “joyful exploration”), the support systems, your underlying assumptions (everything from “If you’re enrolled in this
class, I assume you have taken these classes and feel comfortable with these topics. If any of those are a little hazy, let’s talk to solidify your foundation.” to things like <a href="http://
math.sfsu.edu/federico/">Federico Ardila’s axioms</a>).</li> <li>This kind of syllabus review seeks to make the hidden curricula of college visible to students. It’s about transparency as much as
it’s about what’s going to be covered and how grades are going to be calculated.</li> <li>There’s a lot of stuff in this guide that I think my syllabi already accomplish (or at least, that I have
made an intentional effort to accomplish: welcoming students, describing the support structures, describing my role as a partner in their learning).</li> <li><em>BUT!</em> I have not intentionally
grappled with “affirm[ing] the belonging of racially/ethnically minoritized students in higher education by representing their experiences in the course materials and by deconstructing the
presentation of white students and white experiences as the norm.” This is where my focus is going to be as I revamp my syllabus this time around.</li> </ul> <p>Further actions:</p> <ul> <li class=
"task-list-item"><input checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Complete the first worksheet (“Do I Know My Syllabus?”)</li> <li class="task-list-item"><input
checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Preliminary run through my syllabus to target areas that need work (before reading further); jot down ideas with the big
<em>BUT!</em> in mind (this is kind of like a pre-assessment to see where I’m currently at in my equity-mindedness)</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled=""
type="checkbox" /> Read on critically: “grade” my self-assessment as I work through the rest of the resource and, of course, improve upon my efforts. <ul> <li class="task-list-item"><input class=
"task-list-item-checkbox" disabled="" type="checkbox" /> Related: Finalize standards for SBG</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" />
Revamp final paper assignment yet again</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Work out the blog/vlog/alog (audio log, is that a thing)
structure for contributing to the course in a way that doesn’t add make-work, supports student learning, and doesn’t require writing necessarily</li> </ul> </li> <li class="task-list-item"><input
checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Call in some pals who will be receptive to meeting to work on this stuff</li> <li class="task-list-item"><input checked=
"true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Schedule the zoom for the pals</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox"
/> Don’t forget the original plan: dig out notes from “the liberatory classroom” webinar and put them into place!</li> </ul> <p>Final thoughts – I admit that I skimmed the whole resource even though
I really intended to give myself an honest self-assessment. Let me share my excitement about the section that helps you deconstruct your syllabus to help clarify who the syllabus serves:</p> <ul>
<li>the institution – think student learning outcomes, specific institution-wide boxes the course checks, advertising institutional supports like relevant peer tutoring services;</li> <li>the
department – think “This class builds on… from [earlier courses] and will set you up to continue on in [later courses]”</li> <li>the academic field – think “This course will expose you to some of the
core ideas in modern algebra, like <em>primes</em>, <em>fields</em>, <em>rings</em>, <em>ideals</em>, and <em>groups</em>”;</li> <li>the faculty – think “I’m your partner in learning in all of these
ways” (the commitments you make to your students, like how you’ll assign grades, your deadline policies, how you’ll support student learning…</li> </ul> <h3 id="math-in-society-11am">Math in Society
(11am)</h3> <p><em>Context: we have a “Social, Structural, and Institutional Hierarchies” requirement at Hamilton that embeds questions about equity, access, fairness into each concentration (it’s
really cool and afaik we’re the only place that does this).</em></p> <p><em>So, in our math department, you can fulfill this requirement in a few ways (through a stats class with focused applications
— eg, an age discrimination legal case and how stats can be used to determine there is age discrimination happening; through courses in the education program that deal directly with these issues —
many of our students are interested in teaching). We also have the Math in Society Reading Seminar.</em></p> <p>The Math in Society course brings in books like <em>The Algebra Project</em> by Bob
Moses, <em>Weapons of Math Destruction</em> by Cathy O’Neil; it touches on the mathematics of fairness (gerrymandering, voting theory); and it generates discussions, blog posts, and a final paper.
After a couple years of one steadfast colleague offering this course, we’re moving to sharing the teaching of this course in different modules. I’m taking responsibility for a 2-week ethics module to
go along with WoMD. (<em>Hello, PredPol!</em>)</p> <p>Here are some of my thoughts from my work session:</p> <ol> <li>Sharing the Ethics in Math AWM panel (or related video content depending on if
this panel is publicly available) with students and discussing it <ul> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Find out if this session will
be shared publicly</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Develop questions for discussion</li> <li class="task-list-item"><input class=
"task-list-item-checkbox" disabled="" type="checkbox" /> Decide on a platform for student discussion (Perusall?)</li> <li class="task-list-item"><input checked="true" class="task-list-item-checkbox"
disabled="" type="checkbox" /> Come up with some back-up options if the panel isn’t made available (currently, a playlist with short videos featuring each of the panelists: <a href="https://
www.youtube.com/playlist?list=PLvQ06rWj7XfMBOaihDDFL7vGUWtLZwlY1">https://www.youtube.com/playlist?list=PLvQ06rWj7XfMBOaihDDFL7vGUWtLZwlY1</a>)</li> </ul> </li> <li><em>Weapons of Math Destruction</
em> <ul> <li class="task-list-item"><input checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Develop Jigsaw assignment for presenting the chapters after Ch 2 and before
Conclusion</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Develop a short class component featuring ORCAA and the ethics matrix - what is it,
examples (where can I find some of these? what’s the name of Cathy’s coauthor in philosophy/ethics, are there papers somewhere to see this work through an academic lens? maybe the philosophy
collaborator? This is all follow-up work that I need to break down into actual tasks). <ul> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Read/vet:
<a href="https://www.fastcompany.com/90172734/this-logo-is-like-an-organic-sticker-for-algorithms-that-arent-evil">Fast Company short piece on ORCAA</a></li> <li class="task-list-item"><input class=
"task-list-item-checkbox" disabled="" type="checkbox" /> Read/vet: <a href="https://redtailmedia.org/2018/10/29/
redtail-talks-about-flipping-the-script-on-how-we-value-algorithims-with-the-weapons-of-math-destruction-author/">Red Tail Media interview with Cathy O’Neil</a></li> </ul> </li> </ul> </li> <li><em>
Phi Beta Kappa visiting scholar</em> <ul> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Coordinate PBK visit, including class visit (relates to the
Ramanujan course content) <ul> <li class="task-list-item"><input checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Determine date</li> <li class="task-list-item"><input
checked="true" class="task-list-item-checkbox" disabled="" type="checkbox" /> Determine topic preferences</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type=
"checkbox" /> Set up pre-visit conference call to discuss stuff</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Organize venue stuff (with PBK
chapter help)</li> </ul> </li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Determine how to advertise public event well on campus and to students
in class <ul> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" /> Identify opportunities for related events</li> </ul> </li> </ul> </li> </ol> <h3 id=
"lunch-and-a-break-to-go-outside-and-pet-cats-1215pm">Lunch and a break to go outside and pet cats (12:15pm)</h3> <p><em>The cats were happy to be the center of attention.</em></p> <h3 id=
"quarantine-support-introductory-meeting-130pm">Quarantine Support Introductory Meeting (1:30pm)</h3> <p>I won’t go into too much detail here because this meeting is a little bit of a stretch from my
purpose for the day, but I didn’t have control over its timing. I took the Johns Hopkins online contact-tracing course last fall so I could be part of the support group that checks in on students in
isolation/quarantine in the spring. This meeting went into some details about all that stuff (logistics, tips, etc). I’m placing a mental flag to be conscious of how inequities manifest for students
in isolation/quarantine when I start this work.</p> <h3 id="prison-mathematics-project-correspondence-230pm">Prison Mathematics Project Correspondence (2:30pm)</h3> <p>As of last fall, I correspond
with two math enthusiasts through the <a href="https://www.prisonmathproject.org/">Prison Mathematics Project</a>, both of whom are on self-guided tours of number theory. I caught up on reading their
letters and got a start (but not a finish, damn) on my next letters.</p> <p>Further actions:</p> <ul> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" />
Look into getting books to my penpals (specifically <em>Number Theory Through Inquiry</em>)</li> <li class="task-list-item"><input class="task-list-item-checkbox" disabled="" type="checkbox" />
Finish writing back</li> </ul> <h3 id="break-330pm">Break (3:30pm)</h3> <p>More cats, screen-time break, and a very brief walk outside.</p> <h3 id="bonus-zoom-4pm">Bonus Zoom (4pm)</h3> <p>I clocked
out of work for this one, but I want to mention it anyway. I chatted with twitter math friend @Maryamization about practical stuff (how to add a paper clip to a cloth mask to get it to stop fogging
up your glasses) but, carried away with the zeitgeist, we talked about math and community, and she also explained Moonshine to me (monstrous, umbral, or other). This was a soul-restoring chat that,
in the context of the day, gave me a lot to think about in terms of what I like about math and why that makes the other parts worth doing.</p> <h3 id=
"personal-consciousness-raising-and-commitment-to-actions-630pm">Personal Consciousness Raising and Commitment to Actions (6:30pm)</h3> <p>I’m going to write a separate blog post about this because
there’s just too much to say here.</p> <h3 id="clocking-out-for-real-830pm">Clocking Out For Real! (8:30pm)</h3> Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-70501996831696541072020-05-16T20:42:00.000-04:002020-05-16T20:42:25.095-04:00I have lots of thoughts to jot
down about the end of the semester, but first: would anyone like to join a virtual book club?Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-53752209918832010872020-05-11T18:06:00.000-04:002020-05-11T18:08:11.824-04:00<b>Phew</b><div>Last day of
classes! Finals, and then we're finished up.<div><br /></div><div><b>A couple links to helpful things that I didn't get a chance to share earlier</b><br />Questions for an exam: <a href="https:/
/www.francissu.com/post/7-exam-questions-for-a-pandemic-or-any-other-time">https://www.francissu.com/post/7-exam-questions-for-a-pandemic-or-any-other-time</a></div><div>Scanning documents with your
phone and Google Drive: <a href="https://gvsu.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=ea65ca76-8d92-4fe5-8e9b-ab7c00e40ccc">https://gvsu.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=
ea65ca76-8d92-4fe5-8e9b-ab7c00e40ccc</a></div><div><br /></div><div><b>Tools that have been invaluable for teaching remotely</b></div><div>Gradescope (for grading)</div><div>Slack (for project-based
learning)</div><div>Piazza (for asynchronous class discussions that support LaTeX)</div><div>Overleaf (for collaborative document editing, especially with the track changes panel!)</div><div>Explain
Everything (for explainer videos)</div><div>Zoom (yeah yeah, privacy stuff aside: can't imagine teaching without some face-to-face)</div><div>Google Apps Suite (for all sorts of things but especially
forms)</div><div>Google Calendar with Zoom integration</div><div>Boomerang for Gmail for the "Pause Inbox" function (helpful for focus)</div><div><br /></div><div>To be continued, I'm sure!</div></
div>Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-80333946316695026862020-04-28T13:10:00.002-04:002020-04-28T13:10:24.444-04:00I wrote a short piece for the
Early Career section of the Notices of the American Mathematical Society, and you can read it online: <a href="https://www.ams.org/journals/notices/202005/rnoti-p663.pdf">https://
www.ams.org/journals/notices/202005/rnoti-p663.pdf</a><br /> <br /> <div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/
imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="386" data-original-width="688" height="179" src="https://blogger.googleusercontent.com/img/b/
width="320" /></a></div> <br />Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-39457728515487462592020-04-23T02:17:00.001-04:002020-04-23T02:17:52.309-04:00Due to a combination of too much
screen-time and an out-of-date glasses prescription, I get really bad headaches if I work for too long in front of the computer. (I know, right?) <div><br></div><div>To compensate for the breaks
I take during the day, I've been finding myself up at 1 or 2 am trying to catch up. I used to do this before This Happened, but not as often: I intentionally left work physically at work so I would
break the habit.</div><div><br></div><div>Anyway, a few weeks into this habit confirms for me that it's not actually worth the tiredness the next day even if I do get caught up. But I can't seem to
shake the feeling that I am falling behind. Maybe a better way to say it is that I can't seem to make peace with falling behind.</div>Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com1tag:blogger.com,1999:blog-855168037568740476.post-56854753610600450052020-04-09T16:43:00.002-04:002020-04-09T16:43:05.525-04:00<h3 id=
"last-night-faculty-voted-to-make-hamiltons-spring-2020-grade-policy-universal-creditno-creditincomplete-crnc.">Last night, faculty voted to make Hamilton's Spring 2020 grade policy universal Credit/
No Credit/Incomplete (Cr/NC).</h3> <p>The faculty's formal process for making decisions is (surprise!) complicated, so the purpose of this blog post is to give a little more information: how it
worked and some things I considered while voting on policies.</p> <h2 id="process">Process</h2> <p>The process began with a proposed policy that a committee created based on faculty and student
input. The committee's proposal was opt-in Cr/NC, and grades earned would be included on the transcript but no semester GPA would be calculated.</p> <p>The process for moving from this policy to
others we considered required proposing changes to the policy, voting on those changes, or substituting in a new policy and voting on which one we wanted to move forward. There were several rounds,
and it took an extra long time because we were also adapting to the technical challenges of conducting what is supposed to be an in-person meeting online.</p> <h2 id="my-thoughts">My Thoughts</h2>
<p>For students trying to understand some of the things faculty were weighing, here are some of the elements of my own decision-making. It's not exhaustive; I'm listing only those things that I think
might not already be evident to students.</p> <ul> <li> <p><strong>What happens to students if their professor(s) get sick?</strong> <br> The <a href="https://www.desmos.com/calculator/3oh34wdh3x">
numbers and projections</a> for Oneida County scare me. <br> Writing with my own grade book in mind, if I were incapacitated by COVID19, my students would probably be forced to take their courses Cr/
NC. Because I teach one section of a multi section class, it seems extra unfair that the other section might be able to elect grades because of something that happened to <em>me</em>. My own
back-of-the-envelope estimate for faculty with elevated risk factors and comorbidities reinforced this point.</p> </li> <li> <p><strong>Hamilton's faculty had only 2 weeks to figure out how to adapt
courses to remote instruction.</strong><br> I had to revise how I'm going to assess my students' progress using new types of assignments (including plenty of Professor-introduced-error!), and each of
the policies we considered have consequences not just at Hamilton, but for employment and post-bac degree programs.</p> </li> <li> <p><strong>Many schools have already elected Cr/NC.</strong> <br>
Employers, graduate programs, and med schools are already figuring out how to make that Cr/NC work. Some graduate and medical schools will accept Cr/NC grades <em>only if it is a universal policy</
em>, which means that any policy intended to provide students with flexibility ironically left many students <em>without</em> choice. <br> There were nuanced and unique proposals that came up during
the meeting that weren't as simple as "grades with optional Cr/NC" or "universal Cr/NC." By last night, I thought that if we enacted something unique among colleges, it would come back to bite us.
The work that grad schools and employers are doing to adapt to college grading policies is based on what the most prevalent policies are. This is one time we don't want to stand out from the pack.
<br> Here's an incomplete list of colleges with universal Cr/NC policies:</p> <ul> <li>Harvard</li> <li>Yale</li> <li>Columbia</li> <li>Dartmouth</li> <li>Stanford</li> <li>Johns Hopkins</li> <li>
Duke</li> <li>MIT</li> <li>Williams</li> <li>Smith</li> <li>Wellesley</li> </ul> </li> <li> <p><strong>Hamilton needs a policy now that's <em>still</em> a good policy at the end of the semester when
the true toll of COVID19 on our community is more evident.</strong> That's not an unambiguous point in favor of Cr/NC, but it was compelling to me combined with the points above and with the
perspectives of students who were initially in favor of opt-in policies and changed their minds as they faced unexpected challenges.</p> </li> </ul> Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-14326962587227102492020-04-06T11:24:00.002-04:002020-04-09T16:57:06.621-04:00As I’m figuring this out, here’s
what works for me.<br /> <ol> <li>Have course materials (weekly assignments, textbook with bookmarked pages, a blank Overleaf document) to hand.<br /> <em>I have the luxury of a second monitor at
home, so I pull up the assignments, etc. and tile that monitor with them so I can more easily share them on Zoom.</em></li> <li>Have a way of sharing a view of what I’m writing with students.<br />
<em>I have the luxury of an iPad that I can use with a stylus to create a digital document while sharing its screen on Zoom.</em></li> <li>Post a summary of questions and answers to our online course
space.<br /> <em>Piazza is what I’m using – I was already using it before the online adventure to allow for asynchronous office hours, which I really like. Definitely going to use the iPad in regular
office hours and keep this part of the workflow going.</em></li> <li>Record and (selectively) share the recordings with students.<br /> <em>I feel weird about this part, so I have been editing the
recordings down to just me to share with them. I need to get better at rephrasing their questions if I I plan to continue doing this.</em><br /> At times, I stop the recording to do a more personal
check-in with students if there are only a few of us there. And then I forget to record again. A nice hack: put a post-it on your computer screen/keyboard/mouse to remind you to start recording again
when it’s business time.</li> </ol> <blockquote> Written with <a href="https://stackedit.io/">StackEdit</a>.</blockquote> Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-35996335859467837822020-04-02T11:20:00.002-04:002020-04-02T11:20:40.551-04:00<p>So far, the technologies I’m
using for the remote semester are working pretty well (relative to expectations). Now I’m looking at pedagogy and (a) trying to abandon principles that are noble but irrelevant in the face of a
global pandemic while (b) trying to maximize the joyful opportunities to engage with mathematics.</p> <p>A few ideas so far at various stages of implementation:</p> <ul> <li><em>Mathematics for Human
Flourishing</em>: an invitation to read, reflect, and write about the value of mathematics independent of its applications. This seems to be a particularly timely opportunity for students, especially
math majors, to reconnect with the joy of mathematics. The only hurdle right now is finding a way for students to access Francis Su’s book remotely. I’m honestly tempted to just buy and ship this
book to students interested in this option.</li> <li>Student designed exam (thanks to @katemath on Twitter!): <ul> <li>Choose/create 4 problems whose complete and correct solutions show mastery of
the big ideas in the course</li> <li>Justify your choices</li> <li>Submit complete and correct solutions</li> </ul> </li> <li>Choose Your Own Adventure: <ul> <li>Select from a series of predefined
Adventures</li> <li>Adventure materials include YouTube playlists, books that can be accessed online through Hamilton’s library, and additional content created or curated by me for the students</li>
<li>For the interested: topics include… <ul> <li><strong>Applied Cryptography</strong>. Extra stuff comes from a very nice Udacity course, <a href="https://www.udacity.com/course/
applied-cryptography--cs387">https://www.udacity.com/course/applied-cryptography--cs387</a>.</li> <li><strong>Elliptic Curves and Lenstra’s Algorithm</strong>. Additional content created by me that
gives a very, very brief introduction to elliptic curves over finite fields and projective space.</li> <li><strong>Continued Fractions and Convergence</strong>. Additional content from a fair-use
selection from one of my favorite texts, Hardy and Wright’s <em>Introduction to Number Theory</em>. I’ve also selected a portion of an MIT open courseware Number Theory course for undergraduates.</
li> <li><strong>Quadratic Reciprocity and Polynomial Congruences</strong>. Content follows <em>Number Theory through Inquiry</em>. Still curating fun resources on the internet – send me your
suggestions?</li> <li><strong>Pythagorean Triples to Pell Equations</strong>. Ditto the previous.</li> </ul> </li> <li>Students are also free to pitch me their own adventure and I will help as much
as possible.</li> </ul> </li> </ul> <blockquote> <p>Written with <a href="https://stackedit.io/">StackEdit</a>.</p> </blockquote> Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-58626611663360140532020-03-31T15:32:00.000-04:002020-03-31T15:32:04.289-04:00The whole "lower your
expectations" thing is really hard to do.<div> <br /></div> <div> First day of remote instruction: done. It was weird. I don't know if it was good or bad or just weird. I'm not used to being
bad at my job, and I feel really bad at my job right now. I'm sure this is a universal feeling among faculty and students, but it's also very personal for each of us. I want to help
students find the joyful parts of math, and it seems extra impossible right now.</div> <div> <br /></div> <div> On the "life" front, I've got a lot happening that's also making it hard to focus on
doing the job.</div> <div> <br /></div> <div> The question I keep coming back to is, "How long can I sustain 'all hands on deck' without getting sick?"</div> <div> <br /></div> <div> Sorry -- no
cheerful updates here. Just wishing things could go back to normal. I miss the feeling of community and I didn't realize how hard it would be to do class remotely without it.</div>
Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-80404880648808558352020-03-26T15:44:00.000-04:002020-03-26T16:58:10.396-04:00Do you have a bag of dried beans
and a watch/phone that can time seconds? Then you can talk to your kids about exponential growth as you talk about why COVID-19 is so frightening.<br /> <br /> Materials:<br /> 128 (at least)
dried beans in a pile, bowl, stash, or whatever.<br /> 1 plate or bowl.<br /> Timing device<br /> <br /> One player is the shouter (keeps track of time) and one player is the doubler (counts beans).
<br /> Discuss in advance how often your beans will double (I recommend 5 seconds to start).<br /> <br /> The doubler puts one bean from the stash on the plate. The shouter starts keeping track of
time; when time has elapsed, shouter shouts, "Double!"<br /> <br /> For the doubler:<br /> Round 0.<br /> Put one bean from the stash on the plate. (Plate: 1 bean)<br /> <br /> Round 1.<br /> Put
another bean from the stash on the plate, count the beans. (Plate: 2 beans)<br /> <br /> Round 2.<br /> Put 2 more beans from the stash on the plate. (Plate: 4 beans)<br /> <br /> Round 3.<br /> Put
4 more beans from the stash on the plate. (Plate: 8 beans)<br /> [this should have been pretty leisurely so far]<br /> <br /> Round 4.<br /> Put 8 more beans from the stash on the plate. (Plate: 16
beans)<br /> <br /> Round 5.<br /> Put 16 more beans from the stash on the plate. (Plate: 32 beans)<br /> <br /> Round 6.<br /> Put 32 more beans from the stash on the plate. (Plate: 64 beans)<br />
<br /> ...<br /> <br /> Round \(n\).<br /> Put \(2^{n-1}\) beans from the stash on the plate. (Plate: \(2^n\) beans)<br /> <br /> You get the idea. This game gets frantic pretty quickly, and
that is the kind of overwhelmed state exponential growth should invoke: "this is getting really big really fast!"<br /> <br /> To compare to other modes of growth, you can do growth rates like Round
\(n\): put \(n\) [or \(n^2\) if you have a fast counting doubler] beans on the plate -- see how much longer it takes for this type of growth to get frantic.<br /> <br /> Enjoy!<br /> <br /> <span
style="font-size: x-small;">(This is an activity I developed as part of my Project FULCRUM fellowship at the University of Nebraska-Lincoln when I was a graduate student.)</span><br /> <br /> <br />
<br /> <br />Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-67461614369991302572020-03-26T10:55:00.000-04:002020-03-26T16:57:30.981-04:00<span style="font-size: x-small;
">Advice from Abbi Jutkowitz, Film Editor, who has worked from home on and off for 5+ years, and worked from home exclusively for the past 8 months (<a href="https://www.imdb.com/name/nm1631659/">
https://www.imdb.com/name/nm1631659/</a>), in collaboration with Courtney Gibbons, Math Professor, who has worked from home for the last week (<a href="https://people.hamilton.edu/cgibbons">https://
people.hamilton.edu/cgibbons</a>). To download this advice: <a href="https://people.hamilton.edu/cgibbons/files/RemoteSemesterOrientation.pdf">RemoteSemesterOrientation.pdf</a></span><br />
<h2> </h2> <a name='more'></a><br /> <h3> Guiding Principles </h3> Lower your expectations. Then: <br /> <br /> <ol> <li>Develop a plan to satisfy your expectations!</li> <li>Put the plan into
action–do the ideas.</li> <li>Evaluate which actions worked and which did not; tweak the plan.</li> <li>Repeat steps 1-3 until you are able to raise expectations. Then:</li> </ol> <br /> Repeat steps
1-4 until you feel comfortable.<br /> (The goal is to have a plan that is “good enough” without further tweaking, not to be endlessly tweaking.)<br /> <br /> <h3> Plan Elements</h3> ✮✮✮<br /> <h4>
Distractions / Focus</h4> <br /> <ul> <li>Block out times with specific objectives, and find ways to support your schedule.</li> <ul> <li>Use a napkin, a planner, a web calendar, or whatever works
for you to have a reference for your time management.</li> </ul> <li>Some specific objectives might be... </li> <ul> <li>email time - clear out your inbox; </li> <li>class time - complete homework
problems / write a draft of a paper; </li> <li>social time - have Zoom breakfast with your breakfast crew / update your insta story</li> </ul> <li>Some tech options include</li> <ul> <li>Google Apps
suite - Google Calendar, Google Tasks for organizing deadlines and specific tasks</li> <li>Gmail extensions like Boomerang or Pause Inbox to catch up on existing email without being distracted by new
email</li> </ul> </ul> <br /> <br /> ✮✮✮<br /> <h4> Work Space</h4> <br /> <ul> <li>Adopt the mindset that wherever you work is your workspace and therefore should not be used for anything else while
you are working.</li> <ul> <li>For example, if you want to work in bed, do so, but only if you are able to focus and not sleep or do anything else you’d normally do in a bed</li> </ul> <li>When you
are in your workspace, designate to yourself and family members/housemates that you are working and therefore not to be disturbed. Be creative.</li> <ul> <li>Hang a sock or tie or Hamilton swag on
your door</li> <li>If in a shared room, put a noticeable item on your desk</li> <li>If not using a desk, put a binder clip or something similar clipped to your book or screen</li> </ul> <li>Create
the soundscape you need</li> <ul> <li>If you desire silence, use noise-canceling headphones (if possible)</li> <li>For working with some background noise which is not music, try https://
coffitivity.com/</li> </ul> </ul> <br /> <br /> ✮✮✮<br /> <h4> Routines</h4> <br /> <ul> <li>One of the benefits of taking classes at Hamilton is that its residential setting comes with lots of
routines: classes, meals, sports, clubs, office hours, etc. To the extent possible, identify your Hamilton daily/weekly routine and imagine recreating it at home.</li> <ul> <li>Breakfast with pals →
breakfast with the family?</li> <li>Class time → class time / engage with asynchronous course materials/office hours if that is what your professor has scheduled</li> <li>Sports time → do people
still have ShakeWeights?</li> </ul> </ul> <div> (Remember that this is your first draft of your plan!)</div> <br /> <br /> ✮✮✮<br /> <h4> Managing Conflict</h4> <ul> <li>Family / Roommates</li> <ul>
<li>Set your expectations with your cohabitants and come up with a plan in advance for resolving difficulties. </li> <li>“Do not disturb me when the binder clip is on my laptop; it means I’m ‘at
college’ -- when there’s no binder clip, I can help you around the house if you need me.”</li> </ul> <li>Classmates / Instructors</li> <ul> <li>This is new territory for your classmates and your
instructors, too. Adopt an attitude of compromise, compassion, and positivity before addressing issues with your campus colleagues. Your constructive criticism will be extremely helpful to your
instructors (and collaborator classmates), but the circumstances of the semester may make it difficult for anyone to respond rapidly to even the simplest and best advice.</li> <li>Remember too that
your instructors are invested in your success, even though it may be harder to “see” that investment without in-person meetings.</li> <li>With all that in mind: Express your needs, frustrations, and
concerns clearly, openly, and kindly. </li> <ul> <li>Professor Gibbons says: I would always rather know what’s going on for you, my students, than try to guess or anticipate.</li> </ul> </ul> </ul>
<br /> <br /> ✮✮✮<br /> <h4> Health & Wellness</h4> <br /> <ul> <li>We’re in a global pandemic! Take care of yourself and your loved ones first and foremost.</li> </ul> <br /> <br /> ✮✮✮<br />
<i>Be well!</i>Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-22752073506678718192020-03-25T08:19:00.001-04:002020-03-26T16:57:50.681-04:00I managed to put a sad trombone
sound effect (royalty-free) into a video.<div><br></div><div>That's it, that's the whole news.</div>Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-30759325735187331672020-03-21T13:06:00.001-04:002020-03-26T16:57:50.779-04:00Last summer, I taught in a
correctional facility. There were no iPads, no smart boards or projectors or computers. No internet at all and no smart devices. If I wanted a box of chalk with a few colors, I had to get permission
to bring it in and was required to bring it out.<div><br></div><div>The students in my class didn't have computers. They had a textbook (it had to be approved by the facility), access to a small
library of books contingent on a bunch of circumstances, one or two pencils, and paper they mostly had to purchase for themselves unless I got permission to bring some in.</div><div><br></div><div>
That summer made me rethink my reliance on the bells and whistles: online videos, fancy animations, etc. I made a pledge in my teaching log (yep, still have a paper and pencil teaching log!) to make
sure that whatever class stuff I develop is doable with pencil, paper, and discussion. I also pledged to keep my courses as self-contained as possible: between the class and the textbook, students
would have the necessary tools (physical, like pencil and paper, and intellectual, like strategies, outlines, examples, and suggested steps for starting problems) to complete assignments -- or at
least make progress until the next class meeting.</div><div><br></div><div>I have been <i>just okay</i> at keeping those pledges (better than if I hadn't made them), and I find that it's hard to
keep them with the shift to remote instruction. My litmus test is, "Could I teach this course at a correctional facility with minimal changes?" and suddenly the answer has shifted from "kinda" to
"not even close."</div><div><br></div><div>I'm trying to figure out how to bring my principles into alignment. Suggestions welcome.</div>Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-2623796739433885872020-03-19T13:20:00.001-04:002020-03-26T16:57:50.754-04:00There's nothing like looking at a
finished video and thinking: "Wow. What a piece of sh*t."<div><br></div><div>Between driving 8 hours in the past two days (with another 4-5 tomorrow), trying to work with a delightful baby demanding
more dancing around and being held upside down, telling my parents to take COVID-19 seriously (Mom, that's you: 6 feet from other people, no handing over your phone or taking someone else's, no
matter how cute the baby pictures are), etc etc -- not exactly a highly productive time.</div><div><br></div><div>But! Some small victories: our campus has a license for Adobe Premiere Rush and I
worked through the quick tutorial. I made another few minutes of content. I'm ready for a remote meeting later today. We made french fries for breakfast. I scripted a few really awful math jokes to
intersperse in my course video playlists. When I say really bad, I mean: "Why couldn't the base field have puppies? Because it was fixed." (A little Galois humor! Like gallows humor, but French.)</
div><div><br></div><div>Unfinished business: I've written about 5 drafts of an email to my advisees. I hope to finish it and send it soon. Probably a faux pas to start with, "What the actual eff?!"
but it's the most honest opener I've got at the moment.</div>Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-26213913021568843602020-03-16T08:36:00.000-04:002020-03-26T16:57:50.730-04:00We're all facing challenges as
we adapt to CDC guidelines and their implications for our work and social lives. As I put together my plans for my courses, I'm reminding myself this applies to my students and me.<br /> <br />
The Twitter math scene is a really supportive place. I've found some video lectures appropriate for my modern algebra classes and I'll make some to fill gaps. Best practices, ideas,
remote workshops -- all of these and more are happening as people come together. I'm glad my social media use has led me to this wonderful community. I'll add some folks to follow in my
next post!<br /> <br />Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-6128070754540789292020-03-15T19:34:00.000-04:002020-03-26T16:57:50.902-04:00It seems unlikely that we'll be
allowed to work in our offices for the rest of the semester (which is good, as far as the CDC recommendations go). But that means carting home a bunch of stuff and setting up a comfortable (and
cat-proof, and mostly partner-proof) workspace.<br /> <br /> It will take some time to figure out how to rig up all the things I will be using -- microphone, webcam, iPad, dual monitors, possibly a
really old pen-tablet, chalk board. And then it will take time to fix up a few issues in my office (I need to hang up curtains so I can see my screens midday).<br /> <br /> Meanwhile, in
another reality, I would have been on a flight to Turkey right now, heading off to a vacation with a goal of flying back to the US with my sister and my niece. With everything going on, I'm
obviously not on a plane right now, but my sister and niece will be back in the US on Wednesday. I'll head back to my hometown on Tuesday and help get her house ready for her, then pick her up
Wednesday and help her get settled back in, and then Friday I'll head back to Clinton to keep prepping for remote classes.Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-67703739637525795622020-03-13T12:16:00.001-04:002020-03-13T12:16:33.469-04:00Today in our in-person Modern
Algebra class, my students and I talked about our concerns moving forward and I solicited some student preferences that I found useful to rethink my delivery of course material in the coming weeks.
<br /><br />First, my students expressed a preference for lecture-length [50 minute] videos. Color me surprised! I (naively) thought the TikTok generation would prefer short videos.
I predict I will get overwhelmed with the creation and editing of 50 minute videos, so I'll do short edits. But I will probably use the YouTube playlist feature to put together 50 minutes of content
that will autoplay for students.<br /><br />The apps I'm most familiar with for pencasting (creating digital whiteboard lecture that you speak on top of, either as you create it or as you play back
the penstrokes or both) are Doceri (<a href="https://apps.apple.com/us/app/doceri-interactive-whiteboard/id412443803">https://apps.apple.com/us/app/doceri-interactive-whiteboard/id412443803</a>,
fairly simple to learn, direct post-to-YouTube option that's easy to use) and ExplainEverything (<a href="https://apps.apple.com/us/app/explain-everything-whiteboard/id1020339980">https://
apps.apple.com/us/app/explain-everything-whiteboard/id1020339980</a>, more [maybe too many?] features, a steeper for me learning curve, but it has an infinite canvas!) on an iPad.<br /><br />
I've also dug out my 2013 Wacom Intuos tablet, which still works with the latest Mac OS, for the Zoom whiteboard feature. I know you can theoretically connect yourself and your iPad to Zoom,
but I think the old-skool approach -- a specific pen-tablet that you plug into your computer as an additional input device -- will be easier to manage as I get used to Zoom. I'm already
familiar with how to use a pen-tablet from my math cartoonist days (ha!), so I'm feeling better about online office hours via Zoom than I was when I imagined figuring out how to manage multiple
connections, etc.<br /><br />I'm also cooking up a low-tech solution (h/t to <a href="https://twitter.com/benblumsmith" target="_blank">@benblumsmith</a> on Twitter, who seems to have a similar
low-tech problem-solving strategy) of mounting a (physical) blackboard to my office bookshelves using pulleys so I can raise and lower it and still be in my "recording zone" with the current
configuration of the old mic and webcams I dug out for better quality recording.<br /> <br /> Okay -- time to draft out a first attempt at a plan for Modern Algebra done online!Courtney R
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-75279723525051520292020-03-13T09:43:00.001-04:002020-03-26T16:57:50.952-04:00<iframe allowfullscreen=""
frameborder="0" height="344" src="https://www.youtube.com/embed/RxxmLuVF4MM" width="459"></iframe><br /><br /> <br /><br /> I woke up with the weird earworm "Nothing Every Happens on Mars" from the
equally weird (and charming) <i>Waiting for Guffman</i>.<br /><br /> <br /><br /> Why? I don't know, but in the cold light of day (literally -- it's cold and rainy here) it seems relevant. Both
for its contrast to the state of things: everything is happening all at once! and for its looming, stark relevance: social distancing is going to be, well, boring.* I'm kind of a master of
boredom that comes from not having a whole lot going on (as are many residents of more rural areas).** But I feel for those of you in (or about to be in) big cities who are (a) way more likely to be
dealing with COVID-19 firsthand (a topic that's heavy on my mind but that I have no capacity to write about yet) and (b) further frustrated by the lack of normalcy. No sports. No
concerts. No museums. Yikes. At least here in Clinton I have a hill I can walk to, alone and without seeing anyone else, and from its top I can see the beautiful sunrise and sunset.
<br /><br /> <br /><br /> More thoughts more closely related to the challenge of changing to online courses with a widely dispersed audience later. In the meantime, I'll be thinking about that
poor Martian looking for a little excitement. Right now, we have a little extra to share.<br /><br /> <br /><br /> *And at the same time, absolutely necessary to save lives. The latest
UCSF report forecasts 1 to 1.5 million COVID-19 deaths. Take all of this seriously, okay? Even if you will be okay, you inevitably know lots of people who might not be.<br /><br /> <br />
<br /> *That's not totally fair to Clinton, NY, and the Utica area: there is a bunch happening, but it's not the same as living in a big city where there's a lot of stuff happening and you can just
step outside and find it.Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-53221903358951149512020-03-13T07:01:00.000-04:002020-03-26T16:57:50.876-04:00I think, my dudes, that I
properly configured MathJax. Let's see: \(\displaystyle \sum_{n=1}^\infty 2^{-n}\)Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-52118973678472936222020-03-12T18:36:00.001-04:002020-03-26T16:57:50.851-04:00There's no time like a global
pandemic to (re)start a blog, right?<br /> <br /> Hi! I'm Courtney Gibbons. I'm an Associate Professor of Mathematics at Hamilton College, and I'm writing from within my own personal cloud of
panic about the shift from traditional classes to remotely delivered content to students spread across multiple time zones. When I glimpse a view of the horizon through the panic, though, I
have to admit I'm a little bit excited by the challenge.<br /> <br /> This blog, whatever it becomes, will be a place for me to record what I'm trying, how it's going, and -- inevitably -- some of
the math we're doing along the way.<br /> <br /> <br /> <a name='more'></a><br /><br /> <br /> Tonight I'm testing out some Zoom features with mathematics educators that I know from the Twitter math
world. I'm excited to see some friends I haven't seen in a long time and meet some new people through mutual connections.<br /> <br /> What can you expect as far as the math for the next few
weeks? Currently: I'm eight weeks into Modern Algebra (we're discovering why there's no algebraic quintic formula) and Number Theory and Applications (we cracked Enigma with group theory, just worked
through the implementation of RSA and read about some of the flaws of its implementation, and started talking about primality tests and pseudoprimes). Modern Algebra is a Writing Intensive
course, so my students complete weekly writing assignments (with partners) and a final paper (individually). Number Theory is a Speaking Intensive course, so my students had oral midterms
earlier in the semester and... well, we were going to do a conversation midterm (a future blog post on that, I promise) but now -- who knows?<br /> <br /> Students (past and future, but especially
present!), if you're following this grand adventure and would like to contribute to the blog, I'm absolutely thrilled with that prospect!<br /> <br /> Wishing you health, safety, and just enough
excitement. -CRG<br /> <br /> PS: just because I think I finally got MathJax working... \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2}\)<br /> <br /> <br />Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-74710364131025447462019-12-01T20:33:00.000-05:002020-03-26T16:58:05.239-04:00<div style="background-color:
white; margin-bottom: 0.875em;"> <span style="color: #383838; font-family: Times, Times New Roman, serif;"><span style="font-size: 19px;">My office hours, and generally those of the faculty in the
math department, are drop-in. That means that you can show up and expect me to be there during my posted office hours (plus or minus five minutes if I’m running a little bit late). You don’t need to
schedule an appointment to see me; I usually operate on a first-come, first-served basis.</span></span></div> <div style="background-color: white; margin-bottom: 0.875em;"> <span style="color: #
383838; font-family: Times, Times New Roman, serif;"><span style="font-size: 19px;"></span></span></div> <a name='more'></a><br /> <div style="background-color: white; margin-bottom: 0.875em;"> <span
style="color: #383838; font-family: Times, Times New Roman, serif;"><span style="font-size: 19px;">I try to keep the waiting short by limiting an office hours visit to 5-10 minutes if there’s a long
line, which means you might not get all of your questions answered at once. That’s okay, though! One of the benefits of coming to office hours is that you’ll almost inevitably find someone working on
the same problems you are, and you can team up to ask me more questions or figure out a solution together.</span></span></div> <div style="background-color: white; margin-bottom: 0.875em;"> <span
style="color: #383838; font-family: Times, Times New Roman, serif;"><span style="font-size: 19px;">You might discover that I’m more likely to answer questions like, “Why should I use integration by
parts on this problem instead of partial fractions?” than questions like, “How do I do this problem?” My role is to stoke your intellectual curiosity and give you opportunities to learn how to apply
what we’re doing in class to new problems. But you might find that a classmate is willing to walk you through a problem if you’re really stumped. Take this as a reminder to identify the resources you
have (me, your book, a tutor, other students, … ) and find out how to use them most effectively.</span></span></div> <div style="background-color: white; margin-bottom: 0.875em;"> <span style="color:
#383838; font-family: Times, Times New Roman, serif;"><span style="font-size: 19px;">The very last thing you should know about office hours is that I really like math, and I really like talking to
you about math, so it’s hard to imagine a better way to spend a couple hours in the afternoon. If you can make the regularly scheduled office hours, that’s ideal: I’ve set that time aside for
students, not for other parts of my job. When I’m not holding office hours, I’m working on those other tasks: research, class preparation or grading, or the kind of stuff that keeps the college
chugging along (like serving on committees). Please don’t take it personally if you drop in outside of office hours and I have to turn you away! I’d much rather talk to you about math, but I can’t
neglect my other duties, either.</span></span></div> <div style="background-color: white; margin-bottom: 0.875em;"> <span style="color: #383838; font-family: Times, Times New Roman, serif;"><span
style="font-size: 19px;">Hope to see you in office hours soon!</span></span></div> <div style="background-color: white; color: #383838; font-size: 19px; margin-bottom: 0.875em;"> <span style=
"font-family: Times, Times New Roman, serif;"><span id="more-57"></span></span></div> Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-71048206860338810162018-06-20T20:00:00.003-04:002020-03-26T17:21:07.235-04:00For math exams in general, it
can be useful to form a study group to talk over problems and solutions before the exams. It’s also useful to retry problems you’ve seen on homeworks, quizzes, and writing assignments (without
looking at your previous attempt or the graders’ comments) to figure out what you need to focus on studying.<br /> <hr /> Prof. Gibbons’ Linear Algebra exams should take you about 90 minutes to
complete. The format:<br /> <ul> <li><strong>First Page:</strong> “Example or impossible” and True False problems.</li> <li><strong>Middle 2-3 Pages:</strong> Homework-like problems (~5 of them).</
li> <li><strong>Last Page:</strong> Writing Assignment-like problems (~2 of them).</li> </ul> <hr /> <strong>Studying for the First Page</strong><br /> <ul> <li>At the end of each chapter, the book
has True of False questions and discussion questions. These are great problems to make sure that you have a handle on the theory in the class (meaning, literally, the theorems and other results that
we have proved throughout the semester). If you are studying these and would like solutions, contact Prof. Gibbons.</li> <li>Try to read my mind! Make up questions for this page by looking at the
theorems and examples in the notes and book and seeing if you can find good questions that seem to be like the quiz questions. Often the reason that something is impossible is that a theorem says it
can’t happen.</li> <li>Come up with some examples that show lots of things. For example, the identity matrix and the zero matrix are great examples to keep in mind as you work these problems. The
zero matrix works for all of the following statements:</li> <li>A matrix that is singular</li> <li>A matrix for which \(A\mathbf{x} = \mathbf{0}\) has infinitely many solutions</li> <li>A matrix row
equivalent to a matrix with a row of zeros<br /> and others. The matrix \(\begin{bmatrix} 2& 6 \\ 3 & 1 \end{bmatrix}\) has already come up in class a few times as an example of: a singular
matrix, a singular matrix without a zero row, a matrix that is not row equivalent to \(I_2\), a matrix that has a number other than one or zero in reduced row eschelon form, and so on.</li> </ul>
<strong>Studying for the Middle Pages</strong><br /> <ul> <li>There are additional problems at the end of the chapter if you want a source of more problems. If you are studying these and would like
solutions, contact Prof. Gibbons.</li> <li>There are no caluclators on exams, so practice to be sure that you can do a few simple steps (like row reduce or substitute) by hand. (Prof. Gibbons doesn’t
want to check your arithmetic, so she might try to help you out with some row reduction, etc., already completed.)</li> </ul> <strong>Studying for the Last Page</strong><br /> <ul> <li>These
questions will require you to form a Proposition (that is, a <em>universally quantified implication</em>:</li> <li><strong>Proposition.</strong> For all …, if …, then … .<br /> and then to write a
proof that starts with “Proof” and ends with an end-of-proof symbol like \(\square\) or QED. Your first and last sentences should conform to good style (state your assumptions in the first sentences,
do the math, and then conclude what the proof technique you’re using requires you to conclude).</li> <li>One problem will come from a writing assignment or class groupwork, so you will have seen it
before. Another problem will be new, but it will use the same techniques as in class and on writing assignments (like letting \(r\) and \(s\) be real numbers where \(r+s = 1\) in order to come up
something new from two existing things, or using the logical equivalence \[p \implies (q \lor r) \equiv (p \land \lnot q) \implies r).\]</li> </ul> Courtney R Gibbonshttp://www.blogger.com/profile/
04661173090248877692noreply@blogger.com0tag:blogger.com,1999:blog-855168037568740476.post-33104637700143920102014-05-31T20:00:00.005-04:002020-03-26T18:29:56.843-04:00As you probably know, I wear my
heart on my sleeve:<br /> Well, I took the <em>golden opportunity</em> (ha!) to bring the golden ratio $\Phi = \frac{1+\sqrt{5}}{2}$ into Calc 2 this week, using it (and its little pal $\Psi = \frac
{1-\sqrt{5}}{2}$) to find a closed formula for the $n$-th term of the Fibonacci sequence.<br /> The ubiquitous Fibonacci sequence! It’s something you may have encountered out in the wild. You know,
it goes a little like this:<br /> $$F_0 = 1, F_1 = 1, F_n = F_{n-1} + F_{n-2},$$<br /> so that $$F_2 = 2, F_3 = 3, F_4 = 5, F_5 = 8, F_6 = 13, F_7 = 21, \ldots $$<br /> And let’s say for some reason,
you need to cook up $F_{108}$. I hope you have some time on your hands if you’re planning to add all the way up to that. Instead, wouldn’t it be nice if we had a simple formula that we could use —
i.e., a formula that was not recursive — to figure out the $n$-th Fibonacci number?<br /> Luckily, such a formula exists, and there are lots of ways to find it. In this post, we’ll find it using
power series. Read on, brave blogosphere traveler.<br /> <a name='more'></a>First thing we’re going to do is build a power series:<br /> $$\begin{array}{r c l} F(x) &=& \sum_{n=0}^\infty F_n
x^n \\<br /> &=& 1+ x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + 13x^6 + 21x^7 + \cdots.<br /> \end{array}$$<br /> Remember that $F_n = F_{n-1} + F_{n-2}$ as long as $n \geq 2$. This means that we can
rewrite our power series:<br /> $$\begin{array}{r c l} F(x) &=& \displaystyle\sum_{n=0}^\infty F_n x^n \\<br /> &=& 1 + x + \displaystyle\sum_{n=2}^\infty(F_{n-1} + F_{n-2})x^n \\<br
/> &=& 1 + x + \displaystyle \sum_{n=2}^infty(F_{n-1}x^n + F_{n-2}x^n) \\<br /> &=& 1 + x + \displaystyle \sum_{n=2}^\infty F_{n-1}x^n + \displaystyle \sum_{n=2}^\infty F_{n-1}x^n.<br
/> \end{array}$$<br /> So far so good? We’re not done yet, though. We can do a little reindexing. In particular, note that by tweaking the index $n$ a little bit, we have equalities:<br /> $$\begin
{array}{r c l}<br /> \displaystyle \sum_{n=2}^\infty F_{n-1}x^n &=& \displaystyle \sum_{n=1}^\infty F_n x^{n+1} \quad =\quad x(F(x) - 1),\\<br /> \displaystyle \sum_{n=2}^\infty F_{n-2}x^n &
amp;=& \displaystyle \sum_{n=0}^\infty F_n x^{n+2} \quad = \quad x^2 F(x).<br /> \end{array}<br /> $$<br /> You might want to grab a coffee and some scratch paper to make sure you believe all
those equalities up there. No worries; the rest of the post will be here when you’re ready.<br /> Okay — agree with me? Good! Remember what we figured out above, and substitute in to obtain<br /> $$\
begin{array}{r c l} F(x) &=& 1 + x + \displaystyle \sum_{n=2}^\infty F_{n-1}x^n + \displaystyle \sum_{n=2}^\infty F_{n-1}x^n \\<br /> &=& 1 + x + x(F(x) - 1) + x^2 F(x) \\<br /> &
=& 1 + x F(x) + x^2 F(x).<br /> \end{array}$$<br /> Finally, if we solve all of that for $F(x)$, we get the marvelous rational function $F(x) = \frac{-1}{x^2 + x - 1}$.<br /> All right, kids. Now
the question becomes: (a) do you remember Partial Fraction Decomposition from earlier in the semester? and (b) do you remember the quadratic formula? We’re going to need both of those things. Let me
google that for you… <a href="http://lmgtfy.com/?q=partial+fraction+decomposition" target="_blank" title="Partial Fraction Decomposition">Partial Fraction Decomposition</a> and <a href="http://
lmgtfy.com/?q=quadratic+formula" target="_blank" title="Quadratic Formula">Quadratic Formula</a>.<br /> Now that you’ve had a chance to review, you can double check that $x^2 + x - 1 = (x + \Phi)(x +
\Psi)$ where $\Phi = \frac{1+\sqrt{5}}{2}$ (the Golden Ratio!) and $\Psi = \frac{1-\sqrt{5}}{2}$ (the, uh, Aluminum Ratio? Or something?). And this means that<br /> $$\frac{-1}{x^2 + x - 1} = \frac
{A}{x + \Phi} + \frac{B}{x + \Psi}.$$<br /> I’ll let you check my math here, but I’d wager a cat that $A = \frac{-1}{\sqrt{5}} \Phi$ and $B = \frac{1}{\sqrt{5}}\Psi$.<br /> First, let’s work with $\
frac{A}{x+\Phi}$:<br /> $$\begin{array}{r c l}<br /> \frac{A}{x+\Phi} &=& \frac{-1}{\sqrt{5}}\cdot \frac{\Phi}{x+\Phi} \\<br /> &=& \frac{-1}{\sqrt{5}}\cdot \frac{1}{(1 + \frac{x}{\
Phi})} \\<br /> &=& \frac{-1}{\sqrt{5}} \displaystyle \sum_{n=0}^\infty \left(\frac{x}{\Phi}\right)^n\\<br /> &=& \frac{-1}{\sqrt{5}} x^n \Phi^{-n} \\<br /> &=& \frac{1}{\sqrt
{5}} x^n \Psi^n,\\<br /> \end{array}$$<br /> where that last equality comes from the fact that $\Phi \Psi = -1$, so $-\Phi^{-1} = \Psi^1$ (and vice versa). Similar trickery gives $$\frac{B}{x+\Psi} =
\frac{-1}{\sqrt{5}} \displaystyle \sum_{n=0}^\infty x^n \Phi^n.$$<br /> Putting all of this together, we have $$F(x) = \frac{1}{\sqrt{5}} \displaystyle \sum_{n=0}^\infty (\Phi_n - \Psi_n)x^n.$$
Remember that the coefficient of $x^n$ in $F(x)$ is actually $F_n$, and therefore — drumroll, please! —<br /> $$F_n = \frac{1}{\sqrt{5}}(\Phi^n - \Psi^n).$$<br /> And so, to calculate $F_{108}$, just
pop $\frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2} \right)^{108} - \left(\frac{1-\sqrt{5}}{2}\right)^{108}\right)$ into your into your favorite calculator. For instance, <a href="http://
www.wolframalpha.com/input/?i=1%2Fsqrt%285%29+*%28+%28%281%2Bsqrt%285%29%29%2F2%29%5E%28108%29+-+%28%281-sqrt%285%29%29%2F2%29%5E%28108%29%29" target="_blank" title="F108">wolfram alpha</a>
calculates that this Fibonacci number is merely sixteen sextillion, six hundred forty-one quintillion, twenty-seven quadrillion, seven hundred fifty trillion, six hundred twenty billion, five hundred
sixty-three million, six hundred sixty-two thousand, and ninety-six. $\square$Courtney R Gibbonshttp://www.blogger.com/profile/04661173090248877692noreply@blogger.com0 | {"url":"https://crgibbon.blogspot.com/feeds/posts/default","timestamp":"2024-11-14T18:14:15Z","content_type":"application/atom+xml","content_length":"127244","record_id":"<urn:uuid:2ce86131-2e28-4bad-adc1-55b680c0edab>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00741.warc.gz"} |
MathSciDoc: An Archive for Mathematician
For a real analytic periodic function π : β β β , an integer π β ₯2 and π β (1/π ,1), we prove the following dichotomy for the Weierstrass-type function π (π ₯)=β _{π β ₯0} π ^π
π (π ^π π ₯): Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to 2+logπ π . Furthermore, given b and π , the former alternative only happens for
finitely many π unless π is constant. | {"url":"https://archive.ymsc.tsinghua.edu.cn/pacm_category/0105?show=view&size=3&from=1&target=searchall","timestamp":"2024-11-08T12:31:11Z","content_type":"text/html","content_length":"59720","record_id":"<urn:uuid:79f43602-3153-4452-87d7-5d96440215cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00810.warc.gz"} |
Science in Portugal - Characters
Aniceto Monteiro (1907-1980)
António Aniceto Monteiro was born on 31 May 1907 in Moçamedes, Angola. He was a student at the Military School and at the Faculty of Sciences of Lisbon, from which he obtained a post-graduate degree
in Mathematical Sciences in 1930. Shortly after completing his education, he received a scholarship from the National Education Board in order to study at the University of Paris, where he obtained
the degree of Docteur és Sciences Mathématiques. His thesis was titled Sur l’additivité des noyaux de Fredholm and was supervised by Maurice Fréchet.
After his return from Paris, Aniceto Monteiro began a highly intensive intellectual activity, seeking to promote, in conjunction with his colleagues interested and involved in scientific activities,
the presentation and publication of mathematical works. Together with Manuel Valadares (1904-1982), António da Silveira (1904-1985), Peres de Carvalho (1904-1989) and others, he founded the Núcleo de
Matemática, Física e Química, which promoted several courses, conferences and publications. In 1937, in collaboration with Manuel Zaluar Nunes and José da Silva Paulo, he founded the Portugaliae
Mathematica, the first Portuguese magazine dedicated exclusively to mathematical research. Three years later, he founded a research magazine Gazeta de Matemática in collaboration with Bento Caraça,
J. da Silva Paulo, Hugo Ribeiro and Manuel Zaluar Nunes.
In 1940, he was one of the main driving forces behind the foundation of the Sociedade Portuguesa de Matemática, having been its first Secretary General between 1941 and 1942. In 1943, he founded the
Junta de Investigação Matemática with Mira Fernandes and Ruy Luís Gomes. Between 1940 and 1943, he directed the work of the Centro de Estudos Matemáticos of the Instituto para a Alta Cultura. He
received a scholarship from the National Education Board.
In 1945, he was forced to leave Portugal after his access to a university career was barred for political reasons. He moved to Brazil where he took up office at the National Faculty of Philosophy of
the University of Brazil, currently the Federal University of Rio de Janeiro. He left Brazil in 1964, moving to Argentina, where he became a Professor at the University del Sur, Bahia Blanca.
He was a corresponding member of the Brazilian Academy of Letters and an honorary member of the Mathematics Union of Argentina. In 1978, he was awarded the Prémio Gulbenkian de Ciência e Tecnologia
for his work Algèbres de Heyting Symétriques. He died on 29 October 1980 in Bahia Blanca, in Argentina. On 2 October 2000, the President of the Republic of Portugal awarded him posthumously the
Grã-Cruz da Ordem Militar de Santiago e Espada.
Scientific Activity
After completing his PhD at the University of Paris, Monteiro returned to Lisbon where he developed an intense activity of mathematical research in Portugal, promoted publications dedicated to
mathematics and organised several seminars. He created the magazines Portugaliae Mathematica, dedicated to research in mathematics, and Gazeta de Matemática, directed at students. He also created the
Seminário de Análise Geral, the Centro de Estudos Matemáticos of the Instituto para a Alta Cultura and founded the Sociedade Portuguesa de Matemática.
Gazeta de Matemática “Journal of those applying for the aptitude test and of mathematics students of higher education establishments”
After the creation of the Núcleo de Matemática, Física e Química, the foundation of the Portugaliae Mathematica, of the Seminário de Análise Geral, of the Centro de Estudos Matemáticos de Lisboa, of
the Sociedade Portuguesa de Matemática and of the Gazeta de Matemática in Lisbon, the Centro de Estudos Matemáticos do Porto was organised. To this effect, Aniceto Monteiro was invited by Ruy Luís
Gomes to go to Oporto to explain how this had been achieved in Lisbon, collaborating in the organisation of the centre and drawing up the first work schedule of this Centre of Studies in October
1941. In November 1941, Aniceto Monteiro continued his regular collaboration with this centre with a conference under the title “Introduction to General Topology”. In 1942, he held a course in Oporto
called “Introduction to the Notion of a Continuous Function”, which had a great repercussion with his lessons being published. He also held two conferences called “Finite Geometry” and “Finite
Algebra and Analytical Geometry”, directed at a less specialised public. As a result of these activities, the students of the Faculty of Sciences of Oporto tried to found a Mathematics Club, an
initiative which did not come into being due to a negative intervention of the Ministry of the Interior.
In October 1943, together with Ruy Luís Gomes and Mira Fernandes, he participated in the creation of the Junta de Investigação Matemática. The aim of the promoters of this Junta was to promote the
development of mathematical research and to arouse an interest in mathematical research in young people. This Junta later managed to obtain financial support, making possible the hiring of Monteiro
to stay in Oporto. During his stay, he participated in the organisation of the General Analysis Studies, structured into three fields: Modern Algebra, under the responsibility of António Almeida
Costa, Measure and Integration, under the responsibility of Ruy Luís Gomes, and General Topology, under the responsibility of Aniceto Monteiro. These studies included colloquia and lessons, which
made up a collection of publications called “Cadernos de Análise Geral”.
The Junta de Investigação Matemática later hired Aniceto Monteiro to remain in the city of Oporto and to continue with his activity of promoting mathematical research and of encouraging students
interested in mathematical studies. He organised the General Analysis Studies organised into three fields: Modern Algebra, directed by António Almeida Costa, Measure and Integration, directed by Ruy
Luís Gomes, and General Topology, under the responsibility of Aniceto Monteiro. These studies included colloquia and lessons, which made part of the collection called “Cadernos de Análise Geral”.
In 1945, he moved on to the National Faculty of Philosophy of the University of Brazil, today the Federal University of Rio de Janeiro. He developed research activities and conferences in the areas
of General Topology, Hilbert Spaces, Functional Analysis, Ordered Series and Boolean Algebras and Lattices. Here he continued his activity of promoting mathematical research and encouraging and
supporting students who wished to pursue this area of knowledge. While in Brazil, he started publishing a series of monographs called Notas de Matemática, editing six volumes. This series of
publications was later continued by Leopoldo Nachbin, a former student of his. In 1949, pressured by the Portuguese embassy in Brazil, his four-year contract was not renewed and he left Brazil for
Here Monteiro occupied the position of Mathematics Professor at the University of San Juan, where he started an intense activity of teaching, research and development of research projects and pair
work. He directed research seminars in areas such as Boolean Algebra, Deductive Systems and Browder’s Algebras. He also became interested in certain practical problems, namely in hydrology directed
towards irrigation.
He was the main driving force behind the creation of the Department of Scientific Research of this University in 1951. This department included an institute of mathematical research, which was set up
in Mendoza. This institute congregated a group of mathematicians who played a crucial role in the development of mathematics in Argentina and in other Latin American countries. Although continuing
his regular activity at the University of San Juan, Monteiro directed courses and seminars in the Department of Scientific Research, exchanging ideas and research problems with its members.
Together with M. Cotlar and E. Zarantonello, later joined by Julio Rey Pastor, in 1955 he started the publication of the Revista Matemática Cuyana.The Department of Scientific Research was
subsequently dissolved after the military coup of 1956, which deposed General Perón, and its library was transferred to San Luiz. In the meantime, in July 1957, Monteiro moved to the University of
Bahia Blanca, together with many of the old members of the Department of Scientific Research, where he founded a new institute of mathematical research. He later joined Ruy Luís Gomes, who in the
meantime had moved away from Portugal for political reasons.
Shortly after establishing himself at the University of Bahia Blanca, Monteiro organised a meeting with the União Matemática Argentina and published its minutes, creating in this way a series of
publications called Actas. He also started publishing research texts called Notas de Lógica Matemática and Notas de Algebra y Analisis, giving rise to an interchange of publications with specialised
foreign magazines, thus enriching the collections of the institute’s library. Monteiro invited several renowned mathematicians to develop seminars and courses at this institute.
He launched a new series of publications, called Monografías de Matemática, distributed to all those involved in mathematical research. He also reformulated the post-graduate degree in Mathematics
and made profound changes to the teaching of Mathematics in Argentina, which came to have a profound impact on the teaching in the faculties of science of this country.
At the end of the sixties, the first students entirely educated by the University del Sur drew up their PhD theses while the Mathematics Institute continued its activity with the collaboration of
researchers from other Latin American countries.
In 1966, after a coup d’état, several faculties of the University of Buenos Aires were attacked by militaries who carried outacts of repression on students and teachers alike, which led to the
resignation of many university teachers and to the closing down or paralysation of several research centres.
The University del Sur was not directly affected by this situation, thus becoming the main mathematical research centre of Argentina. The effects of the military regime became noticeable, however, in
the reduction of the financing of teaching and research activities. Monteiro continued with his research work, presenting papers and travelling throughout Europe, visiting Romania, Paris, Belgium and
England, where he contacted several researchers in his area.
After returning from Europe, Monteiro retired from the University of Bahia Blanca, having been its first professor emeritus. In 1974, the União Matemática Argentina bestowed on him an Honorary
Fellowship. In the meantime, he continued to work in the institute. In March 1975, he was prevented from entering the University by representatives of the government, giving rise to a reaction of
revolt throughout the Argentinean scientific community. The Conselho Nacional de Investigações Científicas e Técnicas of Buenos Aires invited Monteiro to form part of this institution, awarding him
the highest existing degree. In the following years, several university teachers were pushed away from their positions and imprisoned, leading to the exile of many of them.
In 1977, Monteiro visited Portugal, at the invitation of the Instituto de Investigação Científica, corresponding in this way to the wishes of many colleagues and friends who wanted him to return
after the re-establishment of the democratic regime on 25 April 1974. He remained in Portugal for two years, as a researcher at the Instituto de Investigação Científica, performing work in the Centro
de Matemática e Aplicações Fundamentais of the University of Lisbon. In this research centre he created a line of research in Logic Algebra and presented several conferences, put forward research
topics and offered guidance. This work led to the publication of several works in various magazines of the speciality and two PhD theses.
At the invitation of the Faculty of Sciences of the University of Oporto, he gave three conferences in July 1977 on “Álgebras de Boole cíclicas”, “Álgebras de Nelson finitas e lineares” and “
Aritméticas dos filtros em espaços topológicos”. He also published a paper in the number dedicated to Aureliano Mira Fernandes in the magazine Técnica, of the Instituto Superior Técnico.
In 1978, he was awarded the Prémio Gulbenkian de Ciência e Tecnologia for the work Sur les Algèbres de Heyting Symétriques, written during his only period in Portugal after his exile.
António Monteiro’s scientific work comprehends more than 50 research papers. The list of these papers is included in annex.
Fernando Reis
MONTEIRO, Luiz F., Contribuição Matemática do Professor Dr. António A. R. Monteiro , in Um dia com o Centro de Estudos Matemáticos do Porto, Actas, Porto, Centro de Matemática da Universidade do
Porto, 2001.
Portugaliae Mathematica, 39, 1980. [número em homenagem a António Aniceto Monteiro]
DIONÍSIO, José Joaquim; OLIVEIRA, Augusto J. Franco de; Matemáticos Portugueses , in STRUIK, Dirk J., História Concisa das Matemáticas, 3.ª ed., Lisboa, Gradiva, 1997, pp. 383-388.
Uma curta viagem pela História da Matemática em Portugal Teresa Monteiro
"Para a História da Sociedade Portuguesa de Matemática" José Morgado
António Aniceto Monteiro
List of Publications
"Sur les noyaux additifs dans la théorie des équations intégrales", C. R. Acad. Sci. Paris, 198 (1934), 1737.
"Sur une classe de noyaux développables" C. R. Acad. Sci. Paris, 200 (1935), 2143.
"Sur l'additivité des noyaux de Fredholm", Thèse ès Sciences Math, Univ. de Paris, Portug. Math., 1 (1937-40), l-174. 4. Sur l'additivité dans un anneau, Portug. Math., 1 (1937-40), 289-292. 5.
(Com. H. Ribeiro) "Sur l'axiomatique des espaces (V)", Portug. Math., , (1937-40), 275-288.
"Caractérisation des espaces de Hausdorff au moyen de l'opération de dérivation, Portug. Math., 1 (1937-40), 333-339.
(Com A. Gibert) Os conjuntos mutuamente conexos e os fundamentos da topologia integral , Las Ciencias, ano 7, nº 2 (1940).
Les ensembles fermés et les fondements de la topologie", Portug. Math., 2 (1941), 56-86.
"La notion de fermeture et les axiomes de séparation", Portug. Math., 2 (1941), 290-298. (Avec H. RIBEIRO) L'opération de fermeture et ses invariants dans les systémes partiellement ordonnés, Portug.
Math., 3 (1942), 171-184.
"Caractérisation de l'Opération de fermeture par un axiome", Portug. Math., 4, (1943-45), 158-160.
(Avec. H. Ribeiro) "La notion de fonction continue", Summa Brasiliensis Math., 1 (1945), 1-8.
"L'arithmétique des filtres premiers", C. R. Acad. Sci. Paris, 225 (1947), 846-848.
Filtros e Ideais, I. Notas de Matemática, n.° 2, Rio de Janeiro, (1948).
Filtros e Ideais, II . Notas de Matemática, nº 5 Rio de Janeiro (1948).
Réticulés distributifs de dimension linéaire n." C. R. Acad. Sci. Paris, 226, (1948), 1658-1660.
(Com M. Peixoto). Note On uniform continuity , Proc. Inter. Congress of !Math., N.º 1 (1950), 385.
"L'Arithmétique des Filtres et les Espaces Topologiques", I Notas de Lógica Mat., Nº 29 (1974). Trabalho realizado em 1950 para o concurso organizado pela Soc. Math. de France em homenagem a M.
Fréchet, cf. Bull. Soc. Math.France, 79 (1951), XXXIX-XL.
(Com M. Peixoto). "Le nombre de Lebesgue et la continuité uniforme", Portug. Math., 10 (1951), 105-113.
Les filtres des espaces compacts, Gaz. Mat., 50 (1951), 95-96.
Propriedades características de los filtros de un Algebra de Boole, Acta Cuyana de Ingenieria, 1 (1951), 1-7.
L'Arithmétique des Filtres et les Espaces Topologiques , Symp. sur les probl. math. étudiés en Amérique Latine, Villavicencio, Unesco, Montevideo (1954), 129-162.
Axiomes indépendents pour les algébres de Brouwer , Rev. Unión Mat Argentina, 17 (1955), 149-160
"Les ensembles ordonnés compacts", Rev. Mat. Cuyana, (1955), 187-194
(Com O. Varsavsky). Algebra's de Heyting monádicas , Actas de las X jornadas de la Union Mat. Arg., Bahía Blanca (1957), 52-59.
Normalidad en las algebras de Heyting monádicas , Actas de las l Jornadas de la Union Mat. Arg., Bahía Blanca (1957), 50-51.
Algebras Monádicas, Conférences réalisées dans le «2.º Colóquio Brasileiro de Matemática» , Poços de Caldas, 1959, Actas do segundo Colóquio Brasileiro de Matemática, São Paulo, Brasil, (1960)
Matrices de Morgan caractéristiques pour le calcul propositionnel classique , Anais Acad. Brasileira Ciencias, 31 (1960), 1-7.
Linéarisation de la logique positive de Hilbert-Bernays , Rev. Unión Mat. Argentina, 20 (1962), 308-309.
(Com. D. Brignole). Caracterización de las Algebras de Nelson por igualdades, Rev. Unión Mat. Argentina, 79 (1962), 36.
(Com. L. Iturrioz). Representación de las Algebras (le Tarski monádicas , Rev. Unión Mat. Argentina, 19 (1962), 361.
Construcción de las Algebras de Nelson finitas , Rev. Unión Mat. Argentina, 19, (1962), 361.
Algebras de Nelson semi-simples , Rev. Unión Mat. Argentina, 21 (1963), 145-146.
Sur la définition des algébres de Lukasiewicz trivalentes , Bull. Math. Soc. Sci. Math. et Phys. R. P. Roumanie, Nouv. Série, 7 (1963), 1-13.
Construction des algébres de Nelson finies , Bull. Acad. Polonaise Sci., 11 (1963), 359-362.
El cálculo proposicional trivalente de J. Lukasiewicz y la lógica clássica , Rev. Unión Mat. Argentina, 22 (1964-65), 43-44.
Relations between Lukasiewicz Three Valued Algebras and Monadic Boolean Algebras , 1964 International Congress for Logic, Methodology and Philosophy of Sciences, Program and Abstracts, The Hebrew
University, Jerusalem (1964), 16-17.
(Com. L. Iturrioz). Calculo proposicional implicativo clássico con n varia bles proposicionales, Rev. Unión Mat. Argentina, 22 (1964-65), 146.
(Com. R. Cignoli). Construcción geometrica de las algebras de Lukasiewicz trivalentes libres, Rear. Unión Mat. Argentina, 22 (1964-65), 152.
Généralisación de un teorema de Sikorski sobre Algebras de Boole , Rev. Unión flat. Argentina, 22 (1964-65), 151.
(Com. R. Cignoli). Boolean elements in Lukasiewicz Algebras , II, Proc. Japan Acad., 41 (1965), 676-680.
Généralisation d'un théorème de Sikorski sur les Algébres de Boole , Bull. Sci. Math., 2è^me Série, 89 (1965), 65-74.
Algebras de Boole involutivas , Rev. Unión Mat. Argentina, 23 (1966-68), 39.
(Com. I. Brignole). Caractérisation des Algébres (le Nelson par des égalités, I et II, Proc. Japan Acad., 43 (1967), 279-283.
"Construction des algébres de Lukasiewicz trivalentes clans les algébres de Boole monadiques, I, Math. Japonicae, 12 (1967), 1-23.
(Avec. L. MONTEIRO). Algebras de Stone con n generadores libres, Rev. Unión Mat. Argentina, 23 (1966-68), 201.
Sobre un cláculo proposicional de Moisil , Rev. Unión .Mat. Argentina, 23 (166-68), 2111.
Generadores de Reticulados distributivos finitos , Actas del Simpósio Panamericano de Matemática Aplicada, Buenos Aires (1968), 465.
(Com. O. Chateaubriand). Les Algébres de Morgan libres , Notas de Lógica Mat., N.° 26 (1969).
La semi-sinnplicité (les Algébres de Boole Topologiques et les systèmes déductifs , Rev. Unión Mat. .Argentina, 23 (1971), 417-448.
L'Arithmétique des Filtres et les Espaces Topologiques , II, Notas de Lógica Mat., N.° 30 (1974).
Séminaire sur les algèbres d'Heyting symmétriques , Notes polycopiées rédigées par A. Figallo (1974-75).
Algèbres de Boole cycliques , Revue Roumaine Math. Pures Appli., 23 (1978), 71-76.
Conjuntos graduados de Zadeh , Técnica (Revista de Engenharia, IST), 4.19-4.50 (1978), 11-34.
Les N-lattice linéaires , Textos e Notas, CMAF, N.º 15 (1978), 11-19.
Les éléments réguliers d'un N-lattice , Anais Acad. Brasileira Ciencias, 54(1980), 653-656.
Sur les algèbres d'Heyting symmétriques , Portug. .Math., 39 (1980). | {"url":"http://cvc.instituto-camoes.pt/ciencia_eng/p43.html","timestamp":"2024-11-04T01:04:27Z","content_type":"text/html","content_length":"95608","record_id":"<urn:uuid:e0933168-96af-4699-90d5-eae13d2471a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00082.warc.gz"} |
Three circles of radius a, b and c touch each other externally. The area of the triangle formed by
Three circles of radius a, b and c touch each other externally. The area of the triangle formed by
joining their centres is :
(D)None of these
1 thought on “Three circles of radius a, b and c touch each other externally. The area of the triangle formed by <br /><br />joining their centr”
1. Answer:
happy mother’s day
Leave a Comment | {"url":"https://wiki-helper.com/three-circles-of-radius-a-b-and-c-touch-each-other-eternally-the-area-of-the-triangle-formed-by-40043219-20/","timestamp":"2024-11-04T06:07:50Z","content_type":"text/html","content_length":"126293","record_id":"<urn:uuid:da81c7fb-210e-4756-a369-1b6461cea1dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00146.warc.gz"} |
Abacus Multiplication Worksheets
Mathematics, specifically multiplication, forms the foundation of numerous academic self-controls and real-world applications. Yet, for numerous learners, understanding multiplication can present a
challenge. To address this hurdle, educators and parents have actually welcomed an effective device: Abacus Multiplication Worksheets.
Intro to Abacus Multiplication Worksheets
Abacus Multiplication Worksheets
Abacus Multiplication Worksheets -
Using the multiplication table we can solve the multiplication problem in the form i e we only have to retrieve the partial products from the multiplication table and add them in the correct places
as we do with paper and pencil 47 23 21 12 10 14 10 8 100 1081 This is absolutely parallel to the multiplication
The abacus is a timeless computing tool that is still applicable in today s classrooms Solving problems on an abacus is a quick mechanical process compared to modern day multifunctional calculators
After learning the necessary counting procedures and memorizing a few simple rules students can use the abacus to solve various problems
Importance of Multiplication Technique Understanding multiplication is pivotal, laying a solid structure for innovative mathematical concepts. Abacus Multiplication Worksheets provide structured and
targeted practice, cultivating a deeper understanding of this basic arithmetic operation.
Evolution of Abacus Multiplication Worksheets
Abacus Multiplication Worksheets Best Kids Worksheets
Abacus Multiplication Worksheets Best Kids Worksheets
Each worksheet has 9 questions in reading 6 digit numbers Level 1 Reading 2 digit 3 digit and 4 digit Count the number of beads in each rod and pen down the number represented by the abacus in this
set of printable worksheets for 2nd grade and 3rd grade kids Level 2 Reading 4 digit 5 digit and 6 digit
Generate Abacus worksheets Unlimited worksheets are available for free on our website Completion of these worksheets will help children be more confident while using the abacus The worksheets include
the numbers Addition Subtraction and Multiplication The worksheets are customized to your child s understanding level
From conventional pen-and-paper workouts to digitized interactive formats, Abacus Multiplication Worksheets have actually developed, accommodating varied discovering styles and preferences.
Types of Abacus Multiplication Worksheets
Fundamental Multiplication Sheets Simple workouts concentrating on multiplication tables, aiding learners develop a solid math base.
Word Issue Worksheets
Real-life situations integrated right into troubles, enhancing vital reasoning and application skills.
Timed Multiplication Drills Examinations made to boost rate and accuracy, assisting in fast mental math.
Advantages of Using Abacus Multiplication Worksheets
Abacus Multiplication Worksheets Best Kids Worksheets
Abacus Multiplication Worksheets Best Kids Worksheets
Dently Observe whether or not they are using the abacus for the second sum It takes time for children to prove to themselves that it is always true 3 For children who have mastered adding 1 to a
number and the commutative law a worksheet with 1 plus a num ber will be a challenge Let each child decide whether or not to use the abacus
Abacus Multiplication 2 through 9 times 2 x 1 digit 3 x 1 digit and 2 digit x 2 digit Worksheet support video where we solve some questions from the worksheets together Access to Support Forums where
you can post any questions you have and review answers to questions that others have posted
Enhanced Mathematical Skills
Regular method develops multiplication efficiency, improving general math abilities.
Boosted Problem-Solving Abilities
Word problems in worksheets develop analytical thinking and method application.
Self-Paced Discovering Advantages
Worksheets accommodate specific learning speeds, cultivating a comfy and versatile knowing atmosphere.
Just How to Develop Engaging Abacus Multiplication Worksheets
Integrating Visuals and Colors Vibrant visuals and colors record interest, making worksheets aesthetically appealing and involving.
Consisting Of Real-Life Situations
Relating multiplication to daily circumstances adds relevance and functionality to workouts.
Customizing Worksheets to Different Skill Degrees Tailoring worksheets based on varying effectiveness degrees makes certain comprehensive knowing. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Gamings Technology-based sources supply interactive understanding experiences, making multiplication appealing and delightful. Interactive Web Sites and Apps On-line
platforms offer varied and available multiplication method, supplementing typical worksheets. Personalizing Worksheets for Different Learning Styles Aesthetic Students Aesthetic aids and
representations help comprehension for students inclined toward visual understanding. Auditory Learners Spoken multiplication issues or mnemonics satisfy students that understand ideas via acoustic
means. Kinesthetic Students Hands-on tasks and manipulatives sustain kinesthetic students in recognizing multiplication. Tips for Effective Implementation in Knowing Consistency in Practice Regular
technique enhances multiplication skills, advertising retention and fluency. Balancing Rep and Range A mix of recurring workouts and varied problem formats maintains interest and comprehension.
Offering Useful Responses Comments aids in identifying locations of improvement, motivating continued progression. Obstacles in Multiplication Technique and Solutions Inspiration and Involvement
Obstacles Boring drills can cause disinterest; cutting-edge methods can reignite inspiration. Conquering Worry of Math Negative understandings around math can prevent development; producing a
positive learning atmosphere is essential. Influence of Abacus Multiplication Worksheets on Academic Performance Studies and Research Findings Study shows a positive relationship between consistent
worksheet usage and enhanced math efficiency.
Abacus Multiplication Worksheets emerge as flexible devices, cultivating mathematical proficiency in students while suiting varied knowing styles. From basic drills to interactive on-line resources,
these worksheets not just enhance multiplication skills yet additionally promote essential reasoning and analytic abilities.
abacus Maths Level 2 worksheets Ucmas Elementary Aucmas Bucmas Primary Math 15095 Criabooks
Practice 1 To 9 With Abacus And Do The Following Activity Identify The Beads In 2020 Abacus
Check more of Abacus Multiplication Worksheets below
Double Digit Multiplication Worksheet Teach Starter
Multiplication 01 Abacus Math 4th Grade Math worksheets Math Poems
Abacus Multiplication Worksheets Best Kids Worksheets
Reading Abacus Worksheets
Multiplication X 5 01 Abacus Math Free Printable worksheets Math Tricks
Complete Guide How to multiply two numbers using Abacus Cuemath
The abacus is a timeless computing tool that is still applicable in today s classrooms Solving problems on an abacus is a quick mechanical process compared to modern day multifunctional calculators
After learning the necessary counting procedures and memorizing a few simple rules students can use the abacus to solve various problems
FREE Abacus Worksheet Generator with Answer Indian Abacus
Unlimited Abacus Practice Question papers Generate unlimited question papers and practice endlessly with the abacus tool Generating unlimited free question papers is now Easy Test your speed and
accuracy by practising loads of questions across multiple topics and categories and difficulty level
The abacus is a timeless computing tool that is still applicable in today s classrooms Solving problems on an abacus is a quick mechanical process compared to modern day multifunctional calculators
After learning the necessary counting procedures and memorizing a few simple rules students can use the abacus to solve various problems
Unlimited Abacus Practice Question papers Generate unlimited question papers and practice endlessly with the abacus tool Generating unlimited free question papers is now Easy Test your speed and
accuracy by practising loads of questions across multiple topics and categories and difficulty level
Abacus Multiplication Worksheets Best Kids Worksheets
Reading Abacus Worksheets
Multiplication X 5 01 Abacus Math Free Printable worksheets Math Tricks
Maths Addition Worksheet abacus Practice Sheet Grade 1 Math Worksheet Generator Crete
How To Use An Abacus To Teach Kids Math WeHaveKids
How To Use An Abacus To Teach Kids Math WeHaveKids
Abacus Multiplication Worksheets Best Kids Worksheets
FAQs (Frequently Asked Questions).
Are Abacus Multiplication Worksheets suitable for all age groups?
Yes, worksheets can be customized to different age and ability degrees, making them versatile for numerous learners.
How often should pupils exercise making use of Abacus Multiplication Worksheets?
Constant method is essential. Normal sessions, ideally a few times a week, can generate substantial improvement.
Can worksheets alone enhance mathematics abilities?
Worksheets are a valuable tool but needs to be supplemented with different understanding methods for comprehensive ability growth.
Exist on the internet systems offering complimentary Abacus Multiplication Worksheets?
Yes, lots of educational internet sites use free access to a wide variety of Abacus Multiplication Worksheets.
Just how can moms and dads support their kids's multiplication practice at home?
Urging regular practice, supplying aid, and creating a positive knowing setting are helpful actions. | {"url":"https://crown-darts.com/en/abacus-multiplication-worksheets.html","timestamp":"2024-11-13T21:25:40Z","content_type":"text/html","content_length":"28372","record_id":"<urn:uuid:20e6a3a7-b5cd-48bd-9f54-f3466f626707>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00561.warc.gz"} |
Precalculus (6th Edition) Blitzer Chapter 2 - Section 2.4 - Dividing Polynomials; Remainder and Factor Theorems - Exercise Set - Page 363 19
quotient $3x-8$ and remainder $r(x)=\frac{20}{x+5}$
Work Step by Step
Step 1. The coefficients of the dividend can be identified as $\{3,7,-20 \}$ and the divisor as $x+5$; use synthetic division as shown in the figure to get the quotient and the remainder. Step 2. We
can identify the result as $\frac{3x^2+7x-20}{x+5}=3x-8+\frac{20}{x+5}$ with the quotient as $3x-8$ and the remainder as $r(x)=\frac{20}{x+5}$ | {"url":"https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-2-section-2-4-dividing-polynomials-remainder-and-factor-theorems-exercise-set-page-363/19","timestamp":"2024-11-02T18:01:22Z","content_type":"text/html","content_length":"84580","record_id":"<urn:uuid:235ef8d9-5bdc-41e9-a380-865215d30c4e>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00730.warc.gz"} |
C1223 – Complex Analysis
be an open subset of
. For any
, we denote
the set of points of
whose distance to the complement of
is larger than
\[ \Omega_r = \{z \in \mathbb{C} \; | \; d(z, \mathbb{C} \setminus \Omega) > r \}. \]
1. Show that \(\Omega_r\) is an open subset of \(\Omega\) and that \(\Omega = \cup_{r>0} \Omega_r\).
2. Assume that \(\Omega\) is connected. Is \(\Omega_r\) necessarily connected? (Hint: consider for example \(\Omega = \{z \in \mathbb{C} \; | \; |\mathrm{Im} \, z| < |\mathrm{Re} \, z| +1 \}\) and \
(r = 1\)).
3. Show that if \(z \in \mathbb{C} \setminus \Omega_r\), there is a \(w \in \mathbb{C} \setminus \Omega\) such that the segment \([w, z]\) is included in \(\mathbb{C} \setminus \Omega_r\). Deduce
from this property that if \(\Omega\) is simply connected, then \(\Omega_r\) is also simply connected. Is the converse true?
4. Show that if \(\Omega\) is bounded and simply connected, \(\mathbb{C} \setminus \Omega\) is connected. (Hint: assume that \(\Omega\) is bounded but that \(\mathbb{C} \setminus \Omega\) is
disconnected, then introduce a suitable dilation of this complement).
From now on, \(\Omega\) is a bounded open subset of \(\mathbb{C}\). Let \(\mathcal{F}\) be a class of holomorphic functions defined on \(\Omega\) (or a superset of \(\Omega\)). A holomorphic function
\(f: \Omega \to \mathbb{C}\) has uniform approximations in \(\mathcal{F}\) if \[ \forall \, \epsilon > 0, \exists \, \hat{f} \in \mathcal{F}, \; \forall \, z \in \Omega, \; |f(z) - \hat{f}(z)| \leq \
epsilon. \]
5. Let \(f: \Omega \to \mathbb{C}\) be a holomorphic function and let \(a \in \mathbb{C} \setminus \Omega\). Assume that \(f\) has uniform approximations in the class of functions defined and
holomorphic on \(\mathbb{C} \setminus \{a\}\). Show that if \(|a|\) is large enough, \(f\) has uniform approximations among polynomials.
6. Show that for any non-empty bounded open subset \(\Omega\) of \(\mathbb{C}\), there is a holomorphic function \(f :\Omega \to \mathbb{C}\) which doesn’t have uniform approximations among
polynomials (Hint: consider \(z\mapsto 1/(z-a)\) for some suitable choice of \(a\)).
A holomorphic function \(f: \Omega \to \mathbb{C}\) has locally uniform approximations in \(\mathcal{F}\) if for any \(r>0\), its restriction to any \(\Omega_r\) has uniform approximations in \(\
mathcal{F}\): \[ \forall \, \epsilon > 0, \forall \, r > 0, \exists \, \hat{f} \in \mathcal{F}, \; \forall \, z \in \Omega_r, \; |f(z) - \hat{f}(z)| \leq \epsilon. \]
7. Show that if \(\Omega\) is not simply connected, there is a holomorphic function \(f: \Omega \to \mathbb{C}\) which has no locally uniform approximations among polynomials (Hint: consider \(f:z\
mapsto 1/(z-a)\) for some suitable choice of \(a\) then compare \[ \int_{\gamma} f(z) \, dz \; \mbox{ and } \; \int_{\gamma} \hat{f}(z) \, dz \] for some suitable closed rectifiable path \(\gamma
From now on, we assume that \(\Omega\) is simply connected.
Let \(f:\Omega \to \mathbb{C}\) and let \(r>0\). We define the function \(\chi_r: \mathbb{C} \setminus \Omega \to \{0,1\}\) by:
• \(\chi_r(z) = 1\) if for every \(\epsilon > 0\), there is a holomorphic function \(\hat{f}_z\) defined on \(\mathbb{C} \setminus \{z\}\) such that \(|f - \hat{f}_z| \leq \epsilon\) on \(\Omega_r
• \(\chi_r(z) = 0\) otherwise.
8. Show that if some points \(z\) and \(w\) of \(\mathbb{C} \setminus \Omega\) satisfy \(\chi_r(z) = 1\) and \(|w - z| < r/2\) then \(\chi_r(w) = 1\) (Hint: first, prove that the open annulus \(A :=
A(w, r/2, +\infty)\) satisfies \(A \subset \mathbb{C} \setminus \{z\}\) and \(\overline{\Omega_r} \subset A\)).
9. Prove that \(\chi_r\) is locally constant then show that if \(\chi_r(a) = 1\) for some \(a \in \mathbb{C} \setminus \Omega\), then \(\chi_r(z) = 1\) for every \(z \in \mathbb{C} \setminus \Omega
10. Assume that \(f: \Omega \to \mathbb{C}\) has locally uniform approximations among holomorphic functions defined on \(\mathbb{C} \setminus \{a\}\) for some \(a \in \mathbb{C} \setminus \Omega\).
Show that \(f\) has locally uniform approximations among polynomials.
11. Let \(\hat{f}: \mathbb{C} \setminus \{a_1,\dots, a_n\}\to \mathbb{C}\) be holomorphic (all the \(a_k\) are distincts). Show that there are holomorphic functions \(\hat{f}_k: \mathbb{C} \setminus
\{a_k\} \to \mathbb{C}\) for \(k=1,\dots,n\) such that \[ \forall \, z \in \mathbb{C} \setminus \{a_1,\dots, a_n\}, \; \hat{f}(z) = \hat{f}_1(z) + \dots + \hat{f}_n(z). \]
Prove the following corollary: if a function \(f: \Omega \to \mathbb{C}\) has locally uniform approximations among holomorphic functions defined on \(\mathbb{C} \setminus \{a_1, \dots, a_n\}\)
for some \(a_1, \dots, a_n \in \mathbb{C} \setminus \Omega\), then \(f\) has locally uniform approximations among polynomials.
The ray with origin \(a\in \mathbb{C}\) and direction \(u \in \mathbb{C}^*\) is the function^1 \[ \gamma: t \in \mathbb{R}_+ \mapsto a + t u. \]
Let \(f:\Omega \mapsto \mathbb{C}\) be a holomorphic function defined on some open subset \(\Omega\) of \(\mathbb{C}\) that contains the image \(\gamma(\mathbb{R}_+)\) of the ray \(\gamma\). The
Laplace transform \(\mathcal{L}_{\gamma}[f]\) of \(f\) along \(\gamma\) at \(s \in \mathbb{C}\) is given by \[ \mathcal{L}_{\gamma}[f](s) = \int_{\mathbb{R}_+} f(\gamma(t)) e^{-\gamma(t) s} \gamma'
(t) \, dt \] (we consider that this integral is defined when its integrand is summable). This definition generalizes the classic Laplace transform \(\mathcal{L}[f]\) since \(\mathcal{L}_{\gamma}[f] =
\mathcal{L}[f]\) when \(\gamma(t) = t\) (that is when \(a=0\) and \(u=1\)).
We assume that there are some \(\kappa > 0\) and \(\sigma \in \mathbb{R}\) such that $$\label{bound} \forall \, z \in \gamma(\mathbb{R}_+), \; |f(z)| \leq \kappa e^{\sigma |z|}.$$
1. Show that if \(\gamma(t) = a + t u\) and \(\mu(t) = a + t(\lambda u)\) for some \(\lambda > 0\), then \[ \mathcal{L}_{\mu}[f] = \mathcal{L}_{\gamma}[f] \] (Reminder: two functions are equal when
the have the same domain of definition and the same values in this shared domain.)
2. Characterize geometrically the set \[ \Pi(u, \sigma) = \left\{ s \in \mathbb{C} \; \left| \; \mathrm{Re} \left( s u \right) > \sigma |u| \right. \right\} \] and show that \(\mathcal{L}_\gamma[f]
\) is defined and holomorphic on \(\Pi(u, \sigma)\).
3. Let \(U\) be an open subset of \(\mathbb{C}^*\). We assume that bound is valid for every \(u \in U\) (for a given origin \(a\) and fixed values of \(\kappa\) and \(\sigma\)). Show that for any \
(s\in \mathbb{C}\), the set \(U_{s}\) of directions \(u \in U\) such that \(s \in \Pi(u,\sigma)\) is open and that the function \(u \in U_s \mapsto \mathcal{L}_{\gamma}[f](s)\) is holomorphic
(Hint: show that the complex-differentiation under the integral sign theorem is applicable).
4. Show that the derivative of \(\mathcal{L}_{\gamma}[f](s)\) with respect to \(u\) is zero (Hint: the result of question 1 may be used).
The exponential integral \(E_1(x)\) is defined for \(x>0\) by \[ E_1(x) = \int_x^{+\infty} \frac{e^{-t}}{t}dt. \]
5. Compute the (classic) Laplace transform \(F\) of \[ t \in \mathbb{R}_+ \to \frac{1}{t+1} \] and give a formula for \(E_1(x)\) that depends on \(F(x)\). Prove that \(E_1\) has a unique holomorphic
extension to the open right half-plane \(\{s \in \mathbb{C} \; | \; \mathrm{Re}(s) > 0\}\).
From now on, we study the case of \[ f: z \in \mathbb{C} \setminus \{-1\} \mapsto \frac{1}{z+1} \] with \(a=0\), \(\sigma=0\) and \(U =\{u \in\mathbb{C} \; | \; \mathrm{Re}(u)>0\}.\)
6. Show that there is a \(\kappa > 0\) such that is valid for every \(u \in U\). Characterize geometrically the set \(U_s\); show that it is non-empty when \(s\in \mathbb{C}\setminus \mathbb{R}_-\).
Let \(s \in \mathbb{C} \setminus \mathbb{R}_-\). We define \[ G(s) := \mathcal{L}_{\gamma}[f](s) \; \mbox{ if } \; u \in U_s \; \mbox{ and } \; \gamma: t \in \mathbb{R}_+ \mapsto tu. \] Note that
this definition is a priori ambiguous since several \(u \in U_s\) exist for a given value of \(s\). For any \(u_0 \in \mathbb{C}^*\) and \(u_1 \in \mathbb{C}^*\) and for any \(\theta \in [0,1]\), we
denote \[u_{\theta} = (1-\theta) u_0 + \theta u_1\] and whenever \(u_{\theta} \neq 0\), we denote \(\gamma_{\theta}\) the ray of origin \(a=0\) and direction \(u_{\theta}\).
7. Let \(s \in \mathbb{C}\setminus \mathbb{R}_-\). Show that if \(u_0 \in U_s\) and \(u_1 \in U_s\) then for every \(\theta \in [0,1]\), \(u_{\theta} \in U_s\) and that \[ \mathcal{L}_{\gamma_1}[f]
(s) - \mathcal{L}_{\gamma_0}[f](s) = \int_0^1 \frac{d}{d\theta} \mathcal{L}_{\gamma_{\theta}}[f](s) \, d\theta. \] Conclude that the definition of \(G\) is unambiguous.
8. We search for a new expression of the difference \(\mathcal{L}_{\gamma_1}[f](s) - \mathcal{L}_{\gamma_0}[f](s)\) to build an alternate proof for the conclusion of the previous question.
Let again \(s \in \mathbb{C}\setminus \mathbb{R}_-\), \(u_0 \in U_s\) and \(u_1 \in U_s\); let \(r \geq 0\) and \(\gamma_0^r\) and \(\gamma_1^r\) be the paths defined by \[ \gamma_{0}^r: t \in
[0,1] \mapsto \gamma_0(tr) \; \mbox{ and } \; \gamma_{1}^r: t \in [0,1] \mapsto \gamma_1(tr) \] Show that \[ \mathcal{L}_{\gamma_1}[f](s) - \mathcal{L}_{\gamma_0}[f](s) = \lim_{r \to +\infty} \
int_{\mu_r} f(z) e^{-sz} \, dz \; \; \mathrm{ where } \; \mu_r = (\gamma_0^r)^{\leftarrow} | (\gamma_1^r) \] Conclude again that the definition of \(G\) is unambiguous (Hint: “close” the path \(\
mu_r\) and use Cauchy’s integral theorem).
9. Prove that \(E_1\) has a unique holomorphic extension to \(\mathbb{C} \setminus \mathbb{R}_-\).
1. (1.5pt) If \(z \in \Omega_r\), then \(d(z, \mathbb{C} \setminus \Omega) > r > 0\). Since any point \(z\) of \(\mathbb{C} \setminus \Omega\) satisfies \(d(z, \mathbb{C} \setminus \Omega) = 0\), we
have \(\mathbb{C} \setminus \Omega \subset \mathbb{C} \setminus \Omega_r\) and thus \(\Omega_r \subset \Omega\).
For any \(z \in \Omega_r\), the number \(\epsilon = d(z, \mathbb{C} \setminus \Omega) - r\) is positive. If \(|w - z| < \epsilon\) then for any \(v \in \mathbb{C} \setminus \Omega\), \(|z - v| \
geq d(z, \mathbb{C} \setminus \Omega)\) and \[ |w - v| \geq |z - v| - |w - z| > d(z, \mathbb{C} \setminus \Omega) - \epsilon = r. \] Consequently \(D(z, \epsilon) \subset \Omega\) and \(\Omega\)
is open.
Finally, if \(z \in \Omega\), since \(\Omega\) is open, \(d = d(z, \mathbb{C} \setminus \Omega) > 0\), thus if \(r = d/2\), the point \(z\) belongs to \(\Omega_r\). Consequently, \(\Omega = \cup_
{r>0} \Omega_r\).
2. (1pt) Let \(\Omega = \{z \in \mathbb{C} \; | \; |\mathrm{Im} \, z| < |\mathrm{Re} \, z| +1 \}\). This set is open: the function \[ \phi: z \in \mathbb{C} \to |\mathrm{Re} \, z| + 1 - |\mathrm{Im}
\, z| \in \mathbb{R} \] is continuous and \(\Omega\) is the pre-image of the open set \(\mathbb{R}_{+}^*\) by \(\phi\). Now, for any purely imaginary number \(iy\) of \(\Omega\), since \(|iy - i|
\leq 1\) or \(|iy + i| \leq 1\), we have \(d(iy, \mathbb{C} \setminus \Omega) \leq 1\). Therefore, no such point belongs to \(\Omega_1\). On the other hand, \(d(-2, \mathbb{C} \setminus \Omega) =
d(-2, \mathbb{C} \setminus \Omega) = 3\sqrt{2}/2 > 1\) and hence \(-2 \in \Omega_1\) and \(2 \in \Omega_1\).
Assume that \(\Omega_1\) is connected. Since it is open, it is path-connected: there is a continuous function \(\gamma:[0,1] \to \Omega_{1}\) that joins \(-2\) and \(2\). By the intermediate
value theorem, there is a \(t\in \left]0,1\right[\) such that \(\mathrm{Re} \, \gamma(t) = 0\). Since \(\gamma(t) \in \Omega_1\), we have a contradiction. Consequently, \(\Omega_1\) is not
3. (2.5pt) By definition of \(\Omega_r\), its complement satisfies \[ \mathbb{C} \setminus \Omega_r = \{ z \in \mathbb{C} \; | \; d(z, \mathbb{C} \setminus \Omega) \leq r \}. \] Thus, if \(z \in \
mathbb{C} \setminus \Omega_r\), since \(\mathbb{C} \setminus \Omega\) is closed, there is a \(w \in \mathbb{C} \setminus \Omega\) such that \(|z - w| \leq r\). Since for any \(\lambda \in [0,1]\)
, the point \(z_{\lambda} = \lambda w + (1-\lambda) z\) satisfies \(|z_{\lambda} - w| = (1-\lambda) |z - w| \leq r\), we also have \(z_{\lambda} \in \mathbb{C} \setminus \Omega_r\). Hence, \([w,
z] \subset \mathbb{C} \setminus \Omega_r\).
Assume that \(\Omega\) is simply connected. Let \(z \in \mathbb{C} \setminus \Omega_r\) and let \(\gamma\) be a closed path of \(\Omega_r\); since \(\Omega_r \subset \Omega\), \(\gamma\) is also
a closed path of \(\Omega\). Let \(w \in \mathbb{C} \setminus \Omega\) such that \([w, z] \subset \mathbb{C} \setminus \Omega_r\). Since \([w, z]\) is connected and the function \(\xi \in \mathbb
{C} \setminus \Omega_r \mapsto \mathrm{ind}(\gamma, \xi)\) is locally constant, \(\mathrm{ind}(\gamma, z) = \mathrm{ind}(\gamma, w)\). Since \(\Omega\) is simply connected, \(\mathrm{ind}(\gamma,
w)= 0\). Hence, \(\Omega_r\) is simply connected.
Alternatively, assume that \(\Omega_r\) is multiply connected. Let \(\mathbb{C} \setminus \Omega_r = K \cup L\) where \(K\) is bounded and non-empty and \(d(K, L) > 0\). Since \(\Omega_r \subset
\Omega\), \(\mathbb{C} \setminus \Omega \subset \mathbb{C} \setminus \Omega_r\). Let \(K' = K \cap (\mathbb{C} \setminus \Omega)\) and \(L' = L \cap (\mathbb{C} \setminus \Omega)\). Clearly, \(\
mathbb{C} \setminus \Omega = K' \cup L'\), \(K'\) is bounded and \(d(K', L') > 0\). Now, let \(z \in K\) and let \(w \in \mathbb{C} \setminus \Omega\) such that \([w, z] \subset \mathbb{C} \
setminus \Omega_r = K \cup L\). Every connected \(C\) subset of \(K \cup L\) that contains a point of \(K\) is included in \(K\) since for any \(r \leq d(K,L)\), the dilation \(C + B(0,r)\)
doesn’t intersect \(L\). Since \([w,z]\) is a connected subset of \(K \cup L\) and \(z \in K\), we have \([w,z] \subset K\), thus \(K'\) is also non-empty. Finally, \(\Omega\) is multiply
The converse result is false: for any open subset \(\Omega\) which is bounded and not simply connected, for \(r\) large enough, \(\Omega_r = \varnothing\) which is simply connected.
4. (2.5pt) Assume that \(\mathbb{C} \setminus \Omega\) is disconnected. Since \(\Omega\) is bounded, for \(r\) large enough, the annulus \(A(0,r+\infty)\), which is connected, is a subset of \(\
mathbb{C} \setminus \Omega\). Consider a dilation of \(\mathbb{C} \setminus \Omega\) which is not (path-)connected; let \(V_{\infty}\) be the component of this dilation that contains the annulus
above, while \(V\) is the (non-empty) union of the other components of the dilation. By construction, \(V\) and \(V_{\infty}\) are open and disjoints and \(V\) is bounded. The set \[K = (\mathbb
{C} \setminus \Omega) \cap V = (\mathbb{C} \setminus \Omega) \setminus V_{\infty}\] is closed, non-empty and bounded; with \[ L = (\mathbb{C} \setminus \Omega) \cap V_{\infty} = (\mathbb{C} \
setminus \Omega) \setminus V, \] which is closed, we have \(\mathbb{C} \setminus \Omega = K \cup L\). Thus \(\mathbb{C} \setminus \Omega\) is multiply connected.
5. (1.5pt) Since \(\Omega\) is bounded, there is a \(r>0\) such that \(\Omega \subset K = \overline{D(0, r)}\). Let \(a \in \mathbb{C}\) such that \(|a|>r\); let \(\epsilon > 0\) and \(\hat{f}: \
mathbb{C} \setminus \{a\}\) be a holomorphic function such that \(|f - \hat{f}| \leq \epsilon/2\) on \(\Omega\). Since \(K\) is a compact subset of \(D(0, |a|) \subset \mathbb{C} \setminus \{a\}
\), the Taylor series expansion \(\sum a_n (z-c)^n\) of \(\hat{f}\) in \(D(0, |a|)\) is uniformly convergent in \(K\), thus there is a \(m \in \mathbb{N}\) such that the polynomial \[ p(z) = \
sum_{n=0}^m a_n (z-c)^n \] satisfies \(|\hat{f} - p| \leq \epsilon/2\) in \(K\) and hence in \(\Omega\). Consequently, \[ \forall \, z \in \Omega, \; |f(z) - p(z)| \leq |f(z) - \hat{f}(z)| + |\
hat{f}(z) - p(z)| \leq \epsilon. \]
6. (1.5pt) Assume that there is a polynomial \(p\) such that \(|f - p| < 1\) on \(\Omega\). Since \(\Omega\) is bounded, the set \(K = \overline{\Omega}\) is compact and since \(p\) is continuous on
\(K\), it is bounded on \(\Omega\). Given that \[ |f(z)| \leq |f(z) - p(z)| + |p(z)| \] the function \(f\) is necessarily bounded on \(\Omega\).
Now all we have to do is to find a holomorphic function on \(\Omega\) which is not bounded. Consider \(f: z\mapsto 1/(z-a)\) where \(a\) is a point of the boundary of \(\Omega\) (such a point
exists since \(\Omega \neq \varnothing\) and \(\Omega \neq \mathbb{C}\)); since \(\Omega\) is open, \(a \not \in \Omega\) and the function \(f\) is defined and holomorphic on \(\Omega\). By
construction there is a sequence \(z_n \in \Omega\) such that \(z_n \to a\) and thus \[|f(z_n)| = \left|\frac{1}{z_n - a}\right| \to +\infty,\] thus \(f\) is not bounded on \(\Omega\).
7. (2pt) If \(\Omega\) is not simply connected, there is a closed rectifiable path \(\gamma\) of \(\Omega\) and a point \(a \in \mathbb{C} \setminus \Omega\) which is in the interior of \(\gamma\),
that is \(\mathrm{ind}(\gamma, a)\) is a non-zero integer. The function \(z \mapsto 1/(z-a)\) is defined and holomorphic on \(\Omega\). Additionally \[ \frac{1}{2\pi} \int_{\gamma} f(z) dz = \
mathrm{ind}(\gamma, a) \in \mathbb{Z}^*. \] Since the distance between \(\gamma([0,1])\) and the complement of \(\Omega\) is positive, there is a \(r>0\) such that \(\gamma([0,1]) \subset \
Omega_r\). Now if \(\hat{f}\) is a polynomial such that \(|f - \hat{f}| \leq \epsilon\) on \(\Omega_r\), \[ \left| \frac{1}{2\pi} \int_{\gamma} f(z) \, dz \right| \leq \left| \frac{1}{2\pi} \int_
{\gamma} \hat{f}(z) \, dz \right| + \left| \frac{1}{2\pi} \int_{\gamma} (f-\hat{f})(z) \, dz \right|. \] By Cauchy’s theorem (the local version, in \(\mathbb{C}\)), the first integral in the
right-hand side is zero. By the M-L inequality, the second one is dominated by \((\epsilon/2\pi) \times \ell(\gamma)\). Thus for \(\epsilon < 2\pi / \ell(\gamma)\), we would have \[ \left| \frac
{1}{2\pi} \int_{\gamma} f(z) \, dz \right| < 1. \] Consequently \(z\mapsto 1/(z-a)\) is holomorphic on \(\Omega\) but cannot be locally uniformly approximated by polynomials.
8. (3pt) The condition \(|w - z| <r/2\) yields \[ \{z\} \subset \overline{D(w, r/2)} = \mathbb{C} \setminus A(w, r/2, +\infty). \] or equivalently, \(A = A(w, r/2, +\infty) \subset \mathbb{C} \
setminus \{z\}\). Additionally, any \(v \in \overline{\Omega}_r\) satisfies \(d(v, \mathbb{C} \setminus \Omega) \geq r\) and in particular \(|v - z| \geq r\). Since \(|w - z| < r/2\), \[ |v - w|
\geq |v - z| - |w - z| > r - r/2 = r/2, \] hence \(v \in A(w, r/2, +\infty)\). Consequently, \(\overline{\Omega_r} \subset A\).
Since \(\chi_r(z) = 1\), for any \(\epsilon > 0\), there is a function \(\hat{f}_z\) holomorphic in \(\mathbb{C} \setminus \{z\}\) such such that \(|f - \hat{f}| \leq \epsilon / 2\) on \(\Omega_r
\). Now, since the annulus \(A\) is included in the domain of definition of \(\hat{f}_z\), we have the Laurent series expansion \[ \forall \, v \in A, \; \hat{f}_z(v) = \sum_{n=-\infty}^{+\infty}
a_n (v - w)^n. \] This expansion is locally uniformly convergent; since \(\overline{\Omega_r}\) is compact and included in \(A\), there is a natural number \(m\) such that the function \(\hat{f}
_w(v) :=\sum_{n=-m}^m a_n (v - w)^n\) – which is holomorphic on \(\mathbb{C} \setminus \{w\}\) – satisfies \(|\hat{f}_z - \hat{f}_w| \leq \epsilon/2\) on \(\overline{\Omega}_r\). Finally, on \(\
Omega_r\), we have \[ |f - \hat{f}_w| \leq |f - \hat{f}_z| + |\hat{f}_z - \hat{f}_w| \leq \epsilon/2 + \epsilon/2 = \epsilon. \] and thus \(\chi(w) = 1\).
9. (1pt) We know that if \(\chi(z) = 1\) and \(|w - z| < r/2\), then \(\chi(w) = 1\). Now, by contraposition of this property, if \(\chi(w) = 0\) and \(|w-z| < r/2\), we have \(\chi(z) = 0\).
Therefore \(\chi\) is locally constant. Now since \(\Omega\) is simply connected and bounded, by question 4, \(\mathbb{C} \setminus \Omega\) is connected. Since \(\chi: \mathbb{C}\setminus \Omega
\to \{0,1\}\) is locally constant, if \(\chi(a) = 1\) for some \(a \in \mathbb{C}\setminus \Omega\), \(\chi = 1\) on \(\mathbb{C} \setminus \Omega\).
10. (1pt) If \(f: \Omega \to \mathbb{C}\) has locally uniform approximations among the holomorphic functions defined on \(\mathbb{C} \setminus \{a\}\), then for any \(r>0\) \(\chi_r(a)=1\). By the
previous question, \(\chi_r(b)=1\) for any \(b \in \mathbb{C} \setminus \Omega\) and thus \(f\) has locally uniform approximations among the holomorphic functions defined on \(\mathbb{C} \
setminus \{b\}\). We can select a \(b\) that is arbitrarily large, thus by question 5, \(f\) has locally uniform approximations among polynomials.
11. (4pt) We proceed by induction. The result is plain if \(n=1\); now assume that the result holds for a given \(n \geq 1\) and let \(\hat{f}: \mathbb{C} \setminus \{a_1,\dots, a_n, a_{n+1}\}\to \
mathbb{C}\) be holomorphic. Since \(a_{n+1}\) is an isolated singularity of \(f\), there is a \(r>0\) such that \(A(a_{n+1}, 0, r)\) is a non-empty annulus included in the domain of definition of
\(f\). Let \(\sum_{k={-\infty}}^{+\infty} b_k (z-a_{n+1})^k\) be the Laurent series expansion of \(\hat{f}\) in this annulus and define \[ \hat{f}_{n+1}(z) = \sum_{n = -{\infty}}^{-1} b_k (z - a_
{n+1})^k \] Since there are only negative powers, the sum is convergent (and holomorphic) in \(\mathbb{C} \setminus \{a_{n+1}\}\). Now since in the annulus \[ \hat{f}(z) - \hat{f}_{n+1}(z) = \
sum_{n=0}^{+\infty} b_k (z - a_{n+1})^k \] the point \(a_{n+1}\) is a removable singularity of the function \(\hat{f} - \hat{f}_k\) that may thus be extended holomorphically to \(\mathbb{C} \
setminus \{a_1,\dots, a_n\}\). We may apply the induction hypothesis to this extension; we get holomorphic functions \(\hat{f}_k: \mathbb{C} \setminus \{a_k\} \to \mathbb{C}\) for \(k=1,\dots,n\)
such that \[ \forall \, z \in \mathbb{C} \setminus \{a_1,\dots, a_{n+1}\}, \; \hat{f}(z) - \hat{f}_{n+1}(z)= \hat{f}_1(z) + \dots + \hat{f}_n(z) \] which is the induction hypothesis at stage \
Now the corollary: assume that for any \(r>0\) and any \(\epsilon>0\), there is a holomorphic function \(\hat{f}: \mathbb{C} \setminus \{a_1, \dots, a_n\} \to \mathbb{C}\) such that \(|f - \hat
{f}| \leq \epsilon / 2\) in \(\Omega_r\). Let \(\hat{f}_k: \mathbb{C} \setminus \{a_k\} \to \mathbb{C}\) for \(k=1,\dots,n\) be such that \[ \forall \, z \in \mathbb{C} \setminus \{a_1,\dots, a_n
\}, \; \hat{f}(z) = \hat{f}_1(z) + \dots + \hat{f}_n(z). \] Since the restriction of every \(\hat{f}_k\) to \(\Omega\) is locally uniformly approximated by holomorphic functions on \(\mathbb{C} \
setminus \{a_k\}\), by question 9, it is locally uniformly approximated by polynomials, so there is a polynomial \(p_k\) such that \(|\hat{f}_k - p_k| \leq \epsilon/2n\) on \(\Omega_r\). Let \(p
= \sum_{k=1}^n p_k\); on \(\Omega_r\), we have \[ |f - p| \leq |f - \hat{f}| + \sum_{k=1}^n |\hat{f}_k - p_k| \leq \epsilon. \] Thus, \(f\) is locally uniformly approximated by polynomials.
1. (1pt) Since \[ \mathcal{L}_{\mu}[f](s) = \int_{\mathbb{R}_+} f(a + t\lambda u) e^{-s(a+ t\lambda u)} (\lambda u) \, dt, \] if \(\mathcal{L}_{\mu}[f](s)\) is defined, the change of variable \(\tau
= \lambda t\) yields \[ \mathcal{L}_{\mu}[f](s) = \int_{\mathbb{R}_+} f(a + \tau u) e^{-s(a+ \tau u)} u \, d\tau , \] thus \(\mathcal{L}_{\lambda}[f](s)\) is defined and \(\mathcal{L}_{\mu}[f](s)
= \mathcal{L}_{\lambda}[f](s)\). Conversely, if \(\mathcal{L}_{\lambda}[f](s)\) is defined, the same argument with \(1/\lambda\) instead of \(\lambda\) shows that \(\mathcal{L}_{\mu}[f](s)\) is
also defined (and obviously \(\mathcal{L}_{\mu}[f](s) = \mathcal{L}_{\lambda}[f](s)\)).
2. (2pt) The set \(\Pi(u, \sigma)\) is an open half-plane: if \(u = |u|e^{i\theta}\) and \(s = x + iy\),
then the condition \(\mathrm{Re}(s u) > \sigma |u|\) is equivalent to \[ x \cos \theta - y \sin \theta > \sigma. \]
The Laplace transform of \(f\) along \(\gamma\) satisfies \[ \begin{split} \mathcal{L}_{\gamma}[f](s) & = \int_{\mathbb{R}_+} f(a+tu) e^{-s(a+tu)}u\, dt \\ & = (e^{-sa}u) \int_{\mathbb{R}_+} f
(a+tu) e^{-(su)t}\, dt \end{split} \] and thus \[ \mathcal{L}_{\gamma}[f](s) = (e^{-sa} u) \mathcal{L}[t \mapsto f(a+tu)](su). \] The function \(t \in \mathbb{R}_+ \mapsto f(a+tu)\) is locally
integrable (it is continuous since \(f\) is holomorphic) and since for any \(t\geq 0\) \[ -|a| + t |u| \leq |a + tu| \leq |a| + t |u|, \] we have \[ |f(a+tu)| \leq \kappa e^{\sigma |a+tu|} \leq \
kappa \max(e^{-\sigma|a|},e^{\sigma|a|}) e^{(\sigma|u|) t} \] therefore \(t \geq 0 \mapsto |f(a+tu)|e^{-\sigma^+ t}\) is integrable whenever \(\sigma^+ > \sigma |u|\). Hence the Laplace transform
\(\mathcal{L}[t \mapsto f(a+tu)]\) is defined and holomorphic on the set \(\Pi(u, \sigma) =\{s \in \mathbb{C} \; | \; \mathrm{Re}(s) > \sigma |u|\}\) and \(\mathcal{L}_{\gamma}[f]\) as well, as a
composition and product of holomorphic functions.
3. (3pt) For a \(s \in \mathbb{C}\), the set of \(u\) such that \(s \in \Pi(u, \sigma)\) is open as the preimage of the open set \(\left]0, + \infty\right[\) by the continuous function \(u \in U \
mapsto \mathrm{Re}(s u) - \sigma|u|\).
Let \(\psi: U_s \times \mathbb{R}_+ \to \mathbb{C}\) be defined as \[ \psi(u, t) = f(a+tu) e^{-(a+tu)s} u. \] Since \[ |f(a+tu)| \leq \kappa e^{\sigma|a+tu|} \leq \kappa_1 e^{t|u| \sigma} \] with
\(\kappa_1 = \kappa \max(e^{-\sigma|a|},e^{\sigma|a|})\) and \[ |e^{-(a+tu)s}| = e^{-\mathrm{Re}((a+tu)s)} = e^{-\mathrm{Re}(as)}e^{-t\mathrm{Re}(su)}, \] we end up with \[ |\psi(u, t)| \leq (\
kappa_1 e^{-\mathrm{Re}(as)}|u|) e^{t(\sigma |u|- \mathrm{Re}(s u))}. \]
Since for any \(u \in U_s\) \[ \mathcal{L}_{\gamma}[f](s) = \int_{\mathbb{R}_+} \psi(u, t) \, dt \] we check the assumptions of the complex-differentiation under the integral sign theorem:
□ For every \(u \in U_s\), the function \(t \in \mathbb{R}_+ \mapsto \psi(u, t)\) is Lebesgue measurable (it is actually continuous since \(f\) is holomorphic).
□ If \(s \in \Pi(u_0, \sigma)\), since \(\epsilon := \mathrm{Re}(s u_0) - \sigma |u_0|> 0\), by continuity there is a \(0<r<|u_0|\) such that \(|u-u_0| \leq r\) ensures \(\mathrm{Re}(s u) - \
sigma |u| \geq \epsilon/2\). Let \(\kappa_2\) be an upper bound of \(\kappa_1 e^{-\mathrm{Re}(as)}|u|\) when \(|u-u_0| \leq r\). The bound on \(\psi\) that we have derived above yields \[ |\
psi(u,t)| \leq \kappa_2 e^{-t \epsilon/2} \] and the right-hand side of this inequality is a Lebesgue integrable function of \(t\).
□ For every \(t \in \mathbb{R}_+\), it is plain that the function \(u \in U_s \mapsto \psi(u, t)\) is holomorphic.
Consequently \(\mathcal{L}_{\gamma}[f](s)\) is a complex-differentiable function of \(u\).
4. (1.5pt) First method: since we know that the complex-derivative of \(\mathcal{L}_{\gamma}[f](s)\) exists with respect to \(u\), we can apply the chain rule to the differentiable function \[ \chi:
\lambda \in \mathbb{R}_+^{*} \to \mathcal{L}_{\mu}[f](s) \; \mbox{ with } \; \mu(t) = a + t(\lambda u) \] at \(t=1\): it yields \[ \frac{d\chi}{d\lambda} (1) = \frac{d}{du}\mathcal{L}_{\mu}[f](s)
\times \frac{d(\lambda u)}{d\lambda} (1) = \frac{d}{du}\mathcal{L}_{\mu}[f](s) \times u. \] Since by question 1 we know that the function \(\chi\) is constant, we conclude that \(\frac{d}{du}\
mathcal{L}_{\mu}[f](s) = 0\).
Second method: we use the result of the differentiation under the integral sign. It provides \[ \frac{d}{du}\mathcal{L}_{\mu}[f](s) = \int_{\mathbb{R}_+} \frac{\partial \psi}{\partial u} (u, t)
\, dt. \] Since \(\psi(u, t) = g(tu) u\) with \(g(z) = f(a+z) e^{-s(a+z)}\), we have \[ \begin{split} \frac{\partial \psi}{\partial u} (u, t) &= \frac{d}{du} (g(tu) u) = g'(tu) tu + g(tu) = \frac
{d g(tu)}{dt} t + g(tu) \\ &= \frac{d}{dt} (g(tu) t) \end{split} \] and thus \[ \begin{split} \int_0^r \frac{\partial \psi}{\partial u} (u, t) \, dt &= \int_0^r \frac{d}{dt} (g(tu) t) \, dt = [g
(tu) t]^r_0 \\ &= f(a+ ru) e^{-s(a+ru)} r \end{split} \] Since \(\epsilon := \mathrm{Re}(s u) - \sigma|u| > 0\) \[ |f(a+ ru) e^{-s(a+ru)} r| \leq |\psi(u, r)r / u| \leq (\kappa_1 e^{-\mathrm{Re}
(as)}) e^{-\epsilon r} r. \] The right-hand side of this inequality converges to \(0\) when \(r\to +\infty\). Therefore, by the dominated convergence theorem \[ \frac{d}{du}\mathcal{L}_{\mu}[f]
(s) = \int_{\mathbb{R}_+} \frac{\partial \psi}{\partial u} (u, t) \, dt = \lim_{r \to +\infty} \int_{0}^r \frac{\partial \psi}{\partial u} (u, t) \, dt = 0. \]
5. (2pt) Since the Laplace transform of \(t \in \mathbb{R}_+ \to 1/(t+1)\) is given by \[ F(s) = \int_{0}^{+\infty} \frac{e^{-s t}}{t+1} \, dt, \] the change of variable \(t+1 \to t\) yields \[ F(s)
= \int_1^{+\infty} \frac{e^{-s (t-1)}}{t} \, dt = e^s \int_1^{+\infty} \frac{e^{-s t}}{t} \, dt. \] If \(s\) is equal to the real number \(x>0\), the change of variable \(xt \to t\) then provides
\[ F(x) = e^x \int_1^{+\infty} \frac{e^{-xt}}{xt} \, d(xt) = e^x \int_x^{+\infty} \frac{e^{-t}}{t} \, dt \] and thus \(E_1(x) = e^{-x} F(x)\). Since \(t \in \mathbb{R}_+ \to 1/(t+1) e^{-\sigma^
+t}\) is integrable whenever \(\sigma^+ > 0\) the Laplace transform \(F\) of \(t \mapsto 1/(t+1)\) is defined and holomorphic on \(\{s \in \mathbb{C} \; | \; \mathrm{Re}(s) > 0\}\). Thus, the
function \(G: s \mapsto e^{-s} F(s)\) extends holomorphically \(E_1\) on this open right-hand plane. Any other function \(\tilde{G}\) with the same property would be equal to \(G\) on \(\mathbb
{R}_+^*\) and thus every positive real number \(x\) would be a non-isolated zero of \(G - \tilde{G}\). By the isolated zeros theorem, since the open right-hand plane is connected, \(G\) and \(\
tilde{G}\) are necessarily equal.
6. (2pt) For any \(u \in U\), since \(\mathrm{Re} \, u > 0\), for any \(t \geq 0\), \(\mathrm{Re}(\gamma(t)) = t \, \mathrm{Re} \, u \geq 0\). Since for any \(z\) such that \(\mathrm{Re} \, z \geq 0
\), \(|z + 1| \geq 1\), we have \(|f(z)| = 1 / |z + 1| \leq 1\) and thus \[ \forall \, z \in \gamma(\mathbb{R}_+), \, |f(z)| = \kappa e^{\sigma |z|} \] with \(\kappa = 1\) and \(\sigma=0\).
By definition, for any \(s \in \mathbb{C} \setminus \mathbb{R}_-\), \(U_s\) is the set of all directions \(u\) such that \(\mathrm{Re}(u) > 0\) and \(\mathrm{Re}(s u) > 0\). To be more explicit,
if \(\alpha\) denotes the argument of \(u\) in \(\left[-\pi, \pi\right[\) and \(\beta\) the argument of \(s\) in \(\left]-\pi,\pi \right[\), this is equivalent to \[ -\pi/2 < \alpha < \pi/2 \; \
mbox{ and } \; - \pi/2 < \alpha + \beta < \pi/2 \] The points \(u = r e^{i\alpha}\) that satisfy these constraints form an open sector: \(r>0\) is arbitrary and if \(\beta \geq 0\), \(\alpha\) is
subject to \[ -\frac{\pi}{2} < \alpha < \frac{\pi}{2} - \beta \] and if \(\beta < 0\) \[ -\frac{\pi}{2} - \beta < \alpha < \frac{\pi}{2}. \] In any case, since \(|\beta| < \pi\), the set of
admissible arguments \(\alpha\) is not empty.
7. (3pt) It is plain that \[ \mathrm{Re}(u_{\theta}) = \mathrm{Re}((1-\theta) u_0 + \theta u_1) = (1-\theta) \mathrm{Re}(u_0) + \theta \mathrm{Re} (u_1). \] and that for any \(s \in \mathbb{C}\) \[
\mathrm{Re}(su_{\theta}) = \mathrm{Re}(s((1-\theta) u_0 + \theta u_1)) = (1-\theta) \mathrm{Re}(su_0) + \theta \mathrm{Re} (su_1). \] Since \(u \in U_s\) if and only if \(\mathrm{Re}(u) > 0\) and
\(\mathrm{Re}(su) > 0\), if \(u_0 \in U_s\) and \(u_1 \in U_s\) then \(u_{\theta} \in U_s\) for any \(\theta \in [0,1]\).
The function \(u \in U_s \mapsto \mathcal{L}_{\gamma} [f](s)\) is holomorphic, hence the composition of \[ \theta \in [0,1] \mapsto u_{\theta} \in U_s \; \mbox{ and } \; u \in U_s \mapsto \
mathcal{L}_{\gamma} [f](s) \] is continuously differentiable, \[ \mathcal{L}_{\gamma_0}[f](s) - \mathcal{L}_{\gamma_1}[f](s) = \int_0^1 \frac{d}{d\theta} \mathcal{L}_{\gamma_{\theta}}[f](s) \, d\
theta \] and \[ \frac{d}{d\theta} \mathcal{L}_{\gamma_{\theta}}[f](s) = \frac{d \mathcal{L}_{\gamma}[f](s)}{du} |_{u=u_{\theta}} \times \frac{d u_{\theta}}{d\theta} \] which is zero by question
4. Hence \(\mathcal{L}_{\gamma_0}[f](s) = \mathcal{L}_{\gamma_1}[f](s)\); the definition of \(G\) is unambiguous.
8. (4pt) For \(k \in \{0,1\}\) and any \(r\geq 0\), \[ \int_{\gamma_k^r} f(z) e^{-sz} \, dz = \int_0^1 f((tr) u_k) e^{-s((tr)u_k)} r u_k \, dt = \int_0^r f(t u_k) e^{-s(t u_k)} u_k \, dt \] thus by
the dominated convergence theorem \[ \mathcal{L}_{\gamma_1}[f](s) - \mathcal{L}_{\gamma_0}[f](s) = \lim_{r\to +\infty} \left( \int_{\gamma_1^r} f(z) e^{-sz} dz - \int_{\gamma_0^r} f(z) e^{-sz} dz
\right). \] Since \((\gamma_0^r)^{\leftarrow}\) and \(\gamma_1^r\) are consecutive, with the path \(\mu_r = (\gamma_0^r)^{\leftarrow} \,|\, \gamma_1^r\), this is equivalent to \[ \mathcal{L}_{\
gamma_1}[f](s) - \mathcal{L}_{\gamma_0}[f](s) = \lim_{r\to +\infty} \int_{\mu_r} f(z) e^{-sz} dz. \]
Now, the path \(\nu_r\) defined by \(\nu_r(\theta) = (1-\theta) r u_0 + \theta r u_1\) is such that \(\mu_r \, | \, \nu_r^{\leftarrow}\) is a closed rectifiable path in \(U_s\) which is simply
connected. Therefore by Cauchy’s integral theorem \[ \int_{\mu_r} f(z) e^{-sz} dz = \int_{\nu_r} f(z) e^{-sz} dz. \]
For any \(\theta \in [0,1]\), we have \(\mathrm{Re}(s u_\theta) = (1 - \theta) \mathrm{Re}(s u_0) + \theta \mathrm{Re}(s u_1)\), thus \[ \epsilon := \min_{\theta \in [0,1]} \mathrm{Re}(s u_{\
theta}) > 0 \] and \[ |f(\nu_r(\theta)) e^{-s\nu_r(\theta)}| \leq \kappa e^{- \mathrm{Re}(s ru_{\theta})} \leq \kappa e^{-\epsilon r}. \] Given that \(\ell(\nu_r) = r |u_1 - u_0|\), the M-L
inequality provides \[ \left| \int_{\nu_r} f(z) e^{-sz} dz \right| \leq \kappa e^{-\epsilon r}r |u_1 - u_0| \] and thus \[ \lim_{r \to +\infty} \int_{\nu_r} f(z) e^{-sz} dz = 0. \]
Finally, \(\mathcal{L}_{\gamma_1}[f](s) = \mathcal{L}_{\gamma_0}[f](s)\): the definition of \(G\) is unambiguous.
9. (1pt) By construction the function \[s \in \mathbb{C} \setminus \mathbb{R}_- \mapsto e^{-s} G(s)\] is holomorphic (since every \(\mathcal{L}_{\gamma}\) is holomorphic, \(G\) is holomorphic); It
extends \(s \mapsto e^{-s}F(s)\), thus it also extends \(E_1\). Since \(\mathbb{C} \setminus \mathbb{R}_-\) is connected, by the isolated zeros theorem, this extension is unique (the argument is
identical to the one used in question 6).
1. This notation emphasizes that the complex number \(\gamma(t)\) depends on \(t\); however it also depends implicitly on some \(a\) and \(u\) that are usually clear from the context. Feel free to
use a more explicit notation if you feel that it is beneficial.↩ | {"url":"https://eul.ink/complex-analysis/Exam%202018/","timestamp":"2024-11-06T04:28:51Z","content_type":"text/html","content_length":"72450","record_id":"<urn:uuid:2e31bfb9-c156-492c-be1f-d68ae7768d8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00280.warc.gz"} |
An Empirical Investigation Of Stock Markets: The Ccf Approach [PDF] [smps08eh8f80]
<STRONG>An Empirical Investigation of Stock Markets: The CCF Approach attempts to make an empirical contribution to the literature on the movements of stock prices in major economies, i.e. Germany,
Japan, the UK and the USA. Specifically, the cross-correlation function (CCF) approach is used to analyze the stock market. This volume provides some empirical evidence regarding the economic
linkages among a group of different countries.
Chapter 2 and Chapter 3 analyze the international linkage of stock prices among Germany, Japan, the UK and the USA. Chapter 2 applies the standard approach, whereas Chapter 3 uses the CCF approach.
Chapter 4 analyzes the relationship between stock prices and exchange rates. Chapter 5 analyzes the relationship among stock prices, exchange rates, and real economic activities. Chapter 6 summarizes
the main results obtained in each chapter and comments on the possible directions of future research.
AN EMPIRICAL INVESTIGATION OF STOCK MARKETS The CCF Approach
Research Monographs in Japan-U.S. Business & Economics series editors
Ryuzo Sato Rama V. Ramachandran Stern School of Business New York University
KazuoMino Kobe University Japan Other books published in the series: Sato and Ramachandran Conservation Laws and Symmetry: Applications to Economics and Finance Sato, Ramachandran, and Hori
Organization, Performance, and Equity: Perspectives on the Japanese Economy Sato, Grivoyannis, Byrne, and Lian Health Care Systems in Japan and the United States: A Simulation Study and Policy
Analysis Sato and Ramachandran Symmetry and Economic Invariance: An Introduction Sato, Ramachandran, and Mino Global Competition and Integration Negishi Developments ofInternational Trade Theory
Ihori and Sato Government Deficit and Fiscal Reform in Japan
AN EMPIRICAL INVESTIGATION OF STOCK MARKETS The CCF Approach
Shigeyuki Hamori Kobe University, Japan
Hamori. Shigeyuki, 1959An empirical investigation of stock markets:the CCF approach/Shigeyuki Hamori p.cm.-(Research monographs in Japan-U.S. business & economics) Includes bibliographical references
and index. ISBN 978-1-4613-4838-2 ISBN 978-1-4419-9208-6 (eBook) DOI 10.1007/978-1-4419-9208-6 1.
Stock exchanges-Statistics. I. Tide. 11. Series.
HG4551.H3132oo3 332.64'2'021-dc22
Copyright @ 2003 Springer Science+Business Media New York Originally published by Kluwer Academic Publisher in 2003 Softcover reptjnt of the hardcover 1st edition 2003 All rights reserved. No part of
this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without the written
permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the
work. Permission for books published in Europe:
[email protected]
Permissions for books published in the United States of America:
[email protected]
Printed on acid-free paper.
To Hitoshi, Makoto and Naoko
List of Figures List of Tables Acknowledgments 1. INTRODUCTION
xi xv
2. STOCK PRICES ACROSS
INTERNATIONAL MARKETS: A TRADITIONAL APPROACH 1 Introduction 2 Selected Literature Review Data 3 Empirical Technique: VAR 4 Empirical Technique: LA-VAR 5 Conclusion 6
3. STOCK PRICES ACROSS
INTERNATIONAL MARKETS: THE CCF APPROACH Introduction 1 Empirical Technique 2 ARCH-type Models 2.1 2.2 Cheung-Ng Test 3 Data Empirical Results 4 4.1 AR-EGARCH Model 4.2 Cheung-Ng Test Conclusion 5
4. STOCK PRICES AND
EFFECTIVE EXCHANGE RATES 1 Introduction 2 Selected Literature Review 3 Data 4 Empirical Results 4.1 AR-EGARCH Model 4.2 Cheung-Ng Test Conclusion 5
5. STOCK PRICES,
EFFECTIVE EXCHANGE RATES, AND REAL ECONOMIC ACTIVITIES 1 Introduction Selected Literature Review 2 3 Data Empirical Technique 4 Empirical Results 5 Results for Germany 5.1 Results for Japan 5.2
Results for the UK 5.3 Results for the USA 5.4 Conclusion 6
6. SUMMARY AND FUTURE RESEARCH DIRECTIONS Summary 1 Future Research Directions 2 REFERENCES INDEX
List of Figures
2.1 2.2 2.3 2.4 2.5
2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1
Logarithmic Stock Price Index: Germany Logarithmic Stock Price Index: Japan Logarithmic Stock Price Index: UK Logarithmic Stock Price Index: USA First Difference of the Logarithmic Stock Price Index:
Germany First Difference of the Logarithmic Stock Price Index: Japan First Difference of the Logarithmic Stock Price Index: UK First Difference of the Logarithmic Stock Price Index: USA Summary of
Causality Test Two-Step Procedure Standardized Residuals: Germany Standardized Residuals: Japan Standardized Residuals: UK Standardized Residuals: USA Squares of Standardized Residuals: Germany
Squares of Standardized Residuals: Japan Squares of Standardized Residuals: UK Squares of Standardized Residuals: USA Summary of Causality in Mean Summary of Causality in Variance Summary of
Causality: Germany
4.2 4.3 4.4 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
Summary of Causality: Japan Summary of Causality: UK Summary of Causality: USA Forward-Looking Behavior Summary of Causality between Summary of Causality between Summary of Causality between Summary
of Causality between Summary of Causality between Summary of Causality between Summary of Causality between Summary of Causality between
78 79 79 84 RSP and IP: Germany 114 REER and IP: Germany 114 RSP and IP: Japan 115 REER and IP: Japan 115 RSP and IP: UK 116 REER and IP: UK 116 RSP and IP: USA 117 REER and IP: USA 117
List of Tables
1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
Characteristics of the CCF Approach Relation of Each Chapter Summary of Literature Summary Statistics Unit Root Test SBIC Diagnostics Cointegration Test Causality Test: VAR Causality Test: LA-VAR
ARCH-type Models Characteristics of the EGARCH Model Empirical results of the AR-EGARCH model Sample Cross-Correlation of Standardized Residuals: Germany and Japan Sample Cross-Correlation of
Standardized Residuals: Germany and UK Sample Cross-Correlation of Standardized Residuals: Germany and USA Sample Cross-Correlation of Standardized Residuals: Japan and UK Sample Cross-Correlation of
Standardized Residuals: Japan and USA Sample Cross-Correlation of Standardized Residuals: UK and USA
4.1 4.2 4.3 4.4 4.5
4.6 4.7 4.8 4.9 4.10 5.1
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11
Summary of Literature Summary Statistics Unit Root Test Empirical results of the AR-EGARCH model for foreign exchange rates Empirical results of the AR-EGARCH model for stock prices Sample
Cross-Correlation of Standardized Residuals: Germany Sample Cross-Correlation of Standardized Residuals: Japan Sample Cross-Correlation of Standardized Residuals: UK Sample Cross-Correlation of
Standardized Residuals: USA Feedback between the Stock Market and the Foreign Exchange Market Summary of Literature Summary Statistics Unit Root Test Empirical results of the AR-EGARCH model: Germany
Sample Cross-Correlation of Standardized Residuals: industrial production and real stock prices: Germany Sample Cross-Correlation of Standardized Residuals: industrial production and real effective
exchange rate: Germany Empirical results of the AR-EGARCH model: Japan Sample Cross-Correlation of Standardized Residuals: industrial production and real stock prices: Japan Sample Cross-Correlation
of Standardized Residuals: industrial production and real effective exchange rate: Japan Empirical results of the AR-EGARCH model: UK Sample Cross-Correlation of Standardized Residuals: industrial
production and real stock prices: UK
List of Tables 5.12
5.13 5.14
Sample Cross-Correlation of Standardized Residuals: industrial production and real effective exchange rate: UK Empirical results of the AR-EGARCH model: USA Sample Cross-Correlation of Standardized
Residuals: industrial production and real stock prices: USA Sample Cross-Correlation of Standardized Residuals: industrial production and real effective exchange rate: USA
I have benefited greatly from the support of many people in writing this volume. Special thanks are due to Ryuzo Sato, Rama V. Ramachandran, and Kazuo Mino for their kindness in giving me the
opportunity to publish this monograph. I would like to thank Koichi Hamada, Kazuhiro Ohtani, Mitoshi Yamaguchi, David A. Anderson and Ramaprasad Bhar for their many helpful comments and suggestions
for my research. Robert Parry, Mariyo Takeuchi and an anonymous referee kindly read an early version of this volume and gave me helpful suggestions. Finally, I would like to thank my family members
Hitoshi, Makoto and Naoko. Without their warm-hearted support I could not have finished writing this volume and I would like to dedicate this monograph to them. This publication is supported by a
Grant-in-aid from the Rokkodai Foundation.
About the Author
Dr. Shigeyuki Hamori is a Professor in the Graduate School of Economics at Kobe University in Japan. He received a Ph.D. from Duke University in 1991, and has published about 40 papers in refereed
journals. He also received the Highest Quality Rating in 1998 and 1999, and the Murao Award for Young Researchers in 2002.
Chapter 1
News of changes in stock prices can be heard almost every day. The news media often report that stock prices have fallen because of bad economic circumstances or political news or, conversely, have
risen because of encouraging economic or political events. Thus, one of the most important areas of finance literature includes studies that have tried to outline the major determinants of stock
prices. Since it is not easy to quantify political events, most studies have concentrated on economic variables for which data are readily available. Professor Richard Roll in his presidential
address to the 47th annual meeting of the American Finance Association stated that: The immaturity of our science is illustrated by the conspicuous lack of predictive content about some of its most
intensely interesting phenomena, particularly changes in asset prices (Roll 1988, p. 541).
In response to this message, many researchers have tried to show that stock returns can be predicted by a variety of macroeconomic and financial indicators such as dividend yields, interest rates,
inflation, exchange rates, and the growth in industrial production. Since most instruments used to forecast stock returns are linked to the business cycle, the evidence on predictability has been
widely interpreted as indicating that expected returns vary systematically over that cycle. This interpretation seems reasonable, since the capacity of the economy to carry aggregate risks, and hence
the premium on the associated risk factors, will likely depend on the state of the economy. The existing studies of the literature on information flows between markets typically examine causality in
the mean relationship between data on different markets. However, there is a growing literature on the relationship of conditional variances across financial markets and its im-
Table 1.1. Characteristics of the CCF Approach (1)
This approach can analyze the causality not only in mean but also in variance.
This approach does not involve simultaneous modeling and is therefore relatively easy to implement.
This approach is useful when the number of series under investigation is large and long lags are expected in the causation pattern.
This approach has a well-defined asymptotic distribution and its asymptotic behavior does not depend on the normality assumption.
This approach provides information on the timing of causation in contrast to the standard causality test.
Since this approach depends on the residuals estimated from the univariate model, we do not have to worry about the omission-of-variables problem.
plications concerning information transmission mechanisms. As clearly demonstrated by Ross (1989), return volatility also provides useful data on information flow. Thus, data on return volatility in
financial markets can provide information in addition to that available in the return data alone. This development suggests that price volatility has significant implications concerning information
linkages between markets. This volume attempts to make an empirical contribution to the literature on the movements of stock prices in major economies, Le. Germany, Japan, the UK and the USA.
Specifically, the cross-correlation function (CCF) approach, developed by Cheung and Ng (1996) is used to analyze the stock market. As stressed by Cheung and Ng (1996), there are several merits in
using what is known as the CCF approach (Table 1.1). First, this approach can analyze the causality not only in mean but also in variance. Second, compared with a multivariate method, the CCF
approach does not involve simultaneous modeling and is therefore relatively easy to implement. Third, the CCF test is especially useful when the number of series under investigation is large and long
lags are expected in the causation pattern. Fourth, the CCF approach has a well-defined asymptotic distribution and its asymptotic behavior does not depend on the normality assumption. Fifth, this
method also provides information on the timing of causation, in contrast to the conventional causality test. Finally, since this method depends on the residuals estimated from the univariate model,
we do not have to worry about the omission-of-
Table 1.2. Relation of Each Chapter Content Interdependence of stock prices
Traditional Approach
CCF Approach
Chapter 2
Chapter 3
Stock prices and exchange rates
Chapter 4
Stock prices, exchange rates and real economic activities
Chapter 5
variables problem in a multivariate method. It is possible to analyze the causality between any set of variables. Considering its merits, it would seem that not enough research has applied this
approach. 1 It is hoped that this study of the temporal precedence in stock market returns and volatilities can provide some empirical evidence regarding the economic linkages among a group of
different countries. The content is organized as follows. Chapter 2 and Chapter 3 analyze the international linkage of stock prices among Germany, Japan, the UK and the USA. Chapter 2 applies the
standard approach, whereas Chapter 3 uses the CCF approach. Chapter 4 analyzes the relationship between stock prices and exchange rates. Chapter 5 analyzes the relationship among stock prices,
exchange rates, and real economic activities. The basic framework of this volume can therefore be summarized as in Table 1.2. Chapter 2 empirically analyzes the interdependence of stock prices in
Germany, Japan, the UK and the USA from 1980 to 2001. The market liberalization that occurred throughout the 1980's and 1990's might have affected the international linkage of stock returns among
countries. The empirical technique used in this chapter is the traditional timeseries approach, or, more specifically, the vector autoregression (VAR) model and the lag-augmented vector
autoregression (LA-VAR) method. This approach is particularly convenient for estimation and forecasting. The popularity of this approach for analyzing the dynamics of economic systems is due to the
influential work by Sims (1980). Sims (1980) developed VAR models for Germany and the USA, and showed the usefulness of this approach for macroeconomic analysis. The VAR modeling technique is an
effective means of characterizing the dynamic interactions among economic variables, as it reduces dependence on the potentially inappropriate theoretical restrictions of structural models. Using
VAR and LA-VAR models, this chapter analyzes the international linkage of stock markets. In Chapter 2, stock prices across Germany, Japan, the UK and the USA are analyzed using the traditional
time-series approach. However, there is a growing literature on the relationship of conditional variances across financial markets and its implications concerning the information transmission
mechanism. As demonstrated by Ross (1989), return volatility also provides useful data on information flow. Thus, data on return volatility in financial markets can provide information in addition to
that available in the return data alone. Chapter 3 extends the analysis to examine the transmission mechanism of the conditional first and second moments in stock prices across international stock
markets allowing for changing conditional variances as well as conditional mean returns. As is analyzed in Chapters 2 and 3, stock markets have developed interdependently and are consequently linked
together internationally. This capital inflow-outflow may affect exchange rate movements. Stock prices and foreign exchange rates play crucial roles in influencing the development of a country's
economy, and thus the dynamic relationships between stock prices and foreign exchange rates have frequently been used in predicting the future trends for each of these figures by investors. Since the
values of financial assets are determined by the present values of their cash flows, expectations of relative currency values play a considerable role in influencing their price movements especially
for internationally held financial assets. Therefore, stock price innovations may affect or be affected by exchange rate dynamics. In Chapter 4, the relationship between foreign exchange rates and
stock prices is analyzed for Germany, Japan, the UK and the USA by using the CCF approach. The relationships among real economic variables, monetary variables and financial variables have been topics
of active research by economists for a long time. Asset prices are generally forward looking and they reflect future economic activities. Thus, they become a leading indicator of future economic
activities. Chapter 5 analyzes two kinds of lead-lag relationships; one is the relationship between the stock price index and real economic activity, and the other is the relationship between the
effective exchange rate and real economic activity. The causality in mean and in variance among variables is empirically analyzed using the CCF approach. In using this causality methodology, we wish
not only to investigate empirically the relationships among variables but· also to analyze the popular hypothesis in the financial press, which claims that asset return is a leading indicator of
future economic activity.
Chapter 6 summarizes the main results obtained in each chapter and comments on the possible directions of future research. I hope that this book will appeal to students, academics, business people,
market participants and all those with an interest in stock markets.
Notes 1 As indicated by Cheung and Ng (1996), the CCF approach also has some limitations. This approach is designed to detect neither causation patterns that yield zero cross-correlations, nor
non-linear causation patterns.
Chapter 2
Market deregulation and the free flow of capital have increased the globalization of financial markets and affected the nature of relationships among equity price movements across markets. The major
world economies are now interdependent through trade and investment. Therefore, news about economic fundamentals in one country has implications for equity prices in other countries. Karolyi and
Stulz (1996, p. 951) pointed out that increased international correlation of stock returns has the following four effects: (a) fewer domestic risks are internationally diversifiable, so portfolio
volatility increases; (b) the risk premium on the world market portfolio increases; (c) the cost of capital increases for individual firms; and (d) the domestic version of the capital asset pricing
model (CAPM) becomes increasingly inadequate. This chapter empirically analyzes the interdependence of stock prices in Germany, Japan, the UK, and the USA. The empirical technique used in this
chapter is the traditional time-series approach, or, more specifically, the vector autoregression (VAR) model and the lag-augmented vector autoregression (LA-VAR) method. This approach is
particularly convenient for estimation and forecasting. The popularity of this approach for analyzing the dynamics of economic systems is due to the influential work by Sims (1980). Sims (1980)
developed VAH. models for Germany and the USA, and showed the usefulness of this approach for macroeconomic analysis. The VAR modeling technique is an effective means of characterizing the dynamic
interactions among economic variables, as it reduces dependence on the potentially inappropriate theoretical restrictions of structural models.
In economic time-series analyses, pretests for a unit root and for cointegration are usually required before estimating the model. Engle and Granger (1987) showed that if two non-stationary variables
are cointegrated, a vector autoregression in the first difference is misspecified. For example, if two variables are both non-stationary in levels, but the first difference of each variable is
stationary, then it is said that the two variables are integrated of order one. Furthermore, if any linear combination of the two variables is stationary, the two variables are said to be
cointegrated. Because the presence of cointegration can cause Granger's causality tests to be misspecified, it is necessary to test for integration and cointegration before running the causality
tests. If cointegration is found, an error-correction model must be constructed. Thus, the VAR model in first-order differences is used if the variables are integrated of order one and there is no
cointegration between variables, while a vector error-correction model (VECM) is used if the variables are integrated of order one and are cointegrated with each other. Nevertheless, the standard
approach to testing economic hypotheses conditioned on the estimation of a unit root, a cointegrating rank, and cointegration vectors may suffer from severe pretest biases. l Toda and Yamamoto (1995)
developed the LA-VAR to overcome this problem. Their approach is appealing, because it is applicable regardless of whether the VAR process is stationary, integrated, or cointegrated. The LA-VAR
method is, however, inefficient in terms of power and should not totally replace conventional hypothesis testing methods, which are conditional on the estimation of unit roots and cointegration
tests. This chapter applies these two approaches and demonstrates the robustness of their empirical results.
Selected Literature Review
Many papers have analyzed the international linkage of stock markets. Some examples include Agman (1972), Eun and Shin (1989), Kasa (1992), Corhay, Rad, and Urbain (1993), Arshanapalli and Doukas
(1993), Chowdhury (1994), Karolyi and Stulz (1996), Engsted and Lund (1997), Hamori and Imamura (2000), and Ahlgren and Antell (2002). Agman (1972) used monthly data from Germany, Japan, the UK, and
the USA from July 1961 to January 1966, and analyzed whether the behavior of stock prices in the four countries was consistent with the hypothesis that these markets constitute four regions in a
perfect single market (single market hypothesis). He claimed that his tests did not allow for the rejection of this hypothesis. 2 Eun and Shin (1989) used daily data from Australia, Canada, France,
Germany, Hong Kong, Japan, Switzerland, the UK, and the USA from
Table f.l.
Summary of Literature Data
Technique and Main Results
July 1961 - January 1966 (monthly data) Germany, Japan, UK, and USA
Regression; The single market hypothesis is statistically supported.
December 31, 1979 - December 20, 1985 (daily data) Australia, Canada, France, Germany, Hong Kong, Japan, Switzerland, UK, and USA
VARj Shocks in the USA are rapidly transmitted to other markets, whereas no single foreign market can significantly explain the US market movements.
January 1974 - August 1990 (monthly and quarterly data) Canada, Germany, Japan, UK, and USA
VECMj There is one common stochastic trend driving countries' stock markets.
March 1, 1975 September 30, 1991 (biweekly data) France, Germany, Italy, Netherlands, and UK
VECMj There are some common stochastic trends among five European stock markets.
January 1980 - May 1990 (daily data) France, Germany, Japan, UK, and USA
VECMj For the post-October 1987 period, the degree of international co-movements among stock price indices has increased substantially
January 2, 1986 - December 30, 1990 (daily data) Japan, Korea, Taiwan, Singapore, Hong Kong, and USA
VARj A significant link exists between the stock markets of Hong Kong and Singapore and those of Japan and the USA.
Karolyi and Stulz
May 31,1988 - May 29, 1992 (daily data) Japan
and USA
Regressionj Large shocks to broadbased market indices positively impact both the magnitude and persistence of the return correlations.
Agman (1972)
Eun Shin
Kasa (1992)
Rad, and Urbain (1993)
Arshanapalli and Doukas (1993)
Chowdhury (1994)
Table £.1 (continued) Summary of Literature Authors
Technique and Main Results
Engsted and Lund (1997)
1950 - 1988 (annual data) Denmark, Germany, Sweden, and UK
VECMj There are common stochastic trends among dividends in the four countries.
Hamori and Imamura (2000)
December 1969 - May 1995 (monthly data) G7 countries
LA-VARj The causal relationship between US stock markets and those in the rest of the world is significant.
Ahlgren and Antell (2002)
January 1980 - February 1997 (monthly and quarterly data) Finland, France, Germany, Sweden, UK, and USA
VECMj International stock prices are not cointegrated.
December 31, 1979 to December 20, 1985. They investigated the international transmission of stock market movements by estimating a nine-market VAR system. They found that shocks in the US stock
market were rapidly transmitted to the rest of the world, although shocks in other national markets did not have much effect on the US market. Kasa (1992) presented evidence concerning a number of
common stochastic trends in the equity markets of Canada, Germany, Japan, the UK, and the USA. Monthly and quarterly data from January 1974 through August 1990 were used to carry out cointegration
tests for common trends. The cointegration test was used to analyze the long-run co-movements in these five stock markets. The results indicated the presence of a single common trend driving these
countries' stock markets. Estimates of the loading factors suggested that this trend was most important in the Japanese market and least important in the Canadian market. Corhay, Rad, and Urbain
(1993) investigated whether European stock markets displayed a cominon long-run trend in behavior. They used biweekly data from France, Germany, Italy, the Netherlands, and the UK, collected between
March 1, 1975, and September 30, 1991. A cointegration test was used for empirical analysis. They found some common stochastic trends among the five countries. The results obtained by Kasa (1992) and
Corhay, Rad, and Urbain (1993) showed that stock prices were cointegrated and that, thus, world
Chapter 2
stock markets were driven, in part, by one or more common stochastic trends. The existence of a common trend can be interpreted as a natural consequence of well-functioning and integrated capital
markets characterized by free accessibility for both domestic and foreign investors. Arshanapalli and Doukas (1993) used daily data from France, Germany, Japan, the UK, and the USA over the period
from the beginning of January 1980 to the end of May 1990. Using the VECM, they found that the degree of international co-movements in stock price indices changed significantly after the crash of
October 1987. In the pre-crash period, the French, German, and UK stock markets were not related to the US stock market, whereas in the post-crash period the three markets were strongly linked to the
US market. Moreover, the US stock market had a substantial impact on the French, German, and UK markets in the post-crash period, whereas stock market innovations in any of the three European stock
markets had no impact on that of the USA. Finally, there was no evidence of interdependence between the stock price indices of the USA and Japan. A similar result was obtained between Japan and the
three European stock markets. Chowdhury (1994) analyzed the relationship among the stock markets of four newly industrialized economies (NIEs) in Asia: Hong Kong, Korea, Singapore, and Taiwan. A
six-variable VAR model, including Japan and the USA, was estimated using daily rates of return on the stock market indices from January 2, 1986 to December 30, 1990. The results indicated that a
significant link existed between the stock markets of Hong Kong and Singapore and those of Japan and the USA. On the other hand, the markets with severe restrictions on cross-country investing, such
as those of Korea and Taiwan, were not responsive to innovations in foreign markets. Finally, the US stock market influenced, but was not influenced by, the four Asian markets. Karolyi and Stulz
(1996) investigated daily return co-movements between Japanese and US stocks using data from May 31, 1988 to May 29, 1992. They found that US macroeconomic announcements, shocks to the yen/dollar
foreign exchange rate, and Treasury bill returns had no measurable influence on US and Japanese return correlations. However, large shocks to broad-based market indices (Nikkei Stock Average and
Standard and Poor's 500 Stock Index) positively impacted both the magnitude and persistence of the return correlations. Engsted and Lund (1997) used annual data from 1950 to 1988 for Denmark,
Germany, Sweden, and the UK. Their empirical results, based on the VECM, indicated the presence of common trends among dividends in these four countries.
Hamori and Imamura (2000) analyzed the interdependence among stock prices in G-7 countries using the LA-VAR method. Monthly data for the period from December 1969 to May 1995 were used, and stock
prices were analyzed, not only in local currencies but also in US dollars. They showed that the causal relationship between US stock prices and those in the rest of the world was significant. This
may be because of the dominant influence of the US economy on the world market. Ahlgren and Antell (2002) re-examined the evidence for cointegration among international stock prices. They applied the
cointegration test to stock price data from Finland, France, Germany, Sweden, the UK, and the USA. They used both monthly and quarterly data from January 1980 to February 1997. Using small-sample
corrections, they found no evidence for cointegration. They pointed out that the cointegration test was sensitive to the lag length specification in the VAR model, and that some previous empirical
results, such as those of Kasa (1992) and Corhay, Rad, and Urbain (1993), can be explained by the small-sample bias and size distortion of the cointegration test. These studies tested for possible
international stock market linkages. This research generally pointed out the importance of the interdependence of stock markets and the dominant effects of the US market, but some found conflicting
evidence (Table 2.1).
Monthly data of stock prices for Germany, Japan, the UK, and the USA are used for empirical analysis. The sample period is January 1980 through May 2001. The source is the International Financial
Statistics of the International Monetary Fund. The rate of return on stocks is calculated as R t = (In St - In St- d x 100, where St is the stock price at time t. Thus, stock returns are obtained for
the period between February 1980 and May 2001. 3 Table 2.2 shows the summary statistics for the rate of return of each country. This table shows the mean, standard deviation (Std. Dev.), skewness,
kurtosis, and Jarque-Bera statistic with its associated probability value (P-value). The mean is the average value of the series. Standard deviation is a measure of dispersion in the series. Skewness
is a measure of asymmetry of the distribution around its mean. The skewness of a symmetric distribution, like the normal distribution, is zero. Positive (negative) skewness means that the
distribution has a long right (left) tail. Kurtosis is a measure of peakedness or flatness of the distribution of the series. The kurtosis of the normal distribution is three. If the kurtosis is
greater than (less than) three, the distribution is peaked or leptokurtic (flat or platykurtic) relative to the normal distri-
Chapter 2
Table 2.2.
Summary Statistics
Mean (%) Std. Dev. Skewness Kurtosis Jarque-Bera P-value
0.773 5.331 -0.882 5.888 122.137 0.000
0.425 4.337 -0.191 3.671 6.356 0.042
0.929 3.859 -1.267 9.844 568.094 0.000
0.964 3.550 -0.685 5.807 104.057 0.000
Note: Jarque-Bera is the Jarque-Bera statistic to test for normality. P-value is the probability value associated with the Jarque-Bera statistic. The null hypothesis of normal distribution is
rejected at the 5 percent significance level if the P-value for the Jarque-Bera test is less than 0.05.
bution. The Jarque-Bera statistic is used for testing whether the series is normally distributed or not (Jarque and Bera 1987). This measures the difference of the skewness and kurtosis of the series
with those from the normal distribution. The reported P-value is the probability that a Jarque-Bera statistic exceeds (in absolute value) the observed value under the null hypothesis. Thus, a small
P-value leads to the rejection of the null hypothesis of a normal distribution. The mean rate of return is 0.773 percent for Germany, 0.425 percent for Japan, 0.929 percent for the UK, and 0.964
percent for the USA, whereas the standard deviation is 5.331 percent for Germany, 4.337 percent for Japan, 3.859 percent for the UK, and 3.550 percent for the USA. Thus, the USA has the highest mean
return and the lowest standard deviation. The skewness is -0.882 for Germany, -0.191 for Japan, -1.267 for the UK, and -0.685 for the USA. The kurtosis is 5.888 for Germany, 3.671 for Japan, 9.844
for the UK, and 5.807 for the USA. The JarqueBera statistic (P-value) is 122.137 (0.000) for Germany, 6.356 (0.042) for Japan, 568.094 (0.000) for the UK, and 104.057 (0.000) for the USA. Thus, the
null hypothesis of normal distribution is statistically rejected for all countries at the 5 percent significance level.
Empirical Technique: VAR
This section empirically analyzes the interdependence of stock prices in Germany, Japan, the UK, and the USA from 1980 through 2001 using the traditional time-series approach. Suppose that an
n-dimensional column vector Yt is generated by the following model: Yt = 10 + lIt + JIYt-1
+ ... + JkYt-k + Et,
t = 1,2, ..., T
where t is the time trend, k is the order of lag length, Et is the vector of error terms with mean zero and variance-covariance matrix EE , and io, ill J 1 ,··., Jk are vectors (matrices) of
parameters. In the traditional time-series approach, pretests for a unit root and cointegration are usually required before estimating the model. If the variables are integrated of order one, 1 (1),
and are cointegrated with each other, the vector error-correction model (VECM) is used for empirical analysis. A Yt vector is said to be cointegrated if each of its elements individually is 1(1) and
if there exists a nonzero (n x 1) vector a such that a'Yt is stationary. When this is the case, a is called a cointegrating vector. The system is said to be in long-run equilibrium if a'Yt = O. The
deviation from long-run equilibrium is called the equilibrium error. Then, a representation for a cointegrated system is obtained as follows: ~Yt = io+il t + II Yt-l +IIl~Yt-l + .. ·+IIk-l~Yt-k+l+Et
where ~ is a difference operator (~Yt = Yt-Yt-l), II = -(1- Ef=l Ji), and II s = -(Js+l + J S +2 + ... + Jk), (8 = 1,2"", k - 1). The third term IIYt-l on the right hand side is the error-correction
term. Equation (2.2) is known as the error-correction representation of the cointegrated system. Any equilibrium relationship among a set of non-stationary variables implies that their stochastic
trends must be linked. The equilibrium relationship means that the variables cannot move independently of each other. This linkage among the stochastic trends necessitates that the variables be
cointegrated. Since the trends of cointegrated variables are linked, the dynamic paths of such variables must bear some relation to the current deviation from the equilibrium relationship. This
connection between the change in variables and the deviation from equilibrium is expressed in the error-correction representation in equation (2.2). If the variables are integrated of order one and
there is no cointegrating relation among variables, the VAR model in first-order differences is used as follows: ~Yt = il
+ Jl~Yt-l + ... +
+ Ut
where Ut = Et - Et-l' Equation (2.3) shows that there is no errorcorrection representation and thus ~Yt does not respond to the previous period's deviation from long-run equilibrium. 4 The unit root
test developed by Phillips and Perron (1988) is applied to the log of each index to carry out the test that the stock price index has a unit root. s The unit root test statistic is the t-value of'Y
obtained from the following regressions: (CT)
llYt =
+ at + 'YYt-l + Ut,
Chapter 2
Table 2.9. Unit Root Test
Test Statistic CT
-2.373 -1.323 -2.110 -2.588
-0.953 -2.105 -1.955 -0.329
2.002 0.979 3.175 3.580
-14.839** -11.227** -12.948** -11.771**
-14.867** -11.123** -12.899** -11.794**
-14.652** -11.074** -12.429** -11.291**
Level Germany Japan
First Difference Germany Japan
Note: • shows that the null hypothesis of a unit root is rejected at the 5 percent significance level. •• shows that the null hypothesis of a unit root is rejected at the 1 percent significance
level. CTcorresponds to the following regression: AYt 1'+6t+"YYt-l +Ut. C corresponds to the following regression: AYt 1'+"YYt-l +Ut. None corresponds to the following regression: AYt = "YYt-l + Ut.
+ 'YYt-l + Ut, + Ut,
where t is the time trend and Ut is a disturbance term. The first equation (CT) includes a constant term and a time trend, the second equation (C) includes a constant term only, and the third
equation (None) includes no deterministic term. The null hypothesis and the alternative hypothesis are respectively shown as follows:
= 0, < O.
In short, the null hypothesis shows that the unit root is included, while the alternative hypothesis indicates that the unit root is not included. In Table 2.3, each equation is applied to both the
level and the first difference of stock price indices. As an example, let us see the results of Germany. For the level of the stock price index, test statistics are -2.373 for CT, -0.953 for C, and
2.002 for None. For the first difference of the
6.0 ......- - - - - - - - - - - - - - - - - - - - . 5.6
5.2 4.8 4.4
4.0 3.6 80
Figure 2.1.
Logarithmic Stock Price Index: Germany
5.6 ......- - - - - - - - - - - - - - - - - - - - - - .
5.2 4.8 4.4 4.0 3.6 3.2 ..... 80
.....-.....-....-.....-.............-..-.....,........--r-..,..-.,--,r--T--r--r--r-.,... 82 84 86 88 90 92 94 96 98 00
Figure 2.2.
Logarithmic Stock Price Index: Japan
5.5-r---------------------, 5.0
4.5 4.0 3.5
3.0 2.5 -t-........,..-,.....-,......,.--r--r--,--r--r--r--r-........,..-,.....-,......,.--r--r--,--r-I 80 82 84 86 88 90 92 94 96 98 00
Figure !.3. Logarithmic Stock Price Index: UK
6.0-r----------------------. 5.5 5.0
4.5 4.0 3.5
3.0 2.5 ;--r-.........,.......,......,..-,.....-r---1r-"'"1:--"1--r--r--,---r-r-....,..-r--r--r--r--r-' 80 82 84 86 88 90 92 94 96 98 00
Figure e.4. Logarithmic Stock Price Index: USA
.4~------------------...., .3 .2 .1 .0~,N.lflJY VNIN, lIl~jM
-.1 -.2
-.3 -.4 .......~~..,.....,.....,.....,...-r--r---.---.-....,.......,.......,.......,...-....-....-....-....-....--.---.-I 80 82 84 86 88 90 92 94 96 98 00
Figure 2.5.
First Difference of the Logarithmic Stock Price Index: Germany
.4-r-------------------------. .3
.2 .1 .0 loJlIUUll..a h~l ~1A.i·
-.1 -.2
-.3 -.4 -f-~_r__r_.,........T""""""'1r___lr__1__,~~_r"""T"_r_r_'""'T"""_r__r_"T""'".,.__'T"'
Figure 2.6.
First Difference of the Logarithmic Stock Price Index: Japan
Chapter 2
.4..,.---------------------, .3
.2 .1 .0
IMIMII~J1It'11Jm.·UI"II'I... ·
-.2 -.3
e. 7.
First Difference of the Logarithmic Stock Price Index: UK
.4-.------------------------, .3 .2
.1 .0 -.1
-.2 -.3
-.4 80
Figure e.8.
First Difference of the Logarithmic Stock Price Index: USA
stock price index, test statistics are -14.839 for CT, -14.867 for C, and -14.652 for None. Thus, the null hypothesis of a unit root is accepted in all specifications for the level of stock prices.
The hull hypothesis is, however, rejected for all specifications for the first difference of stock prices. These results are robust to all countries. Thus each stock price index is found to be 1(1)
process. 6 The data are plotted in levels and first differences in Figure 2.1 through Figure 2.8. All series appear non-stationary with stationary first differences. Thus, the cointegration test
developed by Johansen (1988) and Johansen and Juselius (1990) is applied to the pair of the log of stock price indices. Since the results of the test can be sensitive to the lag length, it is
important to be careful in choosing a model. The common procedure is to estimate a vector autoregressive using the undifferenced data. 7 Then, use the Schwarz Bayesian information criterion (SBIC) to
select the lag length (Schwarz 1978).8 SBIC is often used to select the appropriate model, and smaller values of SBIC are preferred. As is clear from Table 2.4, the lag length (k) is chosen to be two
for all pairs. Table 2.5 shows the results of diagnostic testing for Table 2.4. In this table, LM(12) is the Lagrange multiplier (LM) test statistic for the null hypothesis that there is no
autocorrelation up to order 12. As is clear from this table, there is no autocorrelation for all pairs, and thus the specification in Table 2.4 is empirically supported. Table 2.6 shows the results
of the cointegration test. In this table, two test statistics are reported. One is the trace test statistic (Atrace) and the other is the maximum eigen-value test statistic (Amax). Critical values
are tabulated in Osterwald-Lenum (1992). In the case of a combination of Germany and Japan for an example, the test statistics are 4.752 (Atrace) and 4.408 (A max ) for the null hypothesis of no
cointegration, and they are smaller than the corresponding 5 percent critical value (15.41 for Atrace and 14.07 for Amax). Thus, the null hypothesis of no cointegration is statistically accepted at
the 5 percent significance level. For other combinations similar results are obtained. Therefore, the standard VAR shown in equation (2.3) is estimated and causality among variables is empirically
analyzed. The causal tests of Granger (1969) are essentially tests of the predictive ability of time-series models. Consider the following bivariate VAR model. D.Y1,t = D.Y2,t =
+ J ll (I)D.Y1,t-1 + ... + Jll(k)D.Y1,t-k +J12(1)D.Y2,t-1 + ... + J12(k)D.Y2,t-k + Ult 12 + J21(1)D.Y1,t-l + ... + J2t{k)D.Y1,t-k
+ ... + J22(k)D.Y2,t-k + U2t
Chapter 2
Table 2.4.
Lag (k)
Germany and Japan
Germany and
Germany and
Japan and UK
Japan and
-6.351 -6.381* -6.300 -6.221 -6.159 -6.103 -6.024 -5.964 -5.889 -5.821 -5.760 -5.682
-6.711 -6.739* -6.710 -6.648 -6.595 -6.525 -6.442 -6.369 -6.337 -6.279 -6.204 -6.132
-6.968 -7.000* -6.919 -6.837 -6.762 -6.695 -6.605 -6.518 -6.455 -6.375 -6.323 -6.255
-7.161 -7.211* -7.147 -7.065 -7.009 -6.935 -6.866 -6.786 -6.722 -6.643 -6.556 -6.469
-7.341 -7.463* -7.390 -7.307 -7.249 -7.172 -7.100 -7.025 -6.939 -6.867 -6.780 -6.701
UK and
USA -7.909 -7.960* -7.905 -7.830 -7.755 -7.678 -7.610 -7.534 -7.471 -7.385 -7.303 -7.232
Note: ... indicates the lag order selected by SBIC.
Table 2.5.
LM(12) P-value
UK and
Germany and Japan
Germany and UK
Germany and
Japan and
Japan and
2.799 0.592
1.223 0.874
1.843 0.765
2.906 0.574
3.872 0.424
3.276 0.513
Note: LM(I2) is the Lagrange multiplier test statistic for the null hypothesis that there is no autocorrelation up to order 12. The null hypothesis of no autocorrelation is rejected at the 5 percent
significance level if the P-value is less than 0.05.
The question investigated here is whether a variable 6.Yj,t can help forecast another variable 6.Yi,t (i =/: j). The definition of causality implies that 6.'Y2 is said to cause 6.Yl if some of J12(k)
(j = 1,2,,,,, k) are not equal to zero in equation (2.4). Similarly, 6.Yl is said to cause 6.Y2 if some of J21(k) (j = 1,2"" ,k) are not equal to zero in equation (2.5). If both events occur, there
is a feedback. 9
Table £.6. Cointegration Test
Germany and Japan Germany and UK Germany and USA
Null Hypothesis
r=O r=1
4.752 0.343
4.408 0.343
r=O r=1
8.363 2.599
5.764 2.599
5.686 0.072
5.613 0.072
5.184 0.329
4.855 0.329
4.851 0.012
4.839 0.012
9.247 0.621
8.626 0.621
Japan and UK
Japan and USA
UK and USA
Note: Ho shows the null hypothesis. >'t~ace is the trace statistic. >'ma.. is the maximum eigenvalue statistic. r is the number of cointegrating vector. • shows that the null hypothesis Is rejected
at the 5 percent significance level. Five percent critical value of >'t~ac. for the null hypothesis of r 0 and r 1 is respectively given by 15.41 and 3.76. Five percent critical value of >'mu for the
null hypothesis of r 0 and r 1 is respectively given by 14.07 and 3.76.
Thus, the null hypothesis that a country j does not cause a country i j) is shown as follows:
Ho: Jij(1)
= Jij(2) =... = Jij(k) = 0
The lag length of VAR is chosen as two for all combinations. The results are shown in Table 2.7. As is clear from this table, the P-value of the test for the null hypothesis that Germany does not
cause the UK is 0.000, and thus Germany causes the UK. Similarly, the Pvalue of the test for the null hypothesis that Germany does not cause the USA is 0.031, and thus there is evidence that Germany
causes the USA. It is also found that the P-value of the test for the null hypothesis· that the USA does not cause the UK is 0.000, and thus the USA causes the UK. It is interesting to see that Japan
is independent of other three countries.
Chapter 2
Table 2.7. Causality Test: VAR
Test Statistics
0.525 0.071
0.769 0.965
19.488 3.469
0.000 0.177
6.979 3.436
0.031 0.179
5.175 1.286
0.075 0.526
Japan does not cause the USA. The USA does not cause Japan.
1.873 1.071
0.392 0.585
UK and USA The UK does not cause the USA. The USA does not cause the UK.
0.272 17.790
0.873 0.000
Null Hypothesis Germany and Japan Germany does not cause Japan. Japan does not cause Germany. Germany and UK Germany does not cause the UK. The UK does not cause Germany. Germany and USA Germany does
not cause the USA. The USA does not cause Germany. Japan and UK Japan does not cause the UK. The UK does not cause Japan. Japan and USA
Note: The null hypothesis of no causality is rejected at the 5 percent significance level if the P-value is less than 0.05.
Empirical Technique: LA-VAR
The standard approach to testing economic hypotheses conditioned on the estimation of a unit root and a cointegrating rank may suffer from severe pretest bias. Toda and Yamamoto (1995) developed
LA-VAR to overcome this problem. 10 Suppose an n-dimensional column vector 'lit is generated by equation (2.1). The following hypothesis test is considered:
Ho : /( 0, the GJR model generates higher values for given €t-i < 0, than for a positive shock of equal magnitude. As with the ARCH and GARCH models the parameters of the conditional variance are
subject to non-negativity constraints. As a special case, the GJR(I,I) model is given as follows:
(1; =
+ (a + 'YDt-d€l-1 + .8(1;-1'
Equation (3.5) becomes
(1; = W + a€l-1 + .8(1;-1 > 0),
for a positive shock (€t-l
(1l =
whereas equation (3.5) becomes
+ (a + 'Y)€;-1 + .8(1l-1 < 0). Thus, the presence of a leverage effect
for a negative shock (€t-l can be tested by the hypothesis that 'Y = O. The impact is asymmetric if'Y :f: O. An alternative way of describing the asymmetry in variance is through the use of the
EGARCH (exponential GARCH) model proposed by Nelson (1991). The EGARCH(p, q) model is given by p
log (1l =
(ai IZt-i I + 'YiZt-i)
+ L.8i log (1l-i i=1
where Zt = €t!(1t. Note that the left-hand side of equation (3.6) is the log of the conditional variance. The log form of the EGARCH(P' q) model
Chapter :1
Table 3.1.
ARCH-type Models
Engle (1982)
ARCH model Ut2 -_ W
Bollerslev (1986)
GARCH model Ut2 -_ W
Taylor (1986) and Schwert (1989) Glosten, Jagannathan and Runkle (1993)
Table 3.2.
(1) (2) (3) (4)
+ "" 2 + "LJi=l 9 f.'iUt-i /.I 2 LJi=lOift-i
absolute error model 2 _ W + "" 9 /.I. 2 . LJi=lO.., ft-.., + " LJi=l f.'.Ut-.
Ut -
GJR model Ut2 -_ W
where Nelson (1991)
+ "" 2 LJi=lOift-i
+ "" 2 .+" 9 /.I. 2 . LJi=l (O.. + 'Y..Dt-..) ft_. LJi=l f.'.Ut-.
= 0 for ft-i > 0 and 1 for ft-i < 0
EGARCH model logul = W + Ef=l (oiIZt-il + 'YiZt-i) + E~=l fJi loguL where Zt = ft/Ut
Characteristics of the EGARCH Model
Since the log value of volatility is used as an explained variable, there is no need to impose non-negative constraint on parameters of variance dynamics. The EGARCH model takes into consideration
the asymmetric effect of the volatility. Only the coefficients of the GARCH term govern the persistence of volatility shocks. The possibility of cyclical behavior in volatility is admitted.
ensures the non-negativity of the conditional variance without the need to constrain the coefficients of the model. The asymmetric effect of positive and negative shocks is represented by inclusion
of the term Zt-i. If "Ii > 0 volatility tends to rise (fall) when the lagged standardized shock, Zt-i = ft-i/Ut-i is positive (negative). The persistence of shocks to the conditional variance is
given by El=l f3i. Since negative coefficients are not precluded in the EGARCH model, the possibility of cyclical behavior in volatility is admitted. As a special case, the EGARCH(l,l) model is given
as follows: logu~ = w + alZt-ll
+ "IZt-l + f3logu~_l
For a positive shock (Zt-l
> 0), equation (3.7) becomes
logol = w + (a + 'Y)IZt-ll whereas for a negative shock (Zt-l log
+ /3logql_l'
< 0), equation (3.7) becomes
ql = w + (a - 'Y) IZt- II + ,B log q;-l .
Thus, the presence of a leverage effect can be tested by the hypothesis that 'Y = O. The impact is asymmetric if 'Y ::j: O. Furthermore, the sum of a and ,B governs the persistence of volatility
shocks for the GARCH (1,1) model, whereas only the parameter ,B governs the persistence of volatility shocks for the EGARCH(1,1) model. In summary, there are at least four merits of using the EGARCH
model. First of all, since the log value of volatility is used as an explained variable, there is no need to impose non-negative constraint on parameters of variance dynamics. Second, the EGARCH
model can take into consideration the asymmetric effect of the volatility. Third, only the coefficients of the GARCH term govern the persistence of volatility shocks. Finally, the possibility of
cyclical behavior in volatility is admitted. 9 Table 3.1 summarizes each model and Table 3.2 shows the characteristics of the EGARCH model.
Cheung-Ng Test
Cheung and Ng (1996) developed a testing procedure for causality in mean and variance. This test is based on the residual cross-correlation function (CCF) and is robust to distributional assumptions.
They developed a two-step procedure to test for causality in mean and variance (Figure 3.1). The first step involves the estimation of univariate timeseries models that allow for time variation in
both conditional means and conditional variances. In the second step, the residuals standardized by conditional variances and the squared residuals standardized by conditional variances are
constructed. The CCF of standardized residuals is used to test the null hypothesis of no causality in mean, whereas the CCF of squared-standardized residuals is used to test the null hypothesis of no
causality in variance (Cheung-Ng test). Following Cheung and Ng (1996) and Hong (2001), let us summarize the two-step procedure of testing causality. Suppose that there are two stationary
time-series, X t and l't. Let lit, 12t, and It be three information sets defined by lit = {Xt-jjj ~ O}, 12t = {l't-jjj ~ O}, and It = {Xt-j, Yi-j;j ~ O}. Y is said to cause X in mean if (3.8)
Chapter 9
Estimation of univariate time-series models that allow for time variation in both conditional means and conditional variances (e.g., AR-EGARCH model)
Standardized Residuals => Causality in Mean Squares of Standardized Residuals => Causality in Variance·
Figure 3.1. Two-Step Procedure
Similarly, X is said to cause Y in mean if
(3.9) Feedback in mean occurs if Y causes X in mean and X causes Y in mean. On the other hand, Y is said to cause X in variance if
(3.10) where J1.:J:,t is the mean of X t conditioned on to cause Y in variance if
Similarly, X is said (3.11)
where J.LlI,t is the mean of yt conditioned on 12t-l. Feedback in variance occurs if X causes Y in variance and Y causes X in variance. The causality in variance has its own interest because it is
directly related to volatility spillover across different assets or markets. The concept defined in equations (3.8) through (3.11) is too general to be empirically testable, and thus additional
structure is required to ma.1{e the general causality concept applicable in practice. Suppose X t and yt can be written as
+ Vh:J:,tft J.LlI,t + /hll,t(t
X t = J.L:J:,t
yt =
where {ft} and {(t} are two independent white noise processes with zero mean and unit variance. For the causality in mean test, we have the standardized innovation as follows: ft=
Xt - J-Lx,t
Y'·x,t yt - J-Ly,t Jhy,t
Since both ft and (t are unobservable, their estimates, it and (t, have to be used to test the hypothesis of no causality in mean. Then, the sample cross-correlation coefficient at lag k, rEdk), is
computed from the consistent estimates of the conditional mean and variance of X t and yt, and is given as follows: A
(k) _
cE,(k) -r=======
v!CEE (O)c"(0)
where cEdk) is the kth lag sample cross-covariance given by .
cEd k ) = T ~)it - i)((t-k":" (),
k = 0, ±1, ±2,···
and similarly CEE(O) and c,dO) are respectively defined as the sample variances of ft and (t. Causality in the mean of X t and yt can be tested by examining rE,(k), the univariate standardized
residual CCF. Under the condition ofregularity, it holds that
VT(1\dkd,'" ,1\dkm» ~ N(O,Im.) where kl,"', k m are m different integers, and ~ shows the convergence in distribution. We can test the null hypothesis of no causality in mean using this test
statistic. To test for a causal relationship at a specified lag k, we compute VTrEdk) with the standard normal distribution. If the test statistic is larger than the critical value of normal
distribution, then we reject the null hypothesis. For the causality in variance test, let 'Ut and Vt be the squares of the standardized innovations, given by (Xt - J-Lx,t)2
= (Yi -
hx,t !Jy,t)2 hy,t
= ft, = a.
Chapter 3
Since both Ut and Vt are unobservable, their estimates, Ut and Vt, have to be used to test the hypothesis of no causality in variance. Then, the sample cross-correlation coefficient at lag k, ruv(k),
is computed from the consistent estimates of the conditional mean and variance of X t and Yi, given as follows:
Tuv(k) =
Cuv(k) VCuu(O)cvv(O)
where cl.lv(k) is the kth lag sample cross-covariance given by
= ~ ~)Ut - ~)(Vt-k - ~),
= 0, ±1, ±2,'"
and similarly Cuu(O) and Cvv(O) are respectively defined as the sample variances of Ut and Vt. Causality in the variance of X t and Yi can be tested by examining the squared standardized residual
CCF, Tuv(k). Under the condition of regularity, it holds that
VT(TUV (kt}, "', Tuv(km)) ~ N(O, 1m ) where k1 , ..• , k m are m different integers, and ~ shows the convergence in distribution. We can test the null hypothesis of no causality in variance using
this test statistic. To test for a causal relationship at a specified lag k, we compute VTTuv(k) with the standard normal distribution. If the test statistic is larger than the critical value of
normal distribution, then we reject the null hypothesis.
Monthly data of stock prices for Germany, Japan, the UK and the USA are used for empirical analysis. This is the same data set as was used in the last chapter. The sample period is January 1980
through May 2001. The source is the International Financial Statistics of the International Monetary Fund. The first difference of the log value is used for empirical analysis as follows: Yt = In St
- In St-l, where St is the stock price index at time t. Thus, the data for the period between February 1980 and May 2001 is actually used for empirical analysis.
4. 4.1
Empirical Results AR-EGARCH Model
A two-step procedure proposed by Cheung and Ng (1996) is employed to analyze the mean and variance causal relationships across markets.
The first step involves the estimation of univariate time-series models that allow for time variation in both conditional means and conditional variances. The AR(k)-EGARCH(p, q) specification: is
used for the first step. The AR-EGARCH model is used to model the stock return dynamics because of its success in financial literature. The conditional mean and conditional variance are respectively
specified as follows: k
Yt = 1l"o
+ L 1l"iYt-i + ft
logo} = w +
L (ailzt-il + ')'iZt-i) + 2:,oi logul_i
where Zt = ft/Ut. Equations (3.12) and (3.13) show AR(k) and EGARCH(p, q) models, respectively. Each model is estimated by the method of maximum likelihood. Parameter estimates and their asymptotic
standard. errors, which are robust to departures from normality using the consistent variance-covariance estimator of Bollerslev and Wooldridge (1992), are reported. 10 Though the CCF test results
presented in the next section are robust to distributional assumptions, inferences about the EGARCH parameter estimates may be sensitive to deviations from normality. Hence the Bollerslev-Wooldridge
standard errors are reported. The Schwarz Bayesian information criterion (SBIC) is used to specify the model (Schwarz 1978). SBIC is often used for model selection and smaller values of SBIC are
preferred. l l The choice of k, p and q is carried out among k = 1,2,,,,12, p=l, 2 and q=l,2 using SBle and residual diagnostics. As a result, the AR(l)-EGARCH(l,l) model is chosen for Japan and the
USA, the AR(1)-EGARCH(2,1) model is chosen for Germany and the AR(2)-EGARCH(2,1) model is chosen for the UK. Table 3.3 shows the empirical results of the AR-EGARCH model. As the table indicates, the
coefficient of the GARCH term (fJ) is estimated to be 0.985 for Germany, 0.908 for Japan, 0.976 for the UK and 0.665 for the USA, and they are statistically significant at the 1 percent level. Thus,
the persistence to volatility shock is relatively high for Germany, Japan and the UK, but is relatively low for the USA. The coefficients of the asymmetric effect (')') are estimated to be -0.098 and
0.279 for Germany, -0.085 for Japan, -0.120 and 0.227 for the UK and -0.304 for the USA. Note that this asymmetric parameter is not statistically significant for Japan. Table 3.3 also shows the
diagnostics of the empirical results of the AR-EGARCH model. The Ljung-Box test statistic Q(s), developed by
Chapter 9
Table 3.3.
Empirical results of the AR-EGARCH model
Mean equation: tit 11'0 + 1I',tlt-, + ft Variance equation: logO'l w + E~=l (o,IZt-,1 Zt ftlO'I
+ 1',Zt-,) + E~=l {3i logO'l_i,
0.008** (0.003) 0.020 (0.059)
0.004 (0.002) 0.356** (0.065)
0.010** (0.002) 0.177** (0.066) -0.172** (0.041)
0.007** (0.002) 0.261** (0.069)
w SE(w) 01 SE(ot} 1'1 SE(-Yt} 02 SE(02) 1'2 SE(-Y2) {3 SE({3)
-0.091 (0.084) 0.294* (0.125) -0.098 (0.087) -0.302* (0.145) 0.279** (0.092) 0.985** (0.011)
-0.789* (0.363) 0.248** (0.087) -0.085 (0.061)
-2.;n5 (1.374) 0.089 (0.200) -0.304* (0.124)
0.908** (0.051)
-0.130 (0.090) 0.519** (0.155) -0.120 (0.093) -0.561** (0.154) 0.227* (0.096) 0.976** (0.011)
0.665** (0.184)
Log Likelihood
410.847 6.216 (0.905) 8.622 (0.735)
469.740 13.045 (0.366) 10.178 (0.600)
503.860 10.546 (0.568) 6.844 (0.868)
513.230 7.219 (0.843) 15.015 (0.241)
11'0 SE(1I'0) 1il
SE(:rd 11'2 SE(1I'2)
Q(12) P-value
Note: Numbers in parentheses are Bollerslev-Wooldridge robust standard errors. Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Q(12) is the Ljung-Box
statistic for the null hypothesis that there is no autocorrelation up to order 12 for standardized residuals. Q2(12) is the Ljung-Box statistic for the null hypothesis that there is no
autocorrelation up to order 12 for standardized residuals squared. The null hypothesis is rejected at the 5 percent significance level if the P-value for each test statistic is less than
Ljung and Box (1978), is defined as follows:
Q(8) = T(T + 2)
T -
where p(i) is the ith sample autocorrelation and T is the number of observations. The Q statistic at lag s, Q(8), is a test statistic for the null hypothesis that there is no autocorrelation up to
order s for standardized residuals and is asymptotically distributed as X2 with degrees of freedom equal to the number of autocorrelation less the number of parameters. 12 As is clear from the table,
Q(12) (P-value) is 6.216 (0.905) for Germany, 13.045 (0.366) for Japan, 10.546 (0.568) for the UK, and 7.219 (0.843) for the USA. Thus, the null hypothesis of no autocorrelation up to order 12 for
standardized residuals is accepted for all countries. 13 This table also indicates the Q2(8) statistic and its associated P-value. The Q2 statistic at lag 8, Q2 (8), is a test statistic for the null
hypothesis that there is no autocorrelation up to order 8 for standardized residuals squared. As is clear from the table, Q2(12) (P-value) is 8.622 (0.735) for Germany, 10.178 (0.600) for Japan,
6.844 (0.868) for the UK, and 15.015 (0.241) for the USA. Thus, the null hypothesis of no autocorrelation up to order 12 for standardized residuals squared is accepted for all countries. 14 These
results empirically support the specification of the selected AR-EGARCH model.
Cheung-Ng Test
The second step involves the application of the Cheung-Ng test to analyze the causality in mean and variance based on the empirical results obtained in the previous section. Standardized residuals
obtained from the AR-EGARCH model are plotted in Figure 3.2 through Figure 3.5 for each country. The standardized squared residuals obtained from the AR-EGARCH model are plotted in Figure 3.6 through
Figure 3.9 for each country Sample cross-correlations of standardized residuals and standardized residuals squared are reported in Tables 3.4 through 3.9. As shown, the cross-correlations at
different lags are independently and normally distributed in large samples. The causality pattern is indicated by significant cross-correlations. IS That is, there is no evidence of causality in mean
(variance) when all cross-correlation coefficients calculated from (squares of) standardized residuals, at all possible leads and lags, are not significantly different from zero. In Table 3.4, for
example, lag refers to the number of periods that the German market lags behind the Japanese market, whereas lead refers to the number of periods that
Chapter 9
4-r-----------------------. 3
o -1 -2 -3
-4 -+-_r_.,....,._,__r--1......,--r....,.--r--r--r--r--r-~_r_.,....,._,__r_T' 80 82 84 86 88 90 92 94 96 98 00
Figure 3.2.
Standardized Residuals: Germany
4-,-----------------------. 3
o -1
-2 -3
-4 -+-_r_.,....,._,__r--1......,--r....,.--r--r--r--r--r-~_r_.,....,._,._r_T' 80 82 84 86 88 90 92 94 96 98 00
Figure 3.3.
Standardized Residuals: Japan
4-r-----------------------. 3
o -1
-2 -3 -4;-~~~~~~~~~~~~~~""T"""""T"""""T"""""T"""""T"""""T"""..,...
Figure 9..4-
Standardized Residuals: UK
4 ""T"""-------------------,
o -1 -2 -3
-4 -t-.,........,......,.-..,..-......,r--"I~_..,.._r.....,......,.._r_""T"""~_r_.,__r_I"""""1r_T'" 80 82 84 86 88 90 92 94 96 98 00
Figure 9.5.
Standardized Residuals: USA
Chapter 3
16.,...---------------------, 12
Figure 9.6.
Squares of Standardized Residuals: Germany
16.,...--------------------. 12
Figure 9.7.
Squares of Standardized Residuals: Japan
16_._---------------------. 12
Figure 9.8.
Squares of Standardized Residuals: UK
16 .........- - - - - - - - - - - - - - - - - - - - - - ,
Figure 9.9.
Squares of Standardized Residuals: USA
Chapter 9
Tab!e 9.4.
Sample Cross-Correlation of Standardized Residuals: Germany and Japan
Levels Lead
a.rman)' and Japan(-")
O.rm.ny .nd J.p.n( +/r)
O.rm.ny .nd J.pu(-Ir)
O.rm.ny .nd J.p...(+Ir)
-0.025 0.028 0.051 0.056 0.091 0.001
0.019 -0.019 -0.024 -0.022 0.029 0.016
-0.022 -0.Q11 0.182·· -0.031 0.001 0.197··
-0.015 -0.053
-0.028 -0.025 0.028 -0.010 0.001 -0.052
-0.092 -0.040 -0.010 0.042 -0.027 0.027
0.067 -0.054 -0.004 0.079 -0.040 0.019
-0.073 -0.092 0.026 0.001 -0.004 -0.072
0.048 -0.020 0.004 -0.104 0.000 -0.052
-0.085 0.010 -0.035 0.129· 0.021 -0.013
-0.063 0.029 0.022 0.074 0.005 -0.012
0.050 -0.005 0.001 0.001 -0.101 -0.011
-0.067 -0.043 -0.001 -0.032 0.114 0.026
-0.048 0.006 -0.038 0.031 -0.022 0.095
0.095 -0.019 -0.125· -0.069 0.127 -0.009·
0.018 0.000 -0.025 -0.009 -0.043 -0.008
-0.014 -0.078 0.076 ~0.030
Note: Significance at the 1 percent level and 5 percent level is indicated by·· and ., respectively. Lag refers to the number of periods that the German market lags behind the Japanese market,
whereas Lead refers to the number of periods that the German market leads the Japanese market. Cross-correlation under the Levels column is based on standardized residuals themselves and is used to
test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test for causality in variance.
Table 9.5.
Sample Cross-Correlation of Standardized Residuals: Germany and UK Levels
Germany and UK(-.)
Oermany and UKC+')
Germany and UK(-')
Gormany and UK(+.)
0.022 0.076 -0.072 -0.031 0.038 -0.033
0.198** -0.071 -0.Q11 0.014 -0.047 -0.085
-0.025 -0.015 0.089 -0.023 -0.035 0.031
0.111 -0.113 -0.024 0.065 -0.006 -0.043
0.052 -0.083 0.149* 0.140* -0.007 -0.065
-0.045 -0.083 0.089 -0.021 0.131* -0.021
-0.071 -0.030 0.105 0.004 -0.035 -0.029
-0.087 -0.090 -0.100 0.024 -0.063 0.005
0.055 -0.065 -0.070 -0.063 0.029 -0.099
-0.004 -0.008 -0.111 -0.099 -0.061 0.053
0.031 -0.084 -0.001 0.025 -0.032 0.006
0.062 -0.065 0.103 0.038 -0.073 -0.074
-0.012 -0.002 -0.Q15 -0.014 0.125* 0.039
-0.011 0.027 0.065 0.072 0.011 0.086
0.049 -0.057 -0.097 -0.064 0.084 -0.023
-0.059 0.066 -0.018 0.014 -0.075 0.072
Note: Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Lag refers to the number of periods that the German market lags behind the UK market, whereas
Lead refers to the number of periods that the German market leads the UK market. Cross-correlation under the Levels column is based on standardized residuals themselves and is used to test for
causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test for causality in variance.
Chapter 9
Table 9.6.
Sample Cross-Correlation of Standardized Residuals: Germany and USA Levels
Lag Germany and USA(-lo)
Squares Lead
Oermany and USA(+Io)
Germany and USA(-lo)
Oermany and USA(+Io)
0.098 0.008 0.012 -0.033 0.050 0.009
0.105 -0.060 -0.065 0.015 -0.053 -0.034
0.102 -0.008 0.011 0.086 0.047 0.089
-0.010 -0.132* -0.048 "0.013 -0.022 0.040
0.050 -0.105 0.033 -0.024 0.039 -0.074
0.010 -0.050 0.041 0.113 -0.032 -0.009
-0.075 0.009 -0.010 -0.119 0.004 0.005
-0.109 -0.016 -0.014 -0.013· -0.056 -0.003
0.090 -0.048 -0.065 -0.026 -0.027 -0.039
-0.051 -0.044 -0.057 0.024 0.007 0.15S*
0.037 -0.104 0.046 -0.048 -0.011 0.024
0.028 -0.043 -0.028 0.026 0.009 0.015
-0.081 -0.037 0.035 -0.128* 0.055 0.125*
-0.013 -0.035 -0.016 0.129* -0.060 0.012
-0.021 -0.073 -0.063 -0.038 0.016 0.041
-0.091 -0.096 -0.059 0.028 -0.057 -0.033
Note: Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Lag refers to the number of periods that the German market lags behind the US market, whereas
Lead refers to the number of periods that the German market leads the US market. Cross-correlation under the Levels column is based on standardized residuals themselves and is used to test for
causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test for C&usatity In variance.
Table 9.7. Sample Cross-Correlation of Standardized Residuals: Japan and UK Levels k
Japan and UK(-.)
Japan and UK(+.)
Japan and UK(-.)
Japan and UK(+.)
0.039 0.010 0.014 0.029 0.089 -0.031
0.127* -0.025 -0.034 0.117 0.009 -0.084
-0.043 -0.052 0.008 0.003 0.033 0.071
0.021 -0.010
0.014 -0.062 0.089 0.065 -0.027 0.002
0.001 -0.029 0.046 0.015 0.012 0.006
-0.030 -0.078 0.037 -0.002 -0.108 -0.069
0.084 -0.097 -0.081 0.041 0.010 0.034
-0.079 -0.002 0.023 0.071 0.037 -0.035
0.044 -0.021 -0.039 0.013 -0.067 -0.077
-0.040 -0.027 -0.004 0.086 -0.124* -0.030
-0.038 -0.051 -0.021 -0.072 -0.019 -0.087
0.043 -0.064 0.029 -0.037 0.141* 0.030
-0.039 -0.023 -0.121 0.048 0.126* 0.062
0.014 0.029 -0.073 -0.131* -0.103 -0.042
0.072 0.096 0.046 -0.009 0.004 0.035
0.031 0.002 0.113 ~0.009
Note: Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Lag refers to the number of periods that the Japanese market lags behind the UK market, whereas
Lead refers to the number of periods that the Japanese market leads the UK market. Cross-correlation under the Levels column is based on standardized residuals themselves and is used to test for
causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test for causality in variance.
Chapter 9
Table 3.8.
Sample Cross-Correlation of Standardized Residuals: Japan and USA Squares
Lag Japan and USA(-k)
Japan and USA(+k)
Japan and USA(+k)
Japan and USA( -k)
0.032 0.051 -0.011 0.039 -0.010 -0.089
-0.096 0.032 -0.106 0.058 0.014 -0.096
-0.035 -0.050 -0.016 -0.024 -0.028 0.081
-0.035 0.000 0.192** 0.027 0.079 0.089
0.070 -0.001 0.051 0.001 0.028 0.026
-0.025 -0.075 0.051 -0.004 -0.027 -0.017
-0.057 -0.068 -0.027 -0.077 -0.135* -0.054
0.036 -0.089 0.026 -0.029 0.087 0.146*
-0.079 0.000 -0.042 -0.020 0.037 0.045
0.012 -0.013 -0.129* 0.005 0.019 0.029
-0.027 -0.017 0.072 0.046 -0.117 -0.042
-0.009 -0.062 -0.050 -0.110 -0.042 0.014
-0.058 -0.096 0.007 0.004 0.005 0.030
-0.131* 0.004 -0.075 0.080 0.029 0.024
0.076 0.066 0.000 -0.057 -0.076 -0.009
0.045 -0.038 -0.074 0.024 0.009 -0.095
Note: Significance at the 1 percent level and 5 percent level is indicated by·· and ., reo spectively. Lag refers to the number of periods that the Japanese market lags behind the US market, whereas
Lead refers to the number of periods that the Japanese market leads the US market. Cross-correlation under the Levels column is based on standardized residuals themselves and is used to test for
causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test for causa:Iity in variance.
Table 3.9.
Sample Cross-Correlation of Standardized Residuals: UK and USA Levels
Lag UK and
UK and
Lag UK and
UK and
0.170** 0.003 0.064 0.031 0.003 -0.114
0.016 -0.041 -0.067 -0.009 -0.026 0.019
0.105 0.Q25 -0.034 -0.039 0.015 0.026
-0.001 -0.057 -0.012 .-0.068 0.085 -0.023
-0.017 -0.101 0.039 -0.028 0.114 -0.062
-0.074 -0.029 0.033 0.043 -0.025 -0.048
-0.036 -0.017 0.021 -0.028 -0.073 -0.014
-0.051 0.007 0.053 0.065 -0.014 0.122
-0.047 0.104 -0.111 -0.038 -0.053 0.026
0.019 0.009 -0.100 -0.013 0.054 -0.038
-0.087 -0.050 0.139* 0.133* 0.006 -0.033
-0.066 -0.041 -0.084 -0.039 -0.019 -0.010
-0.018 0.034 -0.141* 0.011 0.046 0.132*
-0.035 -0.150* 0.014 0.021 0.009 0.012
0.035 0.030 0.060 -0.025 0.017 0.068
0.024 -0.088 -0.002 -0.090 -0.024 -0.095
Note: Significance at the 1 percent level and 5 percent level is indicated by *. and ., respectively. Lag refers to the number of periods that the UK market lags behind the US market, whereas Lead
refers to the number of periods that the UK market leads the US market. Cross-correlation under the Levels column is based on standardized residuals themselves and is used to test for causality i~.
Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test for causality in variance.
Chapter 9
the German market leads the Japanese market. In other words, significant cross-correlation at a certain number of lags is interpreted as evidence of the Japanese market affecting the German market,
whereas significant cross-correlation at a certain number of lead is interpreted as evidence of the German market affecting the Japanese market. The column labeled Levels gives the cross-correlation
based on standardized residuals themselves. These are used for testing causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to
test for causality in variance. Cross-correlation at lag 0 (contemporaneous correlation) is 0.236 in mean and is statistically significant at the 1 percent level. As is clear from the table, the
German market causes the Japanese market in mean. The causation pattern is of lag 16 from Germany to Japan. In contrast, the Japanese market causes the German market in variance. Four different
causality in variance patterns are found in these markets. Note that there exists evidence of feedback neither in mean nor in variance. Table 3.5 shows the results of the causality test between
Germany and the UK. Contemporaneous correlation is 0.362 in mean and 0.261 in variance, and both of them are statistically significant at the 1 percent level. There exists feedback in mean between
these two markets. The UK market causes the German market in mean up to 23 lags, whereas the German market causes the UK market in mean up to 11 lags. It is interesting to see there is no causality
in variance between these two markets. The empirical results of the causality test between Germany and the USA are shown in Table 3.6. The contemporaneous correlation is 0.381 in mean and 0.403 in
variance, and both of them are statistically significant at the 1 percent level. This table shows the evidence of feedback in the mean of these two markets. The US market causes the German market in
mean up to 24 lags, whereas the German market causes the US market in mean up to 22 lags. Although there is no evidence of feedback, the German market causes the US market in variance. As seen in
Table 3.7, the cross-correlation of standardized residuals reveals evidence of causality in the mean and variance between Japan and the UK. The contemporaneous correlation is 0.337 in mean and 0.205
in variance, and both of them are statistically significant at the 1 percent level. The table shows the evidence of feedback in the mean of these two markets. The UK market causes the Japanese market
in mean at lag 23, whereas the Japanese market causes the UK market in mean at lag 1 and 23. Although there is no evidence of feedback, the UK market causes the Japanese market in variance.
Table 3.8 shows the empirical results between Japan and the USA. The contemporaneous correlation is 0.373 in mean and 0.260 in variance, and both of them are statistically significant at the 1
percent level. Although there exists no evidence of feedback, the Japanese market causes the US market in mean. Two different causality in mean patterns are found in these data. In contrast, there
appears feedback in variance between these two markets. As seen in Table 3.9, the cross-correlation of standardized residuals reveals evidence of feedback in the mean of the UK market and the US
market. The US market causes the UK market in mean up to lag 24, whereas the UK market causes the US market in mean at lag 20. Although there is no evidence of feedback, the US market causes the UK
market in variance up to lag 16.
This chapter specified the dynamics of stock prices using the AREGARCH model and used the CCF approach to analyze the causality of stock prices between Germany, Japan, the UK, and the USA.
Correlation analysis tested for spillovers in conditional means and volatility across countries. Such volatility spillovers may represent causal relationships across markets and also reflect global
economic changes that concurrently alter stock-return volatility across international stock markets. As emphasized in Chapter 1, there are several merits to the CCF approach. First, this approach
analyzes the causality of both means and variances. Second, it is relatively easy to implement as it does not involve simultaneous modeling, as do multivariate methods. 16 Third, the CCF test is
especially useful when the number of series under investigation is large and long lags are expected in the causation pattern. Fourth, this approach has a well-defined asymptotic distribution and its
asymptotic behavior does not depend on the assumption of normality. Fifth, this method provides information on the timing of causation, in contrast to the standard causality test used in the previous
chapter. Finally, since this method depends on the residuals estimated from the univariate model, we do not have to worry about the omission-of-variables problem - it is possible to analyze the
causality between any set of variables. Compared with the Granger causality results reported in the last chapter, cross-correlation statistics reveal more complex and dynamic causation patterns,
which are evident for both causality in mean and causality in variance. For example, the feedback effects in means involve a high-order lag structure. FUrthermore, evidence shows that causality in
variance goes from one market to another and vice versa. The main
Chapter 9
1 • •
Figure 3.10.
Summary of Causality in Mean
! I
Figure 3.11.
Summary of Causality in Variance
results are summarized in Figure 3.10 and Figure 3.11. Cheung and Fung (1997) noted that a proper account of conditional heteroskedasticity can have significant implications for· the study of price
and volatility
spillovers. The information flow between international markets affects not only price movements, but volatility movements as well. For example, though Japanese stock prices have a relatively weak
causal relation with other stock markets, the uncertainty in their prices might have relatively strong causal relationships with other stock markets.
Notes 1 Some other examples are Hamao, Masulis and Ng (1990), Baillie and Bollerslev (1991), Theodossiou and Lee (1993), Susmel and Engle (1994). 2 See Bollerslev, Chou and Kroner (1992), Campbell,
Lo and MacKinlay (1997, Chapter 12), and Watanabe (2000) for examples. 3 Note that LLd. means independent and identical distribution. 4 Let X be a random variable with mean E(X), and let gO be a
convex function. Then Jensen's inequality implies E[g(X)J ~ g(E[XJ). For the example, note that g(X) = X 2 is convex. Hence E[X 2] ~ (E[X])2. 5 Nelson and Cao (1992) shows that inequality constraints
less severe than commonly imposed are sufficient to keep the conditional variance non-negative. For example, in the GARCH(2,1) case, W > 0, (tl ~ 0, 131 ~ 0, and (131 (tl + (t2) ~ 0 are sufficient to
ensure ul > 0, such that (t2 may be negative. 6 The parameter subscripts are not necessary for the GARCH(1,1), GJR(1,1), and EGARCH(1,1) models and are suppressed for the remainder of this section. 7
Nakagawa and Osawa (2000) is a good example of the application of this model. 8 Also see Zakoian (1994). 9 Some other examples are the NGARCH (nonlinear GARCH) model by Engle and Bollerslev (1986)
and Higgins and Bera (1992), and the QGARCH (quadratic ARCH) model by Sentana (1995). lOIf the model is correctly specified, standardized residuals should be independent and identically distributed
with mean zero and variance one. If standardized residuals are normally distributed, then maximum likelihood estimates are asymptotically efficient. However, even if the residuals are not normally
distributed, the estimates are still consistent under quasi-maximum likelihood. 11 SBIC is defined as follows: SBIC = log(8 2 ) + m IO~T) ,where 82 is the residual variance and m is the number of
parameters in the model.
APPENDIX 3.A
12Ifthe series is not based on the empirical results of ARMA estimation, then under the null hypothesis Q( 8) is approximately distributed as X2 with degrees of freedom equal to the number of
autocorrelations. If the series is the residuals of the ARMA estimation, however, the appropriate degrees of freedom are equal to the number of autocorrelations less the number of AR and MA terms.
13Q(24) also shows that there is no autocorrelation in residuals. 14Q2(24) also shows that there is no autocorrelation in squared residuals. 15Note that cross-correlations are not necessarily
symmetric around zero. 168ee Hamao, Masulis and Ng (1990) and Theodossiou and Lee (1993) as an example of the multivariate approach.
APPENDIX 3.A The stock price indices are obtained from the International Financial Statistics of the International Monetary Fund. The series code of the stock price index in each country is as
follows: Germany: 13462..ZF Japan: 15862... ZF UK: 11262 ZF USA: 11162 ZF
Chapter 4
The rapid expansion in international trade during the 1970s and the adoption of freely floating exchange rate regimes by many industrialized countries in 1973 heralded a new era of increased exchange
rate volatility. Inevitably, firms' exposure to foreign exchange rate risks increased. The literature identifies three types of risk that exist under floating exchange rate regimes: transaction
exposure, economic exposure, and operating exposure. 1 Transaction exposure arises from gains or losses incurred in the settlement of investment transactions stated in foreign currency terms;
economic exposure arises from variation in a firm's discounted cash flow when exchange rates fluctuate; and operating exposure arises from the sensitivity of a firm's home currency value to changes
in exchange rates. The last two chapters demonstrated that stock markets have developed interdependently and are, consequently, linked internationally. As a result, capital inflow-outflow
relationships may affect exchange rate movements. When foreign asset markets become more profitable, domestic capital flows out to foreign markets and results in a depreciation of the domestic
currency. When domestic asset markets become more profitable, foreign capital flows in to the domestic market and results in an appreciation of the domestic currency. Asset market theory states that
the foreign exchange rate, which is the relative price of two currencies, is best analyzed within the same framework as used for other asset prices. On the other hand, domestic currency appreciation
reduces the competitiveness of export markets, which causes a negative effect on the
domestic stock market. For an import-dominated country, however, domestic currency appreciation lowers input costs and generates a positive impact on the stock market. Thus, exchange rates affect
stock prices, but as well, stock prices may affect exchange rates. This chapter uses the CCF approach to empirically analyze the relationship between stock prices and foreign exchange rates for
Germany, Japan, the UK, and the USA. Stock prices and foreign exchange rates are specified using the AR-EGARCH model, and the causality in mean and variance between the two variables is analyzed
Selected Literature Review
Since stock prices and foreign exchange rates play crucial roles in the development of a country's economy, their dynamic relationships have attracted the attention of numerous economists. Moreover,
investors often use the relationships between stock prices and foreign exchange rates to predict their future trends. Some examples include Solnik (1987), Bahmani-Oskooee and Sohrabian (1992), Smith
(1992), Ajayi and Mougoue (1996), Abadalla and Murinde (1997), Ong and Izan (1999), and Nieh and Lee (2001). Table 4.1 is a summary of literature. Solnik (1987) employed a regression analysis on
monthly and quarterly data from 1973 to 1983 for eight industrialized countries (Canada, France, Germany, Japan, the Netherlands, Switzerland, the UK, and the USA) and found a negative relation
between real domestic stock returns and real exchange rate movements. However, for monthly data from 1979 to 1983, he observed a weak but positive relationship between the two variables.
Bahmani-Oskooee and Sohrabian (1992) empirically analyzed the relationship between stock prices as measured by the S&P 500 index and the effective US dollar exchange rate, using monthly observations
from July 1973 to December 1988. Using the Granger causality test, they found a dual causal relationship between stock prices and the effective exchange rate, at least in the short run. However, they
were unable to establish any long-run relationship between the two variables using the cointegration approach. Smith (1992) derived an exchange rate equation based on an optimizing intertemporal
model of asset choice. Using quarterly data from Germany, Japan, and the USA from the 1st quarter of 1974 to the 3rd quarter of 1988, he estimated mark-dollar and yen-dollar exchange rate equations.
In both cases, stocks had a significant impact on the exchange rate. Thus, he showed that stock prices play an important role in empirical models of the exchange rate.
Chapter .4
Table 4.1. Summary of Literature
Technique and Main Results
Solnik (1987)
July 1973 - December 1983 (monthly and quarterly data) Canada, France, Germany, Japan, Netherlands, Switzerland, UK, and USA
Regression Analysis; There is a negative relationship between stock returns and exchange rate movements.
BahmaniOskooee andSohrabian (1992)
July 1973 - December 1988 (monthly data) USA
VAR and Granger causality testj There is a dual causal relationship between the stock prices and effective exchange rate.
Smith (1992)
1st quarter of 1974 - 3rd quarter of 1988 (quarterly data) Germany, Japan, and USA
Instrumental variable method; Stocks have a significant impact on the exchange rate.
Ajayi and Mougoue (1996)
April 1985 - July 1991 (daily data) Canada, France, Germany, Italy, Japan, Netherlands, Switzerland, UK, and USA
VECMj Significant feedback relations exist between stock markets and exchange rate markets
Abdalla and Murinde (1997)
January 1985 - July 1994 (monthly data) India, Korea, Pakistan, and the Philippines
VAR and VECM and Granger causality testj Exchange rates cause stock prices in Korea, Pakistan, and India, whereas stock prices cause exchange rates in the Philippines
Ong and !zan (1999)
October 1986 - December 1992 (weekly data) Australia and G-7 countries
Regression Analysis; The equity parity is achieved within a short time.
Nieh and
October 1993 - February 1996 (daily data) G7 countries
VECMj There is no long-run significant relationship between stock prices ad exchange rates.
Lee (2001)
Ajayi and Mougoue (1996) used data on daily closing stock market indices and exchange rates for Canada, France, Germany, Italy, Japan, the Netherlands, the UK, and the USA from April 1985 to July
1991. Their results revealed significant short-run and long-run feedback relationships between stock market indices and exchange rates. Specifically, an increase in the aggregate domestic stock price
had a negative shortrun effect on the value of domestic currency. In the long-run, however, a rise in stock prices affected domestic currency value positively. On the other hand, currency
depreciation had a negative short-run and long-run effect on the stock market. Abadalla and Murinde (1997) investigated the interactions between exchange rates and stock prices in the emerging
financial markets of India, Korea, Pakistan, and the Philippines. They applied the bivariate vector autoregressive model using monthly observations from January 1985 to July 1994. Their results show
unidirectionary causality from exchange rates to stock prices in all sample countries, except the Philippines. This finding also has policy implications: it suggests that the respective governments
should be cautious in their implementation of exchange rate policies, given that such policies have ramifications for their stock markets. Ong and Izan (1999) used weekly data for Australia and the
G-7 countries from October 1, 1986, to December 16, 1992, to test two hypotheses: (i) a parity relationship exists between foreign exchange rates and equity markets; and (ii) equity parity holds in
the short-term, i.e., reactions of equity markets to foreign exchange rate movements (and/or vice-versa) are quicker than those of general prices, interest rates, and gold. Their results show that
equity parity holds between stock markets and exchange rates to the extent that depreciation in a country's currency would cause its stock market return to rise, while an appreciation would have the
opposite effect. It is also apparent that the reaction of stock prices to exchange rate changes (and/or vice-versa) is a rapid one. Nieh and Lee (2001) analyzed the daily closing stock market indices
and foreign exchange rates for the G-7 countries using the VECM, and a sample period of October 1, 1993, to February 15, 1996. They found no long-run significant relationship between stock prices and
exchange rates in the G-7 countries. Furthermore, their short-run analysis from the VECM revealed only a one-day predicting power for the two financial assets, and only in certain countries. Existing
empirical studies typically examine causality in the mean relationship between stock and foreign exchange markets. As pointed out in Chapter 1, however, Ross (1989) demonstrated that return
volatility also provides useful information on information flow. Thus, data
Chapter 4
Table 4.£. Summary Statistics
Exchange Rates Mean Std. Dev. Skewness Kurtosis Jarque-Bera P-value Stock Prices Mean Std. Dev. Skewness Kurtosis Jarque-Bera P-value Correlation
0.081 1.027 0.224 3.505 4.868 0.088
0.162 2.691 0.405 3.765 13.253 0.001
0.103 1.891 -0.423 4.827 43.243 0.000
0.000 1.889 -0.250 2.847 2.917 0.233
0.773 5.331 -0.882 5.888 122.137 0.000
0.425 4.337 -0.191 3.671 6.356 0.042
0.929 3.859 -1.267 9.844 568.094 0.000
0.964 3.550 -0.685 5.807 104.057 0.000
Note: Jarque-Bera is the Jarque-Bera statistic to test for normality. P-value is the probability value associated with the Jarque-Bera statistic. The null hypothesis of normal distribution is
rejected at the 5 percent significance level if the P-value for the Jarque-Bera test is less than 0.05. Correlation is the correlation coefficient between stock prices and foreign exchange rates.
on return volatility in stock and foreign exchange markets can provide information in addition to that available in the return data alone.
Monthly data of stock prices and effective exchange rates for Germany, Japan, the UK, and the USA are used for empirical analysis in this chapter. The effective exchange rate is the weighted average
of various foreign exchange rates and is used to reflect the economic situation of a country. The sample period is January 1980 through May 2001. The source is the International Financial Statistics
of the International Monetary Fund. Table 4.2 shows the summary statistics for the growth rate of stocks and effective exchange rates for each country. The growth rate of stock prices or foreign
exchange rates is calculated as R t = (In X t -In Xt-d x 100, where X t is the stock price index or the effective exchange rate at time t. Thus, growth rates are obtained for the period between
February 1980 and May 2001. Table 4.2 shows the
Table 4.9. Unit Root Test
Test Statistics
Exchange Rates
-0.404 -2.135 -1.465 -0.883
-0.989 -1.760 -1.232 -1.012
0.945 0.712 0.657 -0.037
-10.012** -10.952** -10.836** -11.399**
-9.992** -10.948** -10.818** -11.407**
-9.976** -10.939** -10.824** -11.431**
-2.373 -1.323 -2.110 -2.588
-0.953 -2.105 -1.955 -0.329
2.002 0.979 3.175 3.580
-14.839** -11.227** -12.948** -11.771**
-14.867** -11.123** -12.899** -11.794**
-14.652** -11.074** -12.429** -11.291**
Level Germany Japan UK USA First Difference Germany Japan UK USA
Stock Prices
Level Germany Japan UK USA First Difference Germany Japan UK USA
Note: * shows that the null hypothesis of a unit root is rejected at the 5 percent significance level. .. shows that the null hypothesis of a unit root is rejected at the 1 percent significance
level. CT corresponds to the following regression: aYt = J.4+6t+'YYt-l +Ut. C corresponds to the following regression: aYt = J.4 + 'YYt-l + Ut. None correspond to the following regression: aYt
= /'Yt-l + Ut·
mean, standard deviation (Std. Dev.), skewness, kurtosis, Jarque-Bera statistic, its associated probability value (P-value), and the correlation coefficient between stock prices and foreign exchange
rates. For exchange rates, the average growth rate is 0.081 percent for Germany, 0.162 percent for Japan, 0.103 percent for the UK, and 0.000 percent for the USA, whereas the standard deviation is
1.027 percent
Chapter .4
for Germany, 2.691 percent for Japan, 1.891 percent for the UK, and 1.889 percent for the USA. Thus, Japan has the highest mean and the highest standard deviation. The skewness is 0.224 for Germany,
0.405 for Japan, -0.423 for the UK, and -0.250 for the USA. The kurtosis is 3.505 for Germany, 3.765 for Japan, 4.827 for the UK, and 2.847 for the USA. The Jarque-Bera statistic (P-value) is 4.868
(0.088) for Germany, 13.253 (0.001) for Japan, 43.243 (0.000) for the UK, and 2.917 (0.233) for the USA. Thus, at the 5 percent significance level the null hypothesis of normal distribution is
rejected for Japan and the UK but not for Germany and the USA. The summary statistics for stock prices are exactly the same as obtained in Table 2.2 in Chapter 2. The average growth rate is 0.773
percent for Germany, 0.425 percent for Japan, 0.929 percent for the UK, and 0.964 percent for the USA, whereas the standard deviation is 5.331 percent for Germany, 4.337 percent for Japan, 3.859
percent for the UK, and 3.550 percent for the USA. Thus, the USA has the highest mean but the lowest standard deviation. The skewness is -0.882 for Germany, -0.191 for Japan, -1.267 for the UK, and
-0.685 for the USA. The kurtosis is 5.888 for Germany, 3.671 for Japan, 9.844 for the UK, and 5.807 for the USA. The Jarque-Bera statistic (P-value) is 122.137 (0.000) for Germany, 6.356 (0.042) for
Japan, 568.094 (0.000) for the UK, and 104.057 (0.000) for the USA. Thus, the null hypothesis of normal distribution is statistically rejected for all countries at the 5 percent significance level.
Furthermore, the correlation coefficient between stock prices and exchange rates is -0.053 for Germany, 0.146 for Japan, -0.037 for the UK, and -0.064 for the USA. The unit root test developed by
Phillips and Perron (1988) is used to test that the stock price index or effective exchange rate for each country has a unit root. The unit root test statistic is the t-value of 'Y obtained from the
following regressions: (CT)
(C) (None)
AYt = AYt =
+ &t + 'YYt-l + Ut, J1. + 'YYt-l + Ut, 'YYt-l + Ut, J.t
where A is a difference operator, Le., AYt = Yt -Yt-l, t is the time trend, and Ut is a disturbance term. The first equation includes a constant term and a time trend, the second equation includes a
constant term only, and the third equation includes no deterministic term. The null hypothesis (Ho) and the alternative hypothesis (HA) are shown as follows:
'Y = 0,
'Y < o.
Thus, the null hypothesis shows that a unit root is included and the alternative hypothesis shows that a unit root is not included. In Table 4.3, each equation is applied to both the level and the
first difference of the log of exchange rates and the log of stock price indices. The results for stock prices are exactly the same as obtained in Table 2.3 in Chapter 2. As an example, let us see
the results of Germany. For the level of the effective exchange rate, test statistics are -0.404 for CT, -0.989 for C, and 0.945 for None. For the first difference of the effective exchange rate,
test statistics are -10.012 for CT, -9.992 for C, and -9.976 for None. For the level of the stock price index, test statistics are -2.373 for CT, -0.953 for C, and 2.002 for None. For the first
difference of the stock price index, test statistics are -14.839 for CT, -14.867 for C, and -14.652 for None. Thus, the null hypothesis of a unit root is not rejected in all specifications for the
level of the effective exchange rate and the stock price index. The hull hypothesis is, however, rejected for all specifications for the first difference of each variable. These results are robust to
all countries. Thus, the exchange rate and stock price index are found to be a 1(1) process for all countries.
4. 4.1
Empirical Results AR-EGARCH Model
The two-step procedure proposed by Cheung and Ng (1996) is employed to analyze the mean and variance causal relationships across markets. The first step involves the estimation of univariate
time-series models that allow for time variation in both conditional means and conditional variances. This section uses the AR(k)-EGARCH(p,q) specification for the first stage. The first difference
of log value is used for empirical analysis as follows: Yt = InXt - InXt -ll where X t is the exchange rate or stock price at time t. The conditional mean and conditional variance are respectively
specified as follows: k
Yt =
+L i=l
log ul =
+ ft,
N(0,u 2 )
+ 'YiZt-i) + L Pi log Ul-i
where Zt = ft/Ut and 1t - l is the information set available at time t 1. Equations (4.1) and (4.2) show AR(k) and EGARCH(p,q) models, respectively.
Chapter 4
Empirical results of the AR-EGARCH model for foreign exchange rates
Mean equation: 1It 11'0 + E~=I 11'.111-. + et Variance equation: logo} w + Ef=1 (odzl-d Zt
= et/O'I
+ ")'iZt-i) + E~=I f3.logO'l_i,
11'0 SE(1I'0) 11'1 SE(1I'1) 11', SE(1I")
0.001 (0.001) 0.446** (0.069) -0.149* (0.065)
0.001 (0.002) 0.336** (0.057) -0.152** (0.049)
0.000 (0.001) 0.344** (0.073)
-0.0004 (0.0008) 0.312** (0.034)
-7.695** (1.730) 0.276 (0.141) 0.061 (0.068)
-18.965** (1.466) 0.318** (0.116) 0.102 (0.059)
-2.109 (1.354) 0.457* (0.192) -0.074 (0.114)
-0.824** (0.127) -0.690** (0.091)
0.784** (0.154)
0.815** (0.186) -0.610** (0.181)
-1.435** (0.076) -0.356** (0.059) 0.038 (0.039) 0.413** (0.060) -0.036 (0.040) 1.695** (0.026) -0.863** (0.029)
Log Likelihood Q(12) P-value Q'(12) P-value
834.574 12.400 0.414 4.155 0.980
585.716 13.707 0.320 13.753 0.317
679.691 9.285 0.678 6.930 0.862
SE(w) 01 SE(oI) ")'1 SE(")'I) 02 SE(02) ")'2 SE(")'2) f31 SE(f3.)
679.439· 10.084 0.609 4.465 0.974
Note: Numbers In parentheses are Bollerslev-Wooldridge robust standard errors. Significance at the 1 percent level is indicated by ** and at the 5 percent level is indicated by *. Q(12) is the
Ljung-Box statistics for the null hypothesis that there is no autocorrelation up to order 12 for standardized residuals. Q'(12) is the Ljung-Box statistic for the null hypothesis that there is no
autocorrelation up to order 12 for standardized residuals squared. P-value is the probability value associated with each test statistic. The null hypothesis is rejected at the 5 percent significance
level if the P·value for each test statistic is less than 0.05.
Table 4.5.
Empirical results of the AR-EGARCH model for stock prices
Mean equation: Ye 71'0 + E~=1 71'iYe-i + fe Variance equation: logO'l w + E~=1 (Oi/Ze-i/ + 1'iZe-i) + E:=1 /3i logO'l_i, Ze fe/O't
71'0 SE(7I'0) 71'1 SE(7I'1) 71'2 SE(7I'2)
0.008** (0.003) 0.020 (0.059)
0.004 (0.002) 0.356** (0.065)
0.010** (0.002) 0.177** (0.066) -0.172** (0.041)
0.007** (0.002) 0.261** (0.069)
w SE(w)
-0.091 (0.084) 0.294(0.125) -0.098 (0.087) -0.302* (0.145) 0.279** (0.092) 0.985** (0.011)
-0.789(0.363) 0.248-(0.087) -0.085 (0.061)
-2.375 (1.374) 0.089 (0.200) -0.304* (0.124)
0.908** (0.051)
-0.130 (0.090) 0.519-* (0.155) -0.120 (0.093) -0.561** (0.154) 0.227* (0.096) 0.976** (0.011)
0.665** (0.184)
410.847 6.216 (0.905) 8.622 (0.735)
469.740 13.045 (0.366) 10.178 (0.600)
503.860 10.546 (0.568) 6.844 (0.868)
513.230 7.219 (0.843) 15.015 (0.241)
SE(ot} 1'1 SE(')'I) 02
SE(02) 1'2 SE(')'2) /3 SE(/3) Leg Likelihood
Q(12) P-value Q2(12)
Note: This is the same table as Table 3.3 in Chapter 3. Numbers in parentheses are BollerslevWooldridge robust standard errors. Significance at the 1 percent level is indicated by -and at the 5
percent level is indicated by -. Q(12) is the Ljung-Box statistic for the null hypothesis that there is no autocorrelation up to order 12 for standardized residuals. Q2(12) is the Ljung-Box statistic
for the null hypothesis that there is no autocorrelation up to order 12 for standardized residuals squared. P-value is the probability value associated with each test statistic. The null hypothesis
is rejected at the 5 percent significance level if the P-value for each test statistic is less than 0.05.
Chapter .4
Each model is estimated by the method of maximum likelihood. Parameter estimates and their asymptotic standard errors, which are robust to departures from normality using the consistent
variancecovariance estimator of Bollerslev and Wooldridge (1992), are reported. Though the Cheung-Ng test results presented in the next section are robust to distributional assumptions, inferences
about the EGARCH parameter estimates may be sensitive to deviations from normality. Hence the Bollerslev-Wooldridge standard errors are reported. SBIC and the Ljung-Box test are used to specify the
model. SBIC is often used for model selection and smaller values of SBIC are preferred (Schwarz 1978). The Ljung-Box test is used to check if there is no serial correlation in residuals. The choice
of k, p, and q is carried out among k = 1,2, ..: 12, p=I,2 and q=I,2 using SBIC and residual diagnostics. As a result, the AR(I)-EGARCH(I,I) model is chosen for the UK and the USA, while the AR(2)
-EGARCH(I,2) model is chosen for Germany and Japan. Table 4.4 shows the empirical results of the AR-EGARCH model for foreign exchange rates. As the table clearly indicates, The coefficient of the
GARCH term (/3) is estimated to be 0.815 and -0.610 for Germany, -0.824 and -0.690 for Japan, 0.784 for the UK, and 1.695 and -0.863 for the USA, and they are statistically significant at the 1
percent level. As indicated in Chapter 3, negative coefficients of GARCH term are not precluded in the EGARCH model, and thus the possibility of cyclical behavior in volatility is admitted. The
coefficients of asymmetric effect (r) are estimated to be 0.061 for Germany, 0.102 for Japan, -0.074 for the UK, and 0.038 and -0.036 for the USA. However, their standard errors are large and, thus,
are not statistically significant. Table 4.4 also shows the diagnostics of the empirical results of the AR-EGARCH model. The Ljung-Box test statistic at lag s, Q(s), is a test statistic for the null
hypothesis that there is no autocorrelation up to order s for standardized residuals and is asymptotically distributed as X2 with degrees of freedom equal to the number of autocorrelation less the
number of parameters. As is clear from the table, Q(12) (P-value) is 12.400 (0.414) for Germany, 13.707 (0.320) for Japan, 9.285 (0.678) for the UK, and 10.084 (0.609) for the USA. Thus, the null
hypothesis of no autocorrelation up to order 12 for standardized residuals is not rejected for all countries. This table also indicates the Q2(s) statistic and its associated P-value. The Q2
statistic at lag s, Q2(s), is a test statistic for the null hypothesis that there is no autocorrelation up to order s for standardized residuals squared. As is clear from the table, Q2(12) (P-value)
is 4.155 (0.980) for Germany, 13.753 (0.317) for Japan, 6.930 (0.862) for the UK, and 4.465
(0.974) for the USA. Thus, the null hypothesis of no autocorrelation up to order 12 for standardized residuals squared is not rejected for all countries. These results empirically support the
specification of the AR-EGARCH model. Table 4.5 shows the empirical results of the AR-EGARCH model for stock prices. Table 4.5 is the same table as Table 3.3 in Chapter 3. The same table is shown
again so that those who have not read Chapter 3 can understand the results without any problem. Needless to say, those who read the last chapter can skip the following explanation. The choice of k,
p, and q is carried out among k = 1, ..., 12, p=1, 2 and q=1, 2 using SBIC and residual diagnostics. As a result, the AR(I)-EGARCH(I,I) model is chosen for Japan and the USA, the AR(1)-EGARCH(2,1)
model is chosen for Germany, and the AR(2)-EGARCH(2,1) model is chosen for the UK. As the table clearly indicates, The coefficient of the GARCH term ({3) is estimated to be 0.985 for Germany, 0.908
for Japan, 0.976 for the UK, and 0.665 for the USA, and they are statistically significant at the 1 percent level. Thus, the persistence to volatility shock is relatively high for Germany, Japan, and
the UK, but is relatively low for the USA. The coefficients of asymmetric effect b') are estimated to be -0.098 and 0.279 for Germany, -0.085 for Japan, -0.120 and 0.227 for the UK, and -0.304 for
the USA. Note that this asymmetric parameter is statistically significant except for Japan. Table 4.5 also shows the diagnostics of the empirical results of the AR-EGARCH model. As is clear from the
table, the null hypothesis of no autocorrelation up to order 12 for standardized residuals is not rejected for all countries. It is also clear that the null hypothesis of no autocorrelation up to
order 12 for standardized residuals squared is not rejected for all countries. These results empirically support the specification of the AR-EGARCH model.
Cheung-Ng Test
The second step of the CCF approach is to analyze the causality in mean and variance based on the empirical results obtained in the previous section. Sample cross-correlations of the standardized and
standardized residuals squared are reported in Tables 4.6 through 4.9. There is no evidence of causality in mean (variance) when all cross-correlation coefficients calculated from (squares of)
standardized residuals, at all possible leads and lags, are not significantly different from zero. The causality pattern is indicated by significant cross-correlations. In each table, lag refers to
the number of periods that stock prices lag behind foreign exchange rates, whereas lead refers to the number of periods
Chapter 4
Table 4.6. Sample Cross-Correlation of Standardized Residuals: Germany
Levels k
Lag SP and EX(-k)
SP and EX(+k)
Lag SP and EX(-k)
SP and EX(+k)
-0.051 0.028 0.021 0.035 -0.137· 0.020
-0.097 0.Q18 -0.025 -0.037 0.055 0.000
-0.079 0.015 0.057 -0.076 -0.055 -0.097
0.020 -0.035 -0.081 -0.054 -0.040 0.005
0.038 -0.011 -0.13S· -0.015 0.039 0.086
0.061 -0.098 -0.021 -0.001 0.003 0.045
-0.119 0.095 0.040 0.001 -0.040 -0.016
-0.099 -0.060 -0.049 -0.021 -0.015 -0.047
-0.080 0.035 -0.020 0.047 -0.017 0.094
-0.071 0.034 0.078 0.024 -0.017 -0.026
0.006 -0.014 0.018 -0.036 -0.037 0.004
-0.002 0.006 0.114 -0.116 -0.031 0.039
-0.069 0.027 0.010 0.008 -0.064 0.005
0.030 0.092 -0.014 0.000 0.009 -0.083
0.007 -0.062 -0.028 -0.072 0.036 0.048
-0.073 -0.030 -0.066 0.119 0.053 -0.018
Note: SP and EX show stock prices and foreign exchange rates, respectively. Significance at the 1 percent level and 5 percent level is indicated by·· and ., respectively. Lag refers to the number of
periods that stock prices lag behind foreign exchange rates, whereas Lead refers to the number of periods that stock prices lead foreign exchange rates. Cross-correlation under the Levels column is
based on standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test
for cau~variance.
Table 4.7. Sample Cross-Correlation of Standardized Residuals: Japan
Levels k
Lag SP and EX( -k)
Lead SP and EX(+k)
Lag SP and EX(-k)
Lead· SP and EX(+k)
0.021 0.017 -0.025 0.078 0.093 0.052
0.006 0.011 -0.009 0.046 -0.008 -0.176**
-0.050 -0.039 0.075 -0.072 0.083 -0.074
0.123* 0.044 -0.046 0.125* 0.044 0.071
0.005 -0.050 0.011 0.098 0,015 0.036
0.092 0.093 0.068 -0.025 0.004 0,015
-0.034 -0.051 0.103 -0.117 -0.054 0.067
-0.064 -0.042 -0.024 0.020 -0.064 -0.003
0.067 0.035 -0.042 -0.014 -0.027 -0.093
-0.027 -0.041 -0.175** -0.030 0.036 -0.002
-0.084 0.020 -0.032 0.011 0.077 0.020
-0.062 -0.095 -0.014 0.020 0.022 -0.112
-0.023 0.042 -0.064 -0.098 -0.005 0.068
0.053 0.026 -0.007 0.089 0.026 -0.103
-0.038 -0.059 -0.006 0.043 -0.112 0.037
0.034 -0.057 0.061 -0.062 0.036 -0.004
Note: SP and EX show stock prices and foreign exchange rates, respectively. Significance at the 1 percent level and 5 percent level is indicated by·· and ., respectively. Lag refers to the number of
periods that stock prices lag behind foreign exchange rates, whereas Lead refers to the number of periods that stock prices lead foreign exchange rates. Cross-correlation under the Levels column is
based on standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test
for causilrtY'1ii'""variance.
Chapter .-I
Table 4.8.
Sample Cross-Correlation of Standardized Residuals: UK
Lag SP and EX(-k)
Lead SP and EX(+k)
Lag SP and EX(-k)
SP and EX(+k)
-0.107 -0.043 0.027 0.026 -0.083 -0.046
0.075 0.090 -0.036 -0.045 -0.057 -0.001
-0.046 -0.016 0.135* -0.063 -0.011 0.029
-0.012 0.037 0.000 0.019 0.058 0.016
0.011 -0.058 0.100 0.089 -0.132* 0.029
0.083 -0.036 -0.033 0.087 0.021 -0.109
0.017 0.198** -0.068 0.108 -0.055 0.039
0.054 -0.053 -0.037 -0.078 -0.069 -0.039
-0.031 -0.140* 0.024 0.077 -0.082 -0.049
-0.002 0.063 0.039 0.001 -0.014 0.032
-0.009 0.031 -0.014 0.020 -0.052 0.007
-0.056 0.050 -0.033 -0.121 -0.067 -0.041
0.036 0.066 0.066 -0.009 -0.023 0.073
-0.070 0.026 -0.098 -0.040 -0.010 0.085
0.034 0.055 0.114 0.026 0.008 -0.036
0.208** 0.106 -0.019 -0.064 -0.012 0.067
Note: SP and EX show stock prices and foreign exchange rates, respectively. Significance at the 1 percent level and 5 percent level Is indicated by ** and *, respectively. Lag refers to the number of
periods that stock prices lag behind foreign exchange rates, whereas Lead refers to the number of periods that stock prices lead foreign exchange rates. Cross-correlation under the Levels column is
based on standardized residuals themselves and is used to test for causality in mean. Cr088-correlatlon under the Squares column is based on the squares of standardized residuals and Is used to test
for causAIItY'1ilvariance.
Table 4.9. Sample Cross-Correlation of Standardized Residuals: USA
Levels k
Lag SP and EX( -k)
Lead SP and EX(+k)
Lag SP and EX(-k) 0.097
Lead· SP and EX(+k)
-0.004 -0.089 -0.059 -0.025 -0.079 -0.067
0.113 0.061 -0.082 0.070 -0.092 0.036
-0.025 -0.053 0.054 0.036 -0.035 -0.066
0.154* 0.106 0.056 -0.013 0.013 -0.043
0.021 0.025 0.112 -0.088 -0.078 0.053
-0.040 -0.103 -0.146* 0.036 0.055 0.109
0.032 0.017 0.070 0.000 0.004 -0.064
0.066 -0.072 0.135* 0.006 -0.088 -0.038
-0.025 0.027 0.079 -0.038 -0.004 0.055
0.017 -0.047 -0.021 - 0.006 -0.042 -0.041
-0.072 0.020 0.108 0.095 -0.062 -0.055
0.016 0.059 0.059 0.031 -0.044 0.046
-0.027 -0.053 0.052 0.031 0.149* 0.031
0.030 0.110 0.056 0.088 -0.012 0.083
0.091 0.070 0.003 -0.015 -0.015 0.080
-0.030 0.009 0.056 0.000 0.028 0.006
Note: SP and EX show stock prices and foreign exchange rates, respectively. Significance at the 1 percent level and 5 percent level is indicated by·· and ., respectively. Lag refers to the number of
periods that stock prices lag behind foreign exchange rates, whereas Lead refers to the number of periods that stock prices lead foreign exchange rates. Cross-correlatlon under the Levels column is
based on standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test
for caus8Jl'iY1ii'""variance.
Chapter 4
that stock prices lead foreign exchange rates. In other words, significant cross-correlation at a certain number of lags is interpreted as evidence of foreign exchange rates affecting stock prices,
whereas significant crosscorrelation at a certain number of leads is interpreted as evidence of stock prices affecting foreign exchange rates. Cross-correlation under the Levels column is based on
standardized residuals themselves and is used to test for causality in the mean. Cross-correlation under the Squares column is based on the squares of standardized residuals and is used to test for
causality in the variance. The results for Germany are shown in Table 4.6. As is clear from the table, cross-correlation at lag 0 (contemporaneous correlation) is not statistically significant either
in mean or in variance. Foreign exchange rates cause stock prices in mean at lags 5 and 9, whereas stock prices do not cause foreign exchange rates in mean. There is no evidence of causality in the
variance between stock prices and foreign exchange rates. Table 4.7 shows the results of the causality test for Japan. Contemporaneous correlation is 0.202 in mean and is statistically significant at
the 1 percent level, although it is not statistically significant in variance. Although there is no evidence of feedback, stock prices cause foreign exchange rates in mean at lags 6 and 15 and in
variance at lags 1 and 4. The empirical results of the causality test for the UK are shown in Table 4.8. The contemporaneous correlation is statistically significant neither in mean nor in variance.
Although there is no evidence of feedback, foreign exchange rates cause stock prices in mean at lags 11 and 14. This table shows the evidence of feedback in the variance of these two markets. Foreign
exchange rates cause stock prices in variance up to lag 8, whereas stock prices cause foreign exchange rates in variance at lag 19. As seen in Table 4.9, the cross-correlation of standardized
residuals reveals evidence of causality in mean and variance for the USA. The contemporaneous correlation is not statistically significant either in mean or in variance. The table shows the evidence
of feedback in the mean of these two markets. Foreign exchange rates cause stock prices in mean at lag 23, whereas stock prices cause foreign exchange rates in mean at lag 9. Although there is no
evidence of feedback, stock prices cause foreign exchange rates in variance at lags 1 and 9.
As mentioned earlier, stock prices and foreign exchange rates play crucial roles in influencing a country's economic development. The dynamic relationship between stock prices and foreign exchange
rates is an important area of study and is also analyzed by investors as a tool for
Causality in Mean
Causality in Variance
Figure 4.1.
Summary of Causality: Germany
Causality in Mean
Causality in Variance
.Figure 4.2.
Summary of Causality: Japan
Chapter .4
Causality in Mean
Causality in Variance
Figure .{3.
Summary of Causality: UK
Causality in Mean
-Causality in Variance
Figure 4.4.
Summary of Causality: USA
predicting future trends in both stock prices and exchange rates. Since the values of financial assets are determined by the present values of their future cash flows, expectations of relative
currency values play a considerable role in stock price movements, especially for internationally held financial assets. Therefore, stock price innovations may affect or be affected by exchange rate
dynamics. Existing empirical studies on information flows between stock and foreign exchange markets typically analyze causality in the mean relationship between data on stock prices and foreign
exchange rates. However, there is a growing literature on the relationship of conditional variances across financial markets and its implications concerning information transmission mechanisms. Ross
(1989), among others, points out that information transmission is primarily related to the volatility of price changes. This chapter attempted to characterize the pattern of information flows between
stock and foreign exchange markets by using price and volatility spillovers. The CCF approach is used to examine mean and variance causal relationships. Empirical results differ from country to
country. For Germany, foreign exchange rates possibly cause stock prices in mean at lags 5 and 9, whereas stock prices do not cause exchange rates in mean. There was no causal relationship between
the two variables in variance (Figure 4.1). For Japan, stock prices cause foreign exchange rates in mean at lags 6 and 15, whereas foreign exchange rates do not cause stock prices in variance. Stock
prices cause foreign exchange rates in variance at lags 1 and 4, whereas foreign exchange rates do not cause stock prices in variance (Figure 4.2). For the UK, foreign exchange rates cause stock
prices in mean at lags 11 and 14 in mean, whereas stock prices do not cause foreign exchange rates in mean. Stock prices cause foreign exchange rates at lag 19 in variance, whereas foreign exchange
rates cause stock prices in variance at lags 3 and 8 (Figure 4.3). For the USA, stock prices cause foreign exchange rates in mean at lags 9, whereas foreign exchange rates cause stock prices in mean
at lag 23. Stock prices cause foreign exchange rates in variance at lags 1 and 9, whereas foreign exchange rates do not cause stock prices in variance (Figure 4.4). From a practical viewpoint, most
investors believe that both stock prices and exchange rates can serve as instruments to predict each other's future path. The different empirical results among countries may be due not only to
observed financial factors, but also to deeper causes. Our empirical results show that it is possible to divide the four countries into two groups: Germany and Japan; and the UK and the USA. There is
no feedback effect between the stock price index and the effective exchange rate for the former group, whereas there is a feedback effect
APPENDIX 4.A
Table 4.10. Feedback between the Stock Market and the Foreign Exchange Market No Feedback
Germany Japan
between the two for the latter group (Table 4.1O). As pointed out by Nieh and Lee (2001), these results might be influenced by differences in each country's economic stage, government policy,
patterns of expectation, etc. Other crucial factors affecting the predicting power of stock prices and exchange rates include differences from country to country in the degree of
internationalization, liberalization, and capital control.
Notes 1 See Abdalla and Murinde (1997) and Jorion (1990).
APPENDIX 4.A The data are obtained from the International Financial Statistics of the International Monetary Fund. The series code of each data is as follows: Germany: stock price: share price
(13462...ZF) exchange rate: nominal effective exchange rate(134..NEUZF) Japan: stock price: share prices (15862...ZF) exchange rate: nominal effective exchange rate(158..NEUZF) UK:
stock price: industrial share prices (11262...ZF) exchange rate: nominal effective exchange rate(1l2..NEUZF) USA: stock price: industrial share prices (11l62...ZF) exchange rate: nominal effective
exchange rate(lll..NEUZF)
Chapter 5
The relationships among real economic variables, monetary variables and financial variables have long been topics of active economic research. A simple discount model shows that the fundamental value
of a firm's stock equals the present value of expected future dividends. Future dividends must ultimately reflect real economic activity. If all currently available information is taken into account,
there should be a close relationship between stock returns and expected future economic activity. To the extent that stock prices react quickly to. new information about the future, stock prices
should be a leading indicator of real economic activity, and the absence of any correlation between stock returns and future production growth rates would suggest that stock prices do not actually
reflect the underlying fundamental value. l For the USA, there is substantial evidence in favor of stock prices as a leading indicator of real economic activity. Similarly, the conditional variance
of stock prices depends On the conditional variances of expected future cash flows and of future discount rates, as well as on the conditional covariance between them. If discount rates are constant
over time, the conditional variance of stock prices is proportional to the conditional variance of expected future cash flows. Since the value of corporate equity at the aggregate level should depend
on the state of the economy, it is plausible that a change in the level of uncertainty about future macroeconomic conditions would produce a change in stock return volatility. 2
Information Asset
• Forward Looking
Figure 5.1. Forward-Looking Behavior
This way of thinking holds true for all asset prices, which are generally forward-looking. They reflect, and thus serve as a leading indicator of, future economic activity (Figure 5.1). This chapter
examines the relationships among the stock price indices, effective exchange rates, and industrial production levels for Germany, Japan, the UK, and the USA, using the CCF approach. Our investigation
was motivated by the Fama-Schwert findings for the USA, which established a relationship between industrial production growth and lagged real stock returns (Fama 1990 and Schwert 1990). Specifically,
we investigate two kinds of lead-lag relationships: the relationship between stock price indices and real economic activity, and the relationship between effective exchange rates and real economic
activity. Stock prices, foreign exchange rates, and real economic activities are first specified using the AR-EGARCH model, then, the causality in mean and in variance between the variables is
empirically analyzed using the two-step procedure developed by Cheung and Ng (1996). In using this causality methodology, we wish not only to investigate empirically the relationships among these
three variables, but also to analyze the hypothesis, which is popular in the financial press, that claims that asset prices are a leading indicator of future economic activity. The empirical evidence
for the USA suggests a statistically significant and economically important role for stock prices as a predictor of economic activity. This chapter sheds further light on the issue by investigating
the link between stock price (or exchange rate) changes and subsequent economic activity.
Selected Literature Review
Numerous researchers have examined the relationships among real, monetary, and financial variables. Examples include Schwert (1989),
Chapter 5
Table 5.1.
Summary of Literature
Technique and Main Results
1857 - 1987 (monthly
data), USA
1953 - 1987 (monthly, quarterly and annual data), USA
Regressionj The degree of correlation between the stock returns and production growth rates increases with the length of the holding period.
1989 - 1988 (monthly, quarterly and annual data), USA
Regressionj Fama's findings are robust for a much longer period.
Malliaris and Urrutia (1991)
January 1970 - June 1989 (monthly data), USA
VARj Fluctuations in stock market returns are a leading indicator of future real economic activity. However, the causal relationships among the rates of change and their volatilities for the three
variables are not as statistically significant as the economic and financial literature suggests.
Lee (1992)
January 1947 - December 1987 (monthly data), USA
VARj Stock returns help explain real economic activity, but not inflation.
Dropsy and NazarianIbrahimi
January 1970 - January 1990 (monthly data), Australia, Canada, France, Germany, Italy, Japan, Netherlands, Sweden, Switzerland, UK, and USA
Regressionj Anticipated macroeconomic policies do not help to predict real sock returns.
1920 - 1991 (monthly
GARCH and VAR; Significant relationships between stock market volatility and macroeconomic volatility are found.
Liljeblom and Stenius (1997)
data), Finland
Absolute residual modelj Macroeconomic volatility helps to predict stock and bond return volatility. Financial asset volatility helps to predict future macroeconomic volatility.
Table 5.1 (continued) Summary of Literature
Darrat and Dickens (1999)
January 1970 - June 1989 (monthly data), US
VECMj Result from multivariate cointegration and error-correction models reveal strong evidence of pronounced linkages among real, monetary, and financial sectors of the US economy.
Chai, Hauser, and Kopecky (1999)
January 1957 - March 1996 (monthly and quarterly data), G-7 countries
VECMj The cointegration test show a long-run equilibrium relationship between IP and real stock prices, while the error-correction models indicate a correlation between IP growth and lagged real
stock returns for all countries except Italy.
Park and Ratti (2000)
January 1955 - March 1998 (monthly data), USA
VARj Contractionary monetary policy shocks generate statistically significant movements in inflation and expected real stock returns.
Aylward and Glen (2000)
1951 - 1993 (annual data), 23 countries including 15 developing countries
Regression; Stock prices generally have predictive ability, but with substantial variation across countries. Moreover, stocks are substantially better leading indicators of investment than either GDP
or consumption.
Technique and Main Results
Fama (1990), Schwert (1990), Malliaris and Urrutia (1991), Lee (1992), Dropsy and Nazarian-Ibrahimi (1994), Liljeblom and Stenius (1997), Darrat and Dickens (1999), Choi, Hauser, and Kopecky (1999),
Park and Ratti (2000), and Aylward and Glen (2000). Table 5.1 is a summary of literature. Schwert (1989) analyzed the relationships between macroeconomic volatility and financial asset volatility
using monthly data for the USA over the period 1857 - 1987. 3 He used the absolute residual model to obtain volatility series and found weak evidence that macroeconomic volatility helps to predict
stock and bond return volatility. Stronger ev-
Chapter 5
idence suggested that financial asset volatility helped to predict future macroeconomic volatility. Fama (1990) showed that monthly, quarterly, and annual stock returns were highly correlated with
future production growth rates for the period 1953 - 1987. Moreover, the degree of correlation increased with the length of the holding period. He argued that the relation between current stock
returns and future production growth reflects the information about future cash flows that stock prices incorporate. Schwert (1990) analyzed the relation between real stock returns and real economic
activity from 1889 to 1988. He replicated Fama's (1990) results for the 1953 - 1987 period, using an additional 65 years of data. Fama's findings, however, were found to be robust for the much longer
period, as future production growth rates explained a larger fraction of the variation in stock returns. Fama (1990) and Schwert (1990) established a relationship between industrial production growth
and lagged real stock returns (Fama-Schwert findings). Malliaris and Urrutia (1991) analyzed the relationships among real, monetary and financial variables for the US economy, using Granger causality
tests on monthly data over the period from January 1970 to June 1989. Their findings suggest that fluctuations in stock market returns are a leading indicator of future real economic activity.
However, their empirical results also indicate that the causal relationships among the rates of change and their volatilities for the three variables are not as statistically significant as the
economic and financial literature suggests. Lee (1992) investigated causal relations and dynamic interactions among asset returns, real economic activity, and inflation in the postwar USA, using a
VAR approach. He used monthly data for the period from January 1947 to December 1987. His major findings were (1) stock returns help explain real economic activity, (2) stock returns explain little
of the variation in inflation, although interest rates do explain a substantial fraction of the variation in inflation, and (3) inflation explains little of the variation in real economic activity.
Dropsy and Nazarian-Ibrahimi (1994) analyzed the influence of underlying macroeconomic policies on stock returns using monthly data from January 1970 to January 1990 for 11 industrialized countries:
Australia, Canada, France, Germany, Italy, Japan, the Netherlands, Sweden, Switzerland, the UK, and the USA. They found that anticipated macroeconomic policies did not help to predict stock returns
in these countries since the advent of floating exchange rates in 1973. Liljeblom and Stenius (1997) investigated whether changes in stock market volatility over time can be attributed to the
time-varying volatility of a set of macroeconomic variables. They used monthly data for Fin-
land from 1920 to 1991. Conditional monthly volatility was measured as a simple weighted moving average and also obtained from GARCH estimations. The relationship between macroeconomic volatility and
stock market volatility was investigated by the estimation ofVAR models. The results were found to be strong, when compared with those based on US data. The results obtained for stock market
volatility as a predictor of macroeconomic volatility, and for the converse, were significant. Darrat and Dickens (1999) employed multivariate cointegration and error-correction modeling to
re-examine the Granger-causal relationships among industrial production, money stock, and the S&P 500 index, using monthly US data from January 1970 to June 1989. Unlike the conclusions of Malliaris
and Urrutia (1991), their multivariate results reveal evidence of cointegration and causal relationships among the three macro variables. This finding rejects the dichotomy thesis and provides a new
piece of evidence confirming the sensitivity of cointegration and Granger-causality tests to the omission-of-variables bias. 4 Choi, Hauser, and Kopecky (1999) examined the relationship between
industrial production growth rates and lagged real stock returns for the G-7 countries, using cointegration and error-correction models. They used monthly and quarterly data from January 1957 to
March 1996. The results showed that the log levels of industrial production and real stock prices were cointegrated in all the G-7 countries. In addition, over a short-term horizon, the
error-correction models indicated that the growth rate of industrial production was correlated with lagged real stock returns at some data frequencies in six of the G-7 countries, with Italy being
the only exception. Park and Ratti (2000) investigated the dynamic interdependencies among real economic activity, inflation, stock returns, and monetary policy, using a VAR model. They used monthly
US data from January 1955 to March 1998 and found that shocks due to monetary tightening generated statistically significant movements in inflation and expected real stock returns, and that these
movements were in opposite directions. Aylward and Glen (2000) empirically examined the extent to which stock market prices predicted future economic growth in income, consumption, and investment.
They used annual data for 23 countries, including 15 developing countries, for the period 1951 - 1993. 5 They found that stock prices generally do have some predictive power, the magnitude of which
may vary substantially across countries. Moreover, stocks were substantially better leading indicators of investment than were either GDP or consumption. 6 The results were stronger for the G-7
countries than for the emerging markets.
Chapter 5
Fama (1990), Malliaris and Urrutia (1991), Lee (1992), Dropsy and Nazarian-Ibrahimi (1994), Darrat and Dickens (1999), Choi, Hauser, and Kopecky (1999), Park and Ratti (2000), and Aylward and Glen
(2000) empirically analyzed the relationship between stock prices and real economic activity in mean. Schwert (1990), Malliaris and Urrutia (1991), and Liljeblom and Stenius (1997) investigated these
relationships in variance. No research has yet analyzed the relationship between stock prices and real economic activities in both mean and variance, in the same framework except for Malliaris and
Urrutia (1991).
The data consists of monthly observations of the aggregate stock price index, industrial production index, real effective exchange rates, and consumer price index for Germany, Japan, the UK, and the
USA from January 1980 to May 2001. The source is the International Financial Statistic of the International Monetary Fund. The sample period runs. Following Fama (1990) and Schwert (1990), industrial
production is used to both measure real economic activity and define each country's business cycle. 7 Real stock prices are obtained by dividing the period's nominal stock price index by the
corresponding consumer price index. Table 5.2 shows the summary statistics for the real growth rate of stocks, effective exchange rates, and industrial production index. The real growth rate of stock
prices, foreign exchange rates, and industrial production are calculated as Rt = (In Xt -In Xt-d x 100, where X t is the stock price index, effective exchange rate, or industrial production index at
time t. Thus, real growth rates are obtained for the period between February 1980 and May 2001. Table 5.2 shows the mean, standard deviation (Std. Dev.), skewness, kurtosis, and the Jarque-Bera
statistic with its associated probability value (P-value). The average growth rate of stocks is 0.564 percent for Germany, 0.300 percent for Japan, 0.527 percent for the UK, and 0.642 percent for the
USA. The standard deviation is 5.335 for Germany, 4.363 for Japan, 3.894 for the UK, and 3.605 for the USA. The skewness is -0.858 for Germany, -0.235 for Japan, -1.278 for the UK, and -0.702 for the
USA. The kurtosis is 5.822 for Germany, 3.727 for Japan, 9.713 for the UK, and 5.778 for the USA. The Jarque-Bera statistic (its associated Pvalue) is 116.383 (0.000) for Germany, 7.998 (0.018) for
Japan, 550.339 (0.000) for the UK, and 103.351 (0.000) for the USA. Thus, the null hypothesis of normal distribution is rejected for every country at the 5 percent significance level. The average
rate of foreign exchange rates is 0.078 percent for Germany, 0.173 percent for Japan, 0.101 percent for the UK, and 0.002 per-
Table 5.£.
Summary Statistics Germany
Std.Dev. Skewness Kurtosis J arque-Bera P-value
0.564 5.335 -0.858 5.822 116.383 0.000
0.300 4.363 -0.235 3.727 7.998 0.018
0.527 3.894 -1.278 9.713 550.339 0.000
,0.642 3.605 -0.702 5.778 103.351 0.000
REER Mean (%) Std.Dev. Skewness Kurtosis Jarque-Bera P-value
0.078 1.028 0.223 3.493 4.716 0.095
0.173 2.690 0.411 3.770 13.510 0.001
0.101 1.891 -0.423 4.834 43.531 0.000
0.002 1.890 -0.250 2.845 2.929 0.231
IP Mean (%) Std. Dev. Skewness Kurtosis Jarque-Bera P-value
0.094 1.770 0.246 11.723 814.128 0.000
0.133 1.670 -0.034 3.269 0.821 0.663
0.091 1.033 -0.382 3.965 16.161 0.000
0.221 0.676 -0.365 4.203 21.108 0.000
p(RSP,IP) p(REER,IP) p(RSP, REER)
0.039 -0.096 -0.047
0.023 0.009 0.133
0.022 -0.123 -0.052
-0.039 0.189 -0.072
Mean (%)
Note: IP is industrial production. REER is the real effective exchange rate. RSP is the real stock price index. Every variable Is measured as the change rate In percent. Jarque-Bera Is the
Jarque-Bera statistic to test for normality. P-value is the probability value associated with the Jarque-Bera statistic. The null hypothesis of normal distribution is rejected at the 5 percent
significance level if the P-value for the Jarque-Bera test is less than 0.05. p(RSP,IP) is the correlation coefficient between RSP and IP. p(REER, IP) is the correlation coefficient between REER and
IP. p(RSP, REER) is the correlation coefficient between RSP and REER.
cent for the USA. The standard deviation is 1.028 for Germany, 2.690 for Japan, 1.891 for the UK, and 1.890 for the USA. The skewness is 0.223 for Germany, 0.411 for Japan, -0.423 for the UK, and
-0.250 for the USA. The kurtosis is 3.493 for Germany, 3.770 for Japan, 4.834 for the UK, and 2.845 for the USA. The Jarque-Bera statistic (its associated P-value) is 4.716 (0.095) for Germany,
13.510 (0.001) for Japan, 43.531 (0.000) for the UK, and 2.929 (0.231) for the USA. Thus, the null
Chapter 5
Table 5.3. Unit Root Test
Test Statistics
Level Germany Japan UK USA First Difference Germany Japan UK USA
Level Germany Japan UK USA First Difference Germany Japan UK USA
Level Germany Japan UK USA First Difference Germany Japan UK USA
-2.301 -1.336 -2.299 -2.520
-1.011 -1.843 -1.510 -0.009
-1.173 -1.887 -2.247 -1.091
-14.782** -11.237** -12.682** -11.530**
-14.810** -11.232** -12.691** -11.539**
-14.708** -11.222** -12.597** -11.398**
-0.311 -2.101 -1.494 -0.864
-0.998 -1.568 -1.269 -1.010
0.913 0.954 0.643 -0.026
-10.110** -10.817** -10.907*· -11.426·*
-10.076** -10.823·* -10.881** -11.430**
-10.088** -10.823·· -10.886** -11.454**
-3.183 -1.027 -3.253 -2.557
-0.391 -1.814 -0.341 0.799
1.331 1.568 1.582 3.467
-24.997** -22.971·* -19.526·* -12.526**
-25.364** -22.692** -19.562** -12.417**
-25.916** -22.441*· -19.288** -12.068**
Note: * shows that the null hypothesis of a unit root is rejected at the 5 percent significance level. •• shows that the null hypothesis of a unit root is rejected at the 1 percent significance
level. CT corresponds to the following regression: ~lIt = 1'+6t+Wt-l +Ut. C corresponds to the following regression: ~lIt = I' + 'YlIl-l + Ul. None correspond to the following regression: ~I!l
= 'Yl!t-l + Ut·
hypothesis of normal distribution is rejected for Japan and the UK, but not for Germany and the USA at the 5 percent significance level. The average growth rate of industrial production is 0.094
percent for Germany, 0.133 percent for Japan, 0.091 percent for the UK, and 0.221 percent for the USA. The standard deviation is 1. 770 for Germany, 1.670 for Japan, 1.033 for the UK, and 0.676 for
the USA. The skewness is 0.246 for Germany, -0.034 for Japan, -0.382 for the UK, and -0.365 for the USA. The kurtosis is 11.723 for Germany, 3.269 for Japan, 3.965 for the UK, and 4.203 for the USA.
The Jarque-Bera statistic (its associated P-value) is 814.128 (0.000) for Germany, 0.821 (0.663) for Japan, 16.161 (0.000) for the UK, and 21.108 (0.000) for the USA. Thus, the null hypothesis of
normal distribution is rejected for every country except Japan at the 5 percent significance level. Table 5.2 also shows the correlation coefficients among variables. The correlation coefficient
between the real growth rate of stock prices and the growth rate of industrial production is 0.039 for Germany, 0.023 for Japan, 0.022 for the UK, and -0.039 for the USA. The correlation coefficient
between the real growth rate of foreign exchange and the growth rate of industrial production is -0.096 for Germany, 0.009 for Japan, -0.123 for the UK, and 0.189 for the USA. The correlation
coefficient between the real growth rate of stock prices and the real growth rate of foreign exchange is -0.047 for Germany, 0.i33 for Japan, -0.052 for the UK, and -0.072 for the USA. The unit root
test developed by Phillips and Perron (1988) is used to test that each variable has a unit root. The unit root test statistic is the t value of , obtained from the following regressions. (CT) (C)
6.Yt 6.Yt 6.Yt
= = =
+ M + ,Yt-l + Ut, J.L + ,Yt-l + Ut, ,Yt-l + Ut, J.L
where 6. is a difference operator, Le., 6.Yt = Yt -Yt-l, t is the time trend, and Ut is a disturbance term. The first equation (CT) includes a constant term and a time trend, the second equation (C)
includes a constant term only, and the third equation (None) includes no deterministic term. The null hypothesis (Ho) and the alternative hypothesis (HA) are shown as follows:
Ho: HA:
, = 0, , < O.
Thus, the null hypothesis shows that a unit root is included and the alternative hypothesis shows that a unit root is not included. Each
Chapter 5
equation is applied to both the level and the first difference of the log of the industrial production index, the log of the real effective exchange rate, and the log of the real stock price index.
In Table 5.3, RSP is the real stock price index, REER is the real effective exchange rate, and IP is industrial production. As an example, let us take a look at the results of Germany. For the level
and the first difference of the real stock price index, test statistics are respectively -2.301 and -14.782 for CT, -1.011 and -14.810 for C, and -1.173 and -14.708 for None. For the level and the
first difference of the real effective exchange rate, test statistics are respectively -0.311 and -10.110 for CT, -0.998 and -10.076 for C, and 0.913 and -10.088 for None. For the level and the first
difference of the industrial production index, test statistics are respectively -3.183 and -24.997 for CT, -0.391 and -25.364 for C, and 1.331 and -25.916 for None. Thus, the null hypothesis of a
unit root is not rejected in all specifications for the level of the real stock price index, real effective exchange rate, and industrial production index. The hull hypothesis is, however, rejected
for all specifications for the first difference of the real stock price index, real effective exchange rate, and industrial production index. These results are robust to all countries. Thus, the real
stock price index, real effective exchange rate, and industrial production are found to be a 1(1) process for all countries.
Empirical Technique
The CCF approach proposed by Cheung and Ng (1996) is employed to analyze the mean and variance causal relationships across markets. The first step involves the estimation of univariate time-series
models that allow for time variation in both conditional means and conditional variances. This section uses the AR(k)-EGARCH(p, q) specification for the first stage. The first difference of log value
is used for empirical analysis as follows: Yt = InXt - InXt - l , where X t is the real effective exchange rate, real stock price index, or industrial production at time t. The conditional mean and
conditional variance are respectively specified as follows:
Yt = 7ro +
L 7riYt-i +
logO'~ =
+ 'YiZt-i) + LA logO'~_i i=l
= Ed crt and It-l is the information set available at time Equations (5.1) and (5.2) show the AR(k) process and the EGARCH(p, q) process, respectively.
t - 1.
Each model is estimated by the method of maximum likelihood. Parameter estimates and their asymptotic standard errors, which are robust to departures from normality using the consistent
variancecovariance estimator of Bollerslev and Wooldridge (1992), are reported. Though the Cheung-Ng test results are robust to distributional assumptions, inferences about the EGARCH parameter
estimates may be sensitive to deviations from normality. Hence the Bollerslev-Wooldridge standard errors are reported. SBIC and the Ljung-Box test are used to specify the model. SBIC is often used
for model selection and smaller values of SBIC are preferred. The Ljung-Box test is used to check if there is no serial correlation in residuals. The choice of k, p, and q is carried out among k =
1,2, .. ,12, p=l,2, and q=l, 2 using SBIC and residual diagnostics. The second step of the CCF approach is to analyze the causality in mean and variance based on the empirical results of the
AR-EGARCH model. There is no evidence of causality in mean when all crosscorrelation coefficients calculated from standardized residuals, at all possible leads and lags, are not significantly
different from zero. Similarly, there is no evidence of causality in variance when all cross-correlation coefficients calculated from squares of standardized residuals, at all possible leads and
lags, are not significantly different from zero. The causality pattern is indicated by significant cross-correlations.
5. 5.1
Empirical Results Results for Germany
Let us examine the empirical results for Germany in this section. Table 5.4 shows the empirical results of the AR(k)-EGARCH(p, q) model for Germany. The choice of p and q is carried out among p=l, 2
and q=l,2 using SBIC and residual diagnostics. As a result, the AR(3)EGARCH(l,2) model is chosen for industrial production, the AR(2)EGARCH(l,2) model is chosen for the real effective exchange rate,
and the AR(1)-EGARCH(2,l) model is chosen for the real stock price index. In Chapter 2 and Chapter 3, the AR(2)-EGARCH(l,2) model was chosen for the nominal effective exchange rate and the AR(I)
-EGARCH(2,1) model was chosen for the nominal stock price index. Thus, model specification for real variables is found to be consistent with the model specification for nominal variables.
Chapter 5
Table 5.4.
Empirical results of the AR-EGARCH model: Germany
Mean equation: Yt 11'0 + E~=1 1I'iYt-i + ft Variance equation: logol = w + L~=1 (aiIZt-il Zt = ft/O't
+ "'YiZt-i) + L~=1 (3i logO'~_i'
11'0 SE(1I'0)
0.001 (0.001) -0.380** (0.046) -0.186*· (0.058) 0.116** (0.026)
0.001 (0.001) 0.452** (0.069) -0.157* (0.066)
0.006* (0.003) 0.030 . (0.058)
-6.762* (2.634) 0.547** (0.137) -0.085 (0.077)
-7.491** (1.930) 0.279 (0.146) 0.059 (0.075)
SE(-Y2) /31 SE(/31 ) /32 SECB2)
-0.285 (0.169) 0.539** (0.149)
0.790** (0.214) -0.563** (0.200)
-0.103 (0.074) 0.240 (0.127) -0.096 (0.087) -0.267 (0.145) 0.275** (0.092) 0.981** (0.010)
Log Likelihood Q(12) P-value Q2(12) P-value
711.448 13.553 (0.330) 11.495 (0.487)
833.380 12.761 (0.387) 4.228 (0.979)
SE(1I'1) 11'2 SE(1I'2) 11'3 SE(1I'3) W
SE(w) 01 SE(at} "'Yl
SE(-Yl ) a2
SE(02) "'Y2
410.847 7.192 (0.845) 9.394 (0.669)
Note: IP is industrial production. REER is the real effective exchange rate. RSP is the real stock price index. Each variable is measured as the first difference of log value. Numbers in parentheses
are Bollerslev-Wooldridge robust standard errors. Significance at the 1 percent level is indicated by .. and at the 5 percent level is indicated by *. Q(12) is the LjungBox statistic for the null
hypothesis that there is no autocorrelation up to order 12 for standardized residuals. Q2(12) is the Ljung-Box statistic for the null hypothesis that there is no autocorrelation up to order 12 for
standardized residuals Ilquared. P-vaJue Is the probability value associated with each test statistic. The null hypothesis is rejected at the 5 percent significance level if the P-value for each test
statistic is less than 0.05.
Table 5.5. Sample Cross-Correlation of Standardized Residuals: industrial production and real stock prices: Germany
Levels Lag Lead IP and RSP( -k) IP and RSP(+k)
Squares Lag Lead IP and RSP( -k) IP and RSP(+k)
-0.006 0.085 0.065 0.087 -0.062 0.213**
0.028 0.041 -0.005 0.069 -0.020 -0.141 *
-0.007 -0.091 -0.089 0.065 0.073 0.013
0.009 0.016 0.012 .0.052 -0.012 -0.006
0.031 0.034 0.050 0.157* 0.027 -0.030
-0.020 -0.066 -0.017 -0.063 0.022 -0.012
-0.081 0.006 -0.025 0.041 -0.020 -0.035
-0.029 -0.072 0.109 -0.040 -0.029 -0.001
0.090 0.050 -0.160* -0.011 -0.D15 -0.031
-0.019 -0.080 0.029 -0.053 -0.065 -0.107
0.045 0.043 0.200** 0.007 -0.007 0.128*
0.013 -0.043 0.044 0.026 -0.048 0.007
-0.011 -0.015 -0.037 0.092 -0.043 0.020
0.056 0.075 -0.004 -0.029 0.037 -0.023
0.055 -0.091 -0.013 0.011 -0.060 -0.067
-0.042 -0.032 -0.066 0.008 -0.040 -0.073
Note: IP is industrial production. RSP is the real stock price index. Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Lag refers to the number of
periods that industrial production lags behind the real stock price index, whereas Lead refers to the number of periods that industrial production leads the real stock price index. Cross-correlation
under the Levels column is based on standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized
residuals and is used to test forcau8ilIiYTri variance.
Chapter 5
Table 5.6. Sample Cross-Correlation of Standardized Residuals: industrial production and real effective exchange rate: Germany Levels k
IP and REER(-k)
IP and REER( +k)
IP and REER(-k)
Lead IP and REER( +k)
-0.162* 0.072 -0.077 0.016 -0.192"'* -0.088 -0.033
0.061 -0.025 0.010 0.080 0.000 -0.017
0.005 0.002 0.075 0.201* -0.073 0.003
-0.046 -0.046 0.015 -0.048 0.006 0.000
-0.008 -0.033 -0.011 0.064 0.025 -0.019
-0.018 -0.024 0.094 -0.010 0.039 0.010
0.031 -0.019 0.037 -0.048 -0.062 -0.096
0.069 0.033 -0.012 0.030 -0.058 0.012
-0.042 0.035 -0.051 -0.006 0.074 -0.010
-0.027 0.027 -0.054 -0.032 0.073 -0.023
0.040 0.007 0.166** -0.011 0.031 0.167**
-0.039 0.030 -0.045 0.125* 0.040 -0.006
-0.107 0.034 0.15S* 0.053 -0.061 0.033
-0.069 -0.044 -0.041 0.111 0.018 -0.043
-0.035 -0.034 -0.032 0.019 -0.077 -0.027
-0.023 -0.018 0.003 -0.041 0.001 -0.019
Note: IP is industrial production. REER is the real effective exchange rate. Significance at the 1 percent level and 5 percent level Is Indicated by ** and *, respectively. Lag refers to the number
of periods that Industrial production lags behind the real effective exchange rate, whereas Lead refers to the number of periods that Industrial production leads the real effective exchange rate.
Cr088-Correlation under the l&Y!l! column Is based on the standardized residuals themselves and is used to test for causality In mean. Cross-correlatlon under the Squares column is based on the
squares of standardized residuals and Is used to test for causality in variance.
As the table indicates, the coefficient of the GARCH term ({3) is estimated to be -0.285 and 0.539 for industrial production, 0.790 and -0.563 for the real effective exchange rate, and 0.981 for the
real stock price index, and most of them are statistically significant at the 1 percent level. The coefficient of the asymmetric effect (,) is estimated to be -0.085 for industrial production, 0.059
for the real effective exchange rate, and -0.096 and 0.275 for the real stock price index. However, their standard errors are large and, thus, are not statistically significant in many cases. Table
5.4 also shows the diagnostics of the empirical results of the AR-EGARCH model. The Ljung-Box test statistic at lag s, Q(s), is a test statistic for the null hypothesis that there is no
autocorrelation up to order s for standardized residuals and it is asymptotically distributed as X2 with degrees of freedom equal to the number of autocorrelation less the number of parameters. As
shown in the table, Q(12) (P-value) is 13.553 (0.330) for industrial production, 12.761 (0.387) for the real effective exchange rate, and 7.192 (0.845) for the real stock price index. Thus, the null
hypothesis of no autocorrelation up to order 12 for standardized residuals is accepted for all variables. This table also indicates the Q2(s) statistic and its associated P-value. The Q2 statistic at
lag s, Q2 (s ), is a test statistic for the null hypothesis that there is no autocorrelation up to order s for standardized residuals squared. As the table shows, Q2(12) (P-value) is 11.495 (0.487)
for industrial production, 4.228 (0.979) for the real effective exchange rate, and 9.394 (0.669) for the real stock price index. Thus, the null hypothesis of no autocorrelation up to order 12 for
standardized residuals squared is accepted for all cases. These results empirically support the specification of the AR-EGARCH model. Sample cross-correlations of standardized residuals and
standardized residuals squared are reported in Table 5.5 and Table 5.6. In Table 5.5, lag refers to the number of periods that industrial production lags behind real stock prices, whereas lead refers
to the number of periods that industrial production leads real stock prices. In other words, significant cross-correlation at a certain number of lags is interpreted as evidence of real stock prices
affecting industrial production, whereas significant cross-correlation at a certain number of leads is interpreted as evidence of industrial production affecting real stock prices. Cross-correlation
under the Levels column is based on standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized
residuals and is used to test for causality in variance.
Chapter 5
As is clear from the table, cross-correlation at lag 0 (contemporaneous correlation) is not statistically significant either in mean or in variance. The real stock price index causes industrial
production in mean at lags 6, 10, and 15, whereas industrial production causes the real stock price index in mean at lag 6. The real stock price index causes industrial production in variance at lags
15 and 18, whereas there is no causality from industrial production to the real stock price index in variance. In Table 5.6, lag refers to the number of periods that industrial production lags behind
the real effective exchange rate, and lead refers to the. number of periods that industrial production leads the real effective exchange rate. Significant cross-correlation at a certain number of
lags implies the real effective exchange rate affecting industrial production, while significant cross-correlation at a certain number of leads is interpreted as evidence of industrial production
affecting the real effective exchange rate. Cross-correlation under the Levels column is based on standardized residuals themselves and is used to test for causality in mean. Cross-correlation under
the Squares column is based on the squares of standardized residuals and is used to test for causality in variance. Contemporaneous correlation is -0.162 in mean and 0.140 in variance, and both of
them are statistically significant at the 5 percent level. Although there is no evidence of feedback, the real effective exchange rate causes industrial production in mean at lags 4 and 21. The real
effective exchange rate causes industrial production in variance at lags 4, 15, and 18, and industrial production causes the real effective exchange rate in variance at lag 16.
Results for Japan
Here we examine the empirical results for Japan. Table 5.7 shows the empirical results of the AR(k)-EGARCH(p, q) model for Japan. As in the previous section, the choice of p and q is carried out
among p=l, 2 and q=l, 2 using SBIC and residual diagnostics. As a result, the AR(4)EGARCH(I,I) model is chosen for industrial production, the AR(2)EGARCH(I,2) model is chosen for the real effective
exchange rate, and the AR(I)-EGARCH(I,I) model is chosen for the real stock price index. The model specification for real variables is found to be consistent with the model specification for nominal
variables, recalling that in Chapter 2 and Chapter 3 the AR(2)-EGARCH(l,2) model was chosen for the nominal effective exchange rate and the AR(I)-EGARCH(I,I) model was chosen for the nominal stock
price index. As shown in the table, the coefficient of the GARCH term (.8) is estimated to be 0.989 for industrial production, -0.820 and -0.648 for
Table 5.7.
Empirical results of the AR-EGARCH model: Japan
= 11"0 + 2:~=1 1I".Yt-. + Variance equation: logal = w + 2:f=1 (a.IZt_.1 + 1'.Zt-;) + 2::=1 /3.logal_i. Zt = ftleT! Mean equation: Yt
11"0 SE(7l"o)
0.001 (0.001) . -0.451 ** (0.061) 0.015 (0.065) 0.408** (0.066) 0.162*· (0.065)
0.001 (0.002) 0.334** (0.057) -0.152** (0.049)
0.002 (0.002) 0.346** (0.066)
w SE(w) at SE(at} 1'1 SE(1'd /31 SE(/31 ) /32 SE(,82)
-0.130 (0.121) 0.057 (0.041) -0.068* (0.029) 0.989** (0.014)
-18.893** (1.500) 0.319** (0.117) 0.102 (0.060) -0.820** (0.129) -0.684** (0.093)
-0.815* (0.396) 0.244·· (0.081) -0.080 (0.058) 0.903** (0.057)
Log Likelihood
722.132 19.209 (0.084) 12.316 (0.421)
585.823 13.730 (0.318) 13.709 (0.320)
467.689 12.330 (0.420) 9.283 (0.679)
SE(7l"I) 11"2 SE(1I"2) To3
SE(1I"3) 11"4 SE(1l"4)
Q(12) P-value Q2(12) P-value
Note: IP is industrial production. REER is the real effective exchange rate. RSP is the real stock price index. Each variable is measured as the first difference of log value. Numbers in parentheses
are Bollerslev-Wooldridge robust standard errors. Significance at the 1 percent level is indicated by·· and at the 5 percent level is indicated by •. Q(12) is the LjungBox statistic for the nun
hypothesis that there is no autocorrelation up to order 12 for standardized residuals. Q2(12) is the Ljung-Box statistic for the null hypothesis that there is no autocorrelation up to order 12 for
standardized residuals squared. P-value is the probability value associated with each test statistic. The null hypothesis Is rejected at the 5 percent significance level if the P-value for each test
statistic is less than 0.05.
Chapter 5
Table 5.8. Sample Cross-Correlation of Standardized Residuals: industrial production and real stock prices: Japan
Levels Lag Lead IP and RSP( -k) IP and RSP(+k)
Squares Lag Lead IP and RSP(-k) IP and RSP(+k)
0.007 0.065 -0.016 0.104 0.053 0.042
0.012 0.081 -0.082 0.023 0.050 -0.014
-0.001 -0.033 0.015 -0.005 0.023 0.056
0.064 -0.070 -0.056 0.031 -0.055 -0.066
0.108 0.058 0.101 0.059 -0.043 0.056
-0.007 -0.018 -0.003 0.040 -0.022 -0.028
0.062 0.003 0.046 0.080 -0.041 -0.016
-0.067 0.055 0.003 0.084 0.032 0.010
0.094 0.025 0.011 0.098 0.038 0.035
0.032 -0.056 -0.105 0.064 0.014 0.010
0.117 -0.007 -0.001 0.046 -0.015 0.057
-0.063 0.008 0.041 -0.011 0.029 0.068
0.018 -0.022 0.113 -0.056 -0.060 -0.016
-0.037 -0.037 0.057 0.069 -0.121 -0.129*
-0.055 -0.074 -0.060 -0.064 0.088 -0.005
0.031 -0.033 -0.156* 0.005 0.012 -0.038
Note: IP is industrial production. RSP is the real stock price Index. Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Lag refers to the number of
periods that industrial production lags behind the real stock price Index, whereas Lead refers to the number of periods that Industrial production leads the real stock price index. Cross-correlation
under the Levels column is based on the standardized residuals themselves and is used to test for causality In mean. Cross-correlation under the Squares column is based on the squares of standardized
residuals and is used to test for causaIi'iY1ii variance.
Table 5.9. Sample Cross-Correlation of Standardized Residuals: industrial production and real effective exchange rate: Japan
Levels Lag Lead IP and REER( -k) IP and REER(+k)
Squares Lag Lead. IP and REER(-k) IP and REER( +k)
-0.057 0.017 -0.098 -0.066 0.104 -0.034
0.092 0.028 0.036 -0.007 -0.058 -0.004
0.003 0.015 -0.047 -0.065 -0.126* 0.072
0.004 -0.058 -0.044 -0.039 -0.030 0.041
0.033 0.066 0.017 0.033 -0.048 -0.010
-0.041 0.006 0.005 -0.137* -0.109 -0.023
0.069 0.075 -0.055 0.129* 0.073 0.113
-0.055 -0.018 -0.090 -0.029 0.032 -0.033
0.004 0.026 -0.061 -0.043 -0.003 0.052
0.019 0.003 -0.060 -0.126* -0.077 -0.033
-0.001 -0.044 -0.027 -0.042 0.002 -0.009
0.042 0.080 -0.046 0.047 0.016 0.084
0.129* -0.034 0.088 0.080 0.000 -0.023
-0.065 0.082 0.000 -0.106 -0.051 -0.031
0.059 0.033 0.092 0.059 0.125* -0.007
-0.021 0.108 0.119 0.043 0.008 0.008
Note: IP is industrial production. REER is the real effective exchange rate. Significance at the 1 percent level and 5 percent level is indicated by ** and *. respectively. Lag refers to the number
of periods that industrial production lags behind the real effective exchange rate, whereas Lead refers to the number of periods that industrial production leads the real effective exchange rate.
Cross-correlation under the ~ column Is based on the standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of
standardized residuals and is used to test for causality in variance.
Chapter 5
the real effective exchange rate, 0.903 for the real stock price index, and all are statistically significant at the 1 percent level. As indicated in Chapter 3, negative coefficients of GARCH term
are not precluded in the EGARCH model, and thus the possibility of cyclical behavior in volatility is admitted. The coefficient of the asymmetric effect (-y) is estimated to be -0.068 for industrial
production, 0.102 for the real effective exchange rate, and -0.080 for the real stock price index. This coefficient is statistically significant for industrial production but not for the real
effective exchange rate and the real stock price index. 8 Table 5.7 also shows the diagnostics of the empirical results of the AR-EGARCH model. Q(s) is a test statistic for the null hypothesis that
there is no autocorrelation up to order s for standardized residuals, while Q2(s) is a test statistic for the null hypothesis that there is no autocorrelation up to order s for standardized residuals
squared. Q(12) (P-value) is 19.209 (0.084) for industrial production, 13.730 (0.318) for the real effective exchange rate, and 12.330 (0.420) for the real stock price index. Thus, the null hypothesis
of no autocorrelation up to order 12 for standardized residuals is accepted for all cases. As shown in the table, Q2(12) (P-value) is 12.316 (0.421) for industrial production, 13.709 (0.320) for the
real effective exchange rate, and 9.283 (0.679) for the real stock price index. Thus, the null hypothesis of no autocorrelation up to order 12 for standardized residuals squared is accepted for all
cases, and these results empirically support the specification of the AR-EGARCH model. Sample cross-correlations of standardized residuals and standardized residuals squared are reported in Table 5.8
and Table 5.9. In Table 5.8, lag refers to the number of periods that industrial production lags behind the real stock price index, and lead refers to the number of periods that industrial production
leads the real stock price index. Significant crosscorrelation at a certain number of lags is considered as an indication of the real stock price index affecting industrial production, while
significant cross-correlation at a certain number of leads indicates industrial production affecting the real stock price index. Cross-correlation at lag 0 (contemporaneous correlation) is not
statistically significant either in mean or in variance..The real stock price index does not cause industrial production in mean, whereas industrial production causes the real stock price index in
mean at lag 24. The real stock price index does not cause industrial production in variance, whereas industrial production causes the real stock price index in· variance at lag 21. In Table 5.9, lag
refers to the number of periods that industrial production lags behind the real effective exchange rate, while lead is the
number of periods that industrial production leads the real effective exchange rate. Significant cross-correlation at a certain number of lags is interpreted as evidence of the real effective
exchange rate affecting industrial production, and vice versa for significant cross-correlation at a certain number of leads. Contemporaneous correlation is 0.066 in mean and 0.017 in variance, but
none of them are statistically significant at the 5 percent level.· The real effective exchange rate causes industrial production in mean at lag 19, whereas industrial production causes the real
effective exchange rate at lags 10 and 16. Although there is no evidence of feedback, the real effective exchange rate causes industrial production in variance at lags 5, 10, and 23.
Results for the UK
Let us now examine the empirical results for the UK. Table 5.10 shows the empirical results of the AR(k)-EGARCH(p, q) model for the UK. The choice of p and q is carried out among p=l, 2 and q=l, 2
using SBIC and residual diagnostics with the resulting selection of models being AR(2)EGARCH(l,l) for industrial production, AR(l)-EGARCH(l,l) for the real effective exchange rate, and AR(2)-EGARCH
(I,I) for the real stock price index. Recalling that in Chapter 2 and Chapter 3, the AR(I)EGARCH (1,1) model was chosen for the nominal effective exchange rate and the AR(2)-EGARCH(2,1) model was
chosen for the nominal stock price index, the model specification for real variables is found to be similar to the model specification for nominal variables. As indicated, the coefficient of the
GARCH term (f3) is estimated to be 0.993 for industrial production, 0.784 for the real effective exchange rate, and 0.463 for the real stock price index. It is statistically significant at the 1
percent level for every case. The coefficient of the asymmetric effect (-y) is estimated to be 0.051 for industrial production, -0.074 for the real effective exchange rate, and -0.183 for the real
stock price index. This coefficient is statistically significant for the real stock price index but not for industrial production and the real effective exchange rate. The diagnostics of the
empirical results of the AR-EGARCH model are also indicated in Table 5.10. Q(s) and Q2(8) are Ljung-Box test statistics for the null hypothesis that there is no autocorrelation up to order s for
standardized residuals and standardized residuals squared, respectively. Q(12) (P-value) is 11.477 (0.489) for industrial production, 9.298 (0.677) for the real effective exchange rate, and 9.815
(0.632) for the real stock price index. Thus, the null hypothesis of no autocorrelation up to order 12 for standardized residuals is accepted for all variables. As indicated in the table, Q2(12)
(P-value) is 14.598 (0.264) for indus-
Chapter 5
Table 5.10.
Empirical results of the AR-EGARCH model: UK
Mean equation: tit 11'0 + E~=l 1I'iYt-i + ft Variance equation: logul w + E~=l (ailzt-il Zt
= ft/Ut
+ "'YiZt-i) + E~=1 Pi logul_i,
0.001* (0.0005) -0.208** (0.064) -0.035 (O.OSS)
0.000 (O.OOl) 0.344** (O.073)
0.005* (0.002) 0.233** (0.064) -0.124* (0.OS8)
al SE(aJ) "'Y1 SE('Y1) (31 SE«(3t)
-0.149 (0.078) 0.083 (0.049) 0.051 (0.029) 0.993** (O.007)
-2.111 (1.353) 0.457* (0.192) -0.074 (0.114) 0.784** (O.lS4)
-4.080** (1.217) 0.571** (0.160) -0.183* (0.084) 0.463** (O.17S)
Log Likelihood Q(12) P-value Q2(12) P-value
837.179 11.477 (0.489) 14.598 (0.264)
679.704 9.298 (O.677) 6.891 (0.86S)
496.895 9.815 (O.632) 12.126 (O.436)
SE( 7To) 7rl SE( 7T l) 11'2
SE( 7T2) w SE(w)
Note: IP is industrial production. REER is the real effective exchange rate. RSP is the real stock price index. Each variable is measured as the first difference of log value. Numbers in parentheses
are Bollerslev-Wooldridge robust standard errors. Significance at the 1 percent level is indicated by ** and at the 5 percent level is indicated by *. Q(12) is the LjungBox statistic for the null
hypothesis that there is no autocorrelation up to order 12 for standardized residuals. Q2(12) is the Ljung·Box statistic for the nun hypothesis that there is no autocorrelation up to order 12 for
standardized residuals squared. P-value is the probability value associated with each test statistic. The null hypothesis is rejected at the 5 percent significance level if the P-value for each test
statistic is less than 0.05.
trial production, 6.891 (0.865) for the real effective exchange rate, and 12.126 (0.436) for the real stock price index. Thus, the null hypothesis of no autocorrelation up to order 12 for
standardized residuals squared is accepted for all variables, and the specification of the AR-EGARCH model is supported by these results.
Table 5.11. Sample Cross-Correlation of Standardized Residuals: industrial production and real stock prices: UK
Levels Lag Lead IP and RSP( -k) IP and RSP(+k)
Squares Lag Lead IP and RSP( -k) IP and RSP(+k)
0.027 -0.031 0.122 0.101 0.103 0.086
0.007 -0.029 0.017 -0.049 -0.004 -0.112
0.038 -0.031 -0.010 -0.025 -0.062 0.001
-0.016 -0.034 0.038 ·0.044 0.021 -0.021
-0.018 0.078 0.003 -0.029 -0.064 0.059
-0.019 0.019 -0.097 0.114 0.1350.073
-0.043 0.010 -0.004 -0.070 0.068 -0.016
-0.011 0.1260.036 -0.070 -0.022 0.003
0.1340.020 0.002 0.145* 0.004 0.038
-0.048 -0.005 0.078 -0.011 -0.030 0.011
-0.025 -0.002 -0.078 0.057 0.044 0.161·
0.077 0.007 0.016 -0.059 0.024 0.026
-0.032 -0.107 0.013 -0.009 -0.059 0.015
-0.130· -0.001 -0.040 0.034 0.000 0.028
0.002 0.008 0.086 0.063 -0.036 -0.049
0.100 0.012 -0.082 0.176·· 0.019 0.022
Note: IP is industrial production. RSP is the real stock price index. Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Lag refers to the number of
periods that industrial production lags behind the real stock price Index, whereas Lead refers to the number of periods that industrial production leads the real stock price index. Cross-correlation
under the Levels column is based on the standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized
residuals and is used to test for cauSiiJIiYTri variance.
Chapter 5
Table 5.12. Sample Cross-Correlation of Standardized Residuals: industrial production and real effective exchange rate: UK
Levels Lag Lead IP and REER( -k) IP and REER(+k)
Squares Lag Lead IP and REER( -k) IP and REER( +k)
-0.100 0.039 -0.006 0.022 -0.018 0.010
0.033 0.035 0.019 -0.026 0.042 0.028
-0.020 0.010 -0.011 0.045 -0.016 -0.058
0.071 -0.018 -0.041 0.093 0.109 0.005
-0.048 -0.020 -0.012 -0.077 -0.028 -0.060
-0.095 -0.022 0.032 -0.061 -0.004 -0.033
-0.054 -0.099 0.064 -0.099 -0.032 -0.062
-0.025 -0.098 0.058 0.029 0.040 0.039
-0.024 -0.054 -0.028 0.071 0.121 -0.070
0.005 0.004 -0.025 -0.126* -0.032 0.115
0.024 0.052 -0.063 -0.014 0.016 -0.077
0.028 0.049 0.064 0.006 -0.001 0.095
-0.090 0.021 0.019 -0.118 -0.107 0.047
-0.002 0.002 0.013 0.051 0.026 -0.036
-0.027 -0.023 -0.020 0.083 0.021 -0.002
-0.054 -0.071 -0.020 -0.032 -0.016 -0.047
Note: IP is industrial production. REER is the real effective exchange rate. Significance at the 1 percent level and 5 percent level is indicated by ** and *, respectively. Lag refers to the number
of periods that industrial production lags behind the real effective exchange rate, whereas Lead refers to the number of periods that Industrial production leads the real effective exchange rate.
Cross-correlation under the ~ column is based on the standardized residuals themselves and is used to test for causality In mean. Cl'OII8-correlation under the Squares column is based on the squares
of standardized residuals and is used to test for causality in variance.
Table 5.11 and Table 5.12 indicate the sample cross-correlations of standardized residuals and standardized residuals squared. In Table 5.11, lag refers to the number of periods that industrial
production lags behind the real stock price index, and lead refers to the number of periods that industrial production leads the real stock price index. Significant cross-correlation at a certain
number of lags is an implication of the real stock price index affecting industrial production, while significant cross- . correlation at a certain number of leads is an indication of industrial
production affecting the real stock price index. As is clear from the table, cross-correlation at lag 0 (contemporaneous correlation) is not statistically significant either in mean or in variance.
The real stock price index causes industrial production in mean at lags 13 and 16, whereas industrial production causes the real stock price index in mean at lags 11 and 19. The real stock price
index causes industrial production in variance at lag 18, whereas industrial production causes the real stock price index in variance at lags 8 and 22. In Table 5.12, lag refers to the number of
periods that industrial production lags behind the real effective exchange rate, while lead is the number of periods that industrial production leads the real effective exchange rate. Significant
cross-correlation at a certain number of lags is interpreted as evidence of the real effective exchange rate affecting industrial production, and the opposite relationship is indicated for
significant cross-correlation at a certain number of leads. Contemporaneous correlation is -0.063 in mean and 0.013 in variance, but none of them are statistically significant at the 5 percent level.
Although there is no feedback, industrial production causes the real effective exchange rate in mean at lag 16. It is interesting to see that there is no causal relationship in the variance between
industrial production and the real effective exchange rate.
Results for the USA
Finally, let us examine the empirical results for the USA. Table 5.13 shows the empirical results of the AR(k)-EGARCH(p, q) model for the USA Following determination of p and q from p=1,2 and q=1,2
using SBIC and residual diagnostics, the resulting selection of models is AR(4)-EGARCH(1,1) for industrial production, AR(1)-EGARCH(2,2) for the real effective exchange rate, and AR(l)-EGARCH(l,l)
model for the real stock price index. Reflecting the selection of the AR(l)EGARCH(2,2) model for the nominal effective exchange rate and the AR(1)-EGARCH(1,1) model for the nominal stock price index
in Chapter 2 and Chapter 3, the model specification for real variables is found to be consistent with the model specification for nominal variables.
Chapter 5
Table 5.13. Empirical results of the AR-EGARCH model: USA
Mean equation: Yt 11'0 + E~=1 1I'iYt-i + Et Variance equation: logol w + Ef=1 (o;!Zt-il Zt
= Etlut
+ "YiZt-i) + E~=1 {3i logu~_i'
11'0 8E(1I'0) 11'1 SE(1I"1) 11'2 8E(1I'2) 11'3 8E(1I'3)
0.001** (0.0005) 0.119* (0.059) 0.158** (0.058) 0.172*· (0.063)
-0.001 (0.001) 0.332** (0.036)
0.005* (0.002) 0.276** (0.067)
-0.567·* (0.213) 0.035 (0.050) -0.143** (0.048)
-1.451·* (0.076) -0.346*· (0.064) 0.044 (0.038) 0.404** (0.063) -0.043 (0.040) 1.704*· (0.024) -0.874** (0.026)
-2.350 (1.272) 0.070 (0.182) -0.310* (0.121)
SE(w) 01 8E(0t} "Y1 8E(-yt} 02 8E(02) "Y2 8E(-yz) {31 SE({3t} {32 8E({32) Log Likelihood Q(12) P-value Q2(12) P-value
0.949** (0.021)
954.587 9.020 (0.701) 11.436 (0.492)
679.187 10.050 (0.612) 5.168 (0.952)
0.666·· (0.172)
510.681 6.933 (0.862) 14.383 (0.277)
Note: IP is industrial production. REER is the real effective exchange rate. RSP is the real stock price index. Each variable Is measured as the first difference of log value. Numbers in parentheses
are Bollerslev-Wooldridge robust standard errors. Significance at the 1 percent level is indicated by ** and at the 5 percent level is indicated by •. Q(12) is the LjungBox statistic for the null
hypothesis that there is no autocorrelation up to order 12 for standardized residuals. Q2(12) is the Ljung-Box statistic for the null hypothesis that there is no autocorrelation up to order 12 for
standardized residuals squared. P-value is the probability value associated with each test statistic. The null hypothesis is rejected at the 5 percent significance level If the P-value for each test
statistic is le88 than 0.05.
Table 5.14. Sample Cross-Correlation of Standardized Residuals: industrial production and real stock prices: USA
Levels Lag Lead IP and RSP{ -k) IP and RSP{+k)
Squares Lag Lead IP and RSP{ -k) IP and RSP{ +k)
-0.005 0.158* 0.162* 0.164** -0.092 0.157*
-0.099 0.096 0.048 -0.023 -0.095 0.007
-0.026 -0.019 0.041 0.012 0.039 -0.038
0.034 -0.001 0.068 -0.018 0.032 -0.064
0.027 0.004 0.021 0.080 0.099 0.092
-0.038 0.081 -0.016 -0.158* -0.029 -0.097
-0.063 -0.037 -0.057 -0.008 0.036 -0.001
0.128* 0.017 0.004 -0.012 -0.003 0.052
-0.052 -0.052 0.082 -0.009 0.054 -0.129*
-0.006 0.025 0.073 0.015 -0.096 0.032
0.022 -O.Oll 0.054 0.234** 0.013 0.002
0.063 -0.029 -0.003 -0.062 -0.007 0.090
-0.104 0.096 0.042 -0.024 -0.044 -0.046
0.026 O.Oll -0.112 0.109 0.131* -0.057
0.025 0.077 -0.076 -0.001 0.004 0.076
0.068 -0.002 0.040 -0.051 0.010 0.109
Note: IP is industrial production. RSP is the real stock price index. Significance at the 1 percent level and 5 percent level is indicated by ** and *. respectively. Lag refers to the number of
periods that industrial production lags behind the real stock price index, whereas Lead refers to the number of periods that industrial production leads the real stock price index. Cross-correlation
under the Levels column is based on the standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column is based on the squares of standardized
residuals and is used to test for causaIity in variance.
Chapter 5
Table 5.15. Sample Cross-Correlation of Standardized Residuals: industrial production and real effective exchange rate: USA
Levels Lag Lead IP and REER( -k) IP and REER(+k)
Squares Lag Lead IP and REER(+k) IP and REER( -k)
0.026 -0.014 -0.088 -0.022 -0.116 -0.039
-0.006 -0.062 -0.012 -0.009 -0.138* 0.018
-0.001 0.006 0.011 0.047 0.040 0.035
0.036 -0.019 0.061 0.159* 0.006 -0.064
0.031 -0.044 0.024 0.012 0.068 -0.079
0.091 0.075 0.046 0.134* -0.019 -0.010
0.064 -0.024 -0.053 -0.058 0.038 -0.060
-0.062 0.019 -0.059 0.126* -0.070 -0.022
-0.041 -0.D18 0.053 -0.046 -0.053 0.091
-0.046 0.069 0.049 0.114 -0.011 0.106
0.029 -0.072 0.221** -0.009 -0.022 0.026
0.024 -0.096 0.069 -0.037 0.025 0.006
0.077 -0.082 -0.D17 0.017 -0.006 0.026
0.046 0.037 0.049 -0.071 -0.034 -0.104
0.067 0.103 -0.008 -0.038 -0.003 -0.038
-0.041 0.074 0.197** -0.024 0.075 0.029
Note: IP is industrial production. REER is the real effective exchange rate. Significance at the 1 percent level and 5 percent level is indicated by ** and *. respectively. Lag refers to the number
of periods that industrial production lags behind the real effective exchange rate, whereas Lead refers to the number of periods that industrial production leads the real effective exchange rate.
Cross-correlation under the ~ column Is based on the standardized residuals themselves and is used to test for causality in mean. Cross-correlation under the Squares column Is based on the squares of
standardized residuals and is used to test for causality in variance.
As the table indicates, the coefficient of the GARCH term ({3) is estimated to be 0.949 for industrial production, 1.704 and -0.874 for the real effective exchange rate, and 0.666 for the real stock
price index. It is statistically significant at the 1 percent level for every case. The coefficient of the asymmetric effect (-y) is estimated to be -0.143 for industrial production, 0.044 and -0.043
for the real effective exchange rate, and -0.310 for the real stock price index. This coefficient is statistically significant for industrial production and the real stock price index but not for the
real effective exchange rate. Also shown are the diagnostics of the empirical results of the AREGARCH model. Q(s) is the Ljung-Box test statistic for the null hypothesis that there is no
autocorrelation up to order 8 for standardized residuals, and Q2(s) is likewise for standardized residuals squared. Q(12) (P-value) is 9.020 (0.701) for industrial production, 10.050 (0.612) for the
real effective exchange rate, and 6.933 (0.862) for the real stock price index. Thus, the null hypothesis of no autocorrelation up to order 12 for standardized residuals is accepted for all
variables. As indicated in the table, Q2(12) (P-value) is 11.436 (0.492) for industrial production, 5.168 (0.952) for the real effective exchange rate, and 14.383 (0.277) for the real stock price
index. Thus, the null hypothesis of no autocorrelation up to order 12 for standardized residuals squared is accepted for all variables. These results empirically support the specification of the
AR-EGARCH model. Sample cross-correlations of standardized residuals and standardized residuals squared are reported in Table 5.14 and Table 5.15. In Table 5.14, lag refers to the number of periods
that industrial production lags behind the real stock price index, and lead refers to the opposite relationship. Significant cross-correlation at a certain number of lags is interpreted as evidence
of the real stock price index affecting industrial production, and vice versa for significant cross-correlation at a certain number of leads. As is clear from the table, cross-correlation at lag 0
(contemporaneous correlation) is -0.069 in mean and 0.164 in variance, and is statistically significant in variance but not in mean. The real stock price index causes industrial production in mean at
lags 2, 3, 4, 6, and 18, whereas industrial production causes the real stock price index in mean at lags 10 and 23. The real stock price index causes industrial production in variance at lag 16,
whereas industrial production causes the real stock price index in variance at lag 7. In Table 5.15, lag refers to the number of periods that industrial production lags behind the real effective
exchange rate, whereas lead indicates the number of periods that industrial production leads the real
Chapter 5
effective exchange rate. Significant cross-correlation at a certain number of lags is an implication of the real effective exchange rate affecting industrial production, whereas significant
cross-correlation at a certain number of leads signifies the opposite. Contemporaneous correlation is 0.191 in mean and 0.029 in variance and is statistically significant in mean but not in variance
at the 5 percent level. Although there is no feedback, industrial production causes the real effective exchange rate in mean at lags 5 and 10. The real effective exchange rate causes industrial
production in variance at lag 15, and industrial production causes the real effective exchange rate in variance at lags 4, 10, and 21.
Many asset-pricing theories suggest that asset prices are forwardlooking and reflect market expectations of future earnings. When market prices are aggregated across companies, they may be used as
leading indicators of future growth in aggregate income. If this theory provides a correct empirical description of stock market (or foreign exchange rate market) behavior, lagged asset returns will
contain forward-looking forecasts of industrial production and should, therefore, be useful in predicting the growth rate of industrial production. In this chapter, data for Germany, Japan, the UK,
and the USA were used to examine the ability of stock market prices and foreign exchange rates to predict future economic growth in industrial production. Since all of these countries have
well-developed asset markets and high levels of per capita output, asset prices set by rational investors should exhibit patterns of correlation with the future growth of industrial production within
each country. For Germany, the real stock price index and the real effective exchange rate were found to be leading indicators of real economic activity in both mean and variance (Figure 5.2 and
Figure 5.3). For Japan, the real stock price index was not a leading indicator of real economic activity in either mean or variance, but the real effective exchange rate was a leading indicator of
real economic activity in both (Figure 5.4 and Figure 5.5). For the UK, the real stock price index was a leading indicator of real economic activity in both mean and variance, while the real
effective exchange rate was not a leading indicator of real economic activity in either mean or variance (Figure 5.6 and Figure 5.7). For the USA, the real stock price index was a leading indicator
of real economic activity in both mean and variance, while the real effective exchange rate was a leading indicator of real economic activity only in variance (Figure 5.8 and Figure 5.9).
Causality in Mean
-Causality in Variance
Figure 5.2.
Summary of Causality between RSP and IP: Germany
Causality in Mean
Causality in Variance
-Figure 5.9.
Summary of Causality between REER and IP: Germany
Chapter 5
Causality in Mean
Causality in Variance
Figure 5.4.
Summary of Causality between RSP and IP: Japan
Causality in Mean
-Causality in Variance
Figure 5.5.
Summary of Causality between REER and IP: Japan
Causality in Mean
-Causality in Variance
-Figure 5.6. Summary of Causality between RSP and IP: UK
Causality in Mean
Causality in Variance
Figure 5.7. Summary of Causality between REER and IP: UK
Chapter 5
Causality in Mean
Causality in Variance
-Figure 5.8.
Summary of Causality between RSP and IP: USA
Causality in Mean
Causality in Variance
-Figure 5.9.
Summary of Causality between REER and IP: USA
Thus, the findings of Fama and Schwert appear to hold true for Germany, the UK, and the USA but not for Japan. On the other hand, industrial production growth was significantly correlated with lagged
exchange rate market information for Germany, Japan, and, in part, the USA, but not for the UK. Thus, stock prices were substantially better indicators of real economic activity than exchange rates
for the UK and the USA. However, exchange rates were substantially better indicators of real economic activity than stock prices for Japan. Both were good indicators of real economic activity for
Germany. The result for Japan might be interpreted in one of two ways: either Japanese stock market expectations are too uncertain or too volatile to be of systematic assistance in forecasting future
industrial production growth, or the variance of innovations in other determinants of stock prices (e.g., the risk premium or risk-free rate) is so high that it overwhelms the information value of
real stock returns for industrial production growth. 9
Notes 1 Breeden (1986) developed a variant of the consumption-smoothing model in which expected returns were positively correlated with expected output growth. 2 A large number of papers have
investigated the excessive volatility of stock market returns and questioned the validity of the hypothesis that financial markets are efficient. Shiller (1989) argued that the volatility of
speculative asset prices is excessive, relative to the volatility of real or monetary variables. This evidence increases the challenge to business-cycle theorists, who must now explain not only the
potential relationships among changes in the levels of real, monetary, and financial variables, but also the relationships among their volatilities. 3 Also see Nakagawa and Osawa (2000). 4 This is a
good example of the omission-of-variables problem discussed in Chapter 2. 5 For some emerging economies, shorter sample periods were used, e.g., 1956 - 1993 for Brazil; 1956 - 1992 for Greece; 1964 -
1993 for Korea; 1961 - 1993 for Pakistan; and 1965 - 1993 for Taiwan. 6 As a benchmark, a ten-percent rise in stock prices was usually followed by an increase in GDP of one-half to one percent in the
following year. Similar results were found for consumption, but the effect on investment was stronger.
APPENDIX 5.A
7 In our empirical work we use industrial production as a proxy for aggregate corporate earnings because industrial production is the only aggregate data series available on a monthly basis. It
should be noted that our findings may be sensitive to the choice of industrial production as the aggregate proxy. 8 Hamori (2000b) empirically analyzed the quarterly data of Japan, the UK, and the
USA, and found that volatility was symmetric to real growth rates in real GDP. 9 See Choi, Hauser, and Kopecky (1999).
APPENDIX 5.A The data are obtained from the International Financial Statistics of the International Monetary Fund. The series code of each data is shown as follows: Germany: stock price: share price
(13462...ZF) price: consumer prices (13464.. ZF) production: industrial production (seasonally adjusted): 13466..ZF exchange rate: real effective exchange rate (134..REUZF) Japan: stock price: share
price (15862...ZF) price: consumer prices (15864.. ZF) production: industrial production (seasonally adjusted): 15866..ZF exchange rate: real effective exchange rate (158..REUZF) UK: stock price:
share price (11262... ZF) price: consumer prices (11264.. ZF) production: industrial production (seasonally adjusted): 11266..ZF exchange rate: real effective exchange rate (112..REUZF) USA: stock
price: share price (11162... ZF) price: consumer prices (11164.. ZF) production: industrial production (seasonally adjusted): 11166..ZF exchange ra.te: real effective exchange rate (lll .. REUZF)
Chapter 6
This chapter summarizes the contents of the other chapters in this volume and comments on the possible directions of future research. Chapter 2 analyzed the international transmission of stock prices
in Germany, Japan, the UK and the USA using two time series procedures, the standard VAR approach and the LA-VAR approach. The existing literature generally points out the importance of the
interdependence of stock markets, as well as the dominant effect of the US market. In the analysis conducted in Chapter 2, Germany, the UK and the USA turned out to be closely linked to each other,
whereas Japan tended to remain somewhat independent from the other three countries. Chapter 3 began by specifying the dynamics of stock prices using the AR-EGARCH model, and afterwards analyzed the
causality of stock prices in Germany, Japan, the UK and the USA, based on the test of causality in variance developed by Cheung and Ng (1996). The CCF approach was used to test for spillovers in the
conditional mean and volatility across countries. Volatility spillovers of this type might represent a causal phenomenon across markets, as well as global economic changes that concurrently alter
stock-return volatility across international stock markets. Compared with the Granger causality results reported in Chapter 2, cross-correlation statistics revealed more complex and dynamic causation
patterns. For example, the feedback effects in the means involve a high-order lag structure. There is also evidence that the causality in variance goes from one market to another and vice versa.
These results show that a proper account of conditional heteroskedasticity can have signif-
icant implications for the study of price and volatility spillovers. The information flow between international markets affects not only price movements, but also volatility movements. The use of the
CCF approach helped elucidate several interesting points. Although Japan was passive for causality in mean, it was active for causality in variance. Thus, while Japanese stock prices themselves had
no effects on other stock markets, uncertainty in Japanese stock prices might have had significant effects. Chapter 4 attempted to characterize the pattern of information flows between stock and
foreign exchange markets using price and volatility spillovers. A two-step procedure proposed by Cheung and Ng (1996) was used to examine the mean and variance causal relationships. From a practical
viewpoint, most investors believe that stock prices can serve as a useful instrument to predict the path of exchange rates, and vice versa. The different empirical results among countries might be
due to deeper underlying causes, in addition to observed financial factors. According to our empirical results, the four countries can be grouped into two pairs, namely the UK and the USA, and
Germany and Japan. The first two countries are subject to feedback effects between their stock price indices and effective exchange rates, while the second two countries are subject to no such
effects. As pointed out by Nieh and Lee (2001), these results might stem from differences in each country's economic stage, government policy, patterns of expectation, etc. The differences between
countries in their levels of internationalization, liberalization, and capital control might also be crucial factors affecting the predictive power of stock prices and exchange rates. Many asset
pricing theories suggest that asset prices are forward looking and reflect market expectations of future earnings. By aggregating across companies, aggregate market prices may be used as leading
indicators of the future growth in aggregate income. If this theory provides a correct empirical description of asset market behavior, lagged asset returns may prove useful for forecasts of the
growth in industrial production. In Chapter 5, the author used data from Germany, Japan, the UK and the USA to examine the ability of stock prices and foreign exchange rates to predict future
economic growth in industrial production. Given that all of these countries have well-developed asset markets and high levels of output per capita, the asset prices set by rational investors should
exhibit patterns of correlation with the future growth in industrial production within each country. The Fama-Schwert findings appear to hold true for Germany, the UK and the USA but not for Japan, a
country where industrial production is not significantly correlated with lagged stock market information. When
Chapter 6
we test for correlations between the growth in industrial production and lagged exchange rate market information, the correlations are very significant in Germany and Japan, somewhat significant in
the USA, and not significant in the UK. Thus, stock prices are a substantially better indicator of real economic activity than exchange rates in the UK and USA, whereas exchange rates are a
substantially better indicator than stock prices in Japan. Stock prices and exchange rates are both good indicators of real economic activity in Germany. The result for Japan might be interpreted in
one of two ways: either Japanese stock market expectations are too uncertain or too volatile to be of systematic assistance in forecaSting future industrial production growth, or the variance of
innovations in other determinants of stock prices (e.g., the risk premium or risk-free rate) is so high that it overwhelms the information value of real stock returns for· industrial production
growth. 1
Future Research Directions
Future research may move ahead in several directions. Specifically, I would like to point out three important directions that it has not been possible to address in this volume. Firstly, this volume
mainly uses the AR-EGARCH model as the first step of the CCF approach. While this model considers the prospect of occasional changes in the economic system, it omits specific consideration of
structural changes in the system. The GARCH effects are highly significant with daily and weekly data, but tend to be much milder in data that is sampled less frequently. The Markov switching
heteroskedasticity model has recently been adopted as an alternative method for dealing with GARCH effects in economic data. The main difference between the ARCH-type model and Markov switching model
is the treatment of the unconditional variance, which remains constant in the former model but, in the latter, changes in step with changes in the economy. As indicated by Perron (1987), if
researchers ignore the effects of structural change on the empirical analysis, they tend to find that the economic variables appear to have a high persistency. Diebold (1986) suggests that the high
persistence of volatility might be attributable to the regime shifts in the conditional variance. Lamoureux and Lastrapes (1990) provided corroborative evidence discounting the possibility that
simple structural shifts in unconditional volatility could lead to the spurious GARCH movements within regimes with an unconditional jump occurring between regimes. If this hypothesis is true, then
the Markov switching variance model should give us good empirical results. This is a promising line of research. 2
Secondly, this volume mainly analyzes the causality among variables without touching on the important issue of the economic structure behind the causality. The information on causality is useful for
constructing an explicit economic model to study. It will be left for future research to study an explicit economic model, consistent with the empirical findings in this volume. 3 Finally, this
volume focuses on the performance of four developed countries without taking up the important challenge of analyzing emerging markets. It would be meaningful to learn how economic performance affects
the movements of emerging markets and vice versa. As the linkage of the international capital markets grows in importance, it will not be appropriate to separate the analysis of developed and
developing markets. This line of research is also interesting and promising.
Notes 1 See Choi, Hauser, and Kopecky (1999). 2 Kim and Nelson (1999) is a good reference for the Markov switching model. 3 See Canova and Nicolo (1995), which is a good example of a structural
Abdalla, I. S. A. and Murinde, V. (1997) Exchange rate and stock price interactions in emerging financial markets: evidence on India, Korea, Pakistan and the Philippines, Applied Financial Economics
1, 25-35. Adler, M and Horesh, R. (1974) The relationship among equity markets: comment, Journal of Finance 29, 1311-1317. Agman, T. (1972) The relationship among equity markets: a study of share
price comovements in the United States, United Kingdom, Germany and Japan, Journal of Finance 21, 839-855. Agman, T. (1974) The relationship among equity markets: reply, Journal of Finance 29,
1318-1319. Ahlgren, N. and Antell, J. (2002) Testing for cointegration between international stock prices, Applied Financial Economics 12,851-861. Ajayi, R. A. and Mougoue, M. (1996) On the dynamic
relation between stock prices ad exchange rates, Journal of Financial Research 19, 193-207. Arshanapalli, B. and Doukas, J. (1993) International stock market linkages: evidence from the pre- and
post-October 1987 period, Journal of Banking and Finance 11, 193-208. Aylward, A. and Glen, J. (2000), Some international evidence on stock prices as leading indicators of economic activity, Applied
Financial Economics 10, 1-14 Bahmani-Oskooee, M. and Sohrabian, A. (1992) Stock prices and the effective exchange rate of the dollar, Applied Economics 24,459-464. Baillie, R. T. and Bollerslev, T.
(1991) Intra-day and inter-market volatility in foreign exchange markets, Review of Economic Studies 58, 565-585. Beaulieu, J. J. and Miron, J. A. (1993) Seasonal unit roots in aggregate U.S. data,
Journal of Econometrics 55, 305-328. Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307-327. Bollerslev, T., Chou, R. Y. and Kroner, K.
F. (1992) ARCH modeling in finance, Journal of Econometrics 52, 5-59. Bollerslev, T. and Wooldridge, J. M. (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time varying
covariances, Econometric Reviews 11, 143-172. Breeden, D. (1986) Consumption, production, inflation and interest rates: a synthesis, Journal of Financial Economics 16,3-39.
Canova, F. and Nicola, G. De. (1995) Stock returns and real activity: a structural approach, European Economic Review 39, 981-1015. Campbell, J. Y, Lo, A. W. and MacKinlay, A. C. (1997) The
Econometrics of Financial Markets, Princeton University Press. Cheung, Y-W. and Ng, L. K. (1996) A causality-in-variance test and its application to financial market prices, Journal of Econometrics
12,33-48. Cheung, YoW. and Fung, H.-G. (1997) Information flows between Eurodollar spot and futures markets, Multinational Finance Journal 1, 255-271. Choi, J. J., Hauser, S. and Kopecky, K. J.
(1999) Does the stock market predict real activity? time series evidence from the G-7 countries, Journal of Banking & Finance 23, 1771-1792. Chowdhury, A. R. (1994) Stock market interdependence:
evidence from the Asian NIEs, Journal of macroeconomics 16,629-651. Christie, A. A. (1982) The stochastic behavior of common stock variances: value, leverage and interest rate effects, Journal of
Financial Economics 10,407-432. Corhay, A., Rad, A. T., and Urbain, J. P. (1993) Common stochastic trends in European stock markets, Economics Letters 42, 385-390. Darrat, A. F. and Dickens, R. N.
(1999) On the interrelationships among real, monetary, and financial variables, Applied Financial Economics 9, 289-293. Dickey, D. A. and Fuller, W. A. (1979) Distribution of the estimators for
autoregressive time series with a unit root, Journal of the American Statistical Association 14, 427-431. Dickey, D. A. and Fuller, W. A. (1981) Likelihood ratio statistics for autoregressive time
series with a unit root, Econometrica 49, 1057-1072. Diebold, F. (1986) Comment on modelling the persistence of conditional variance, Econometric Reviews 5, 51-56. Dropsy, V. and Nazarian-Ibrahimi,
F. (1994) Macroeconomic policies, exchange rate regimes and national stock markets, International Review of Economics and Finance 3, 195-220. Enders, W. (1995) Applied Econometric Time Series, John
Wiley & Sons. Engle, R. F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, 987-1008. Engle, R. F. and Bollerslev, T.
(1986) Modeling the persistence of conditional variances, Econometric Reviews 5, 1-50. Engle, R. F. and Granger, C. W. J. (1987) Co-integration and error correction: representation, estimation, and
testing, Econometrica 55, 251-276. Engsted, T. and Lund, J. (1997) Common stochastic trends in international stock prices and dividends: an example of testing overidentifying restrictions on multiple
cointegration vectors, Applied Financial Economics 1,659-665. Eun, C. S. and Shin, S. (1989) International transmission of stock market movements, Journal of Financial and Quantitative Analysis 24,
241-256. Fama, E. F. (1990) Stock returns, expected returns, and real activity, Journal of Finance 45, 1089-1108. Franses, P. H. (1996) Periodicity and Stochastic 7rends in Economics Time Series,
Oxford University Press. Franses, P. H. and Hobijn, B. (1997) Critical values for unit root tests in seasonal time series, Journal of Applied Statistics 24, 25-47. Geweke, J., Meese, R. and Dent, W.
(1983) Comparing alternative tests of causality in temporal systems: analytic results and experimental evidence, Journal of Econometrics 21, 161-194.
Glosten, L. R., Jagannathan, R. and Runkle, D. E. (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance 48, 1779-1801.
Granger, C. W. J. (1969) Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, 424-438. Granger, C. W. J. (1986) Developments in the study of cointegrated
economic variables, Oxford Bulletin of Economics and Statistics 16, 199-211. Guilkey, D. K. and Salemi, M. K. (1982) Small sample properties of three tests for Granger-causal ordering in a bivariate
stochastic system, Review of Economics and Statistics 64, 668-680. Hamao, Y., Masulis, R. W. and N'g, V. (1990) Correlations in price changes and volatility across international stock markets, Review
of Financial Studies 3, 281307. Hamilton, J. D. (1994) Time Series Analysis, Princeton University Press. Hamilton, J. D. and Lin, G. (1996) Stochastic volatility and the business cycle, Journal of
Applied Econometrics 11, 573-593. Hamori, S. (2000a) The transmission mechanism of business cycles among Germany, Japan, the UK, and the USA, Applied Economics 32, 405-410. Hamori, S. (2000b)
Volatility ofreal GDP: some evidence from the United States, the United Kingdom and Japan, Japan and the World Economy 12, 143-152. Hamori, S. (2001) Seasonality and stock returns: some evidence from
Japan, Japan and the World Economy 13,463-481. Hamori, S. and Imamura, Y. (2000) International transmission of stock prices among G7 countries: LA-VAR approach, Applied Economics Letters 7,613-618.
Hamori, S. and Tokihisa, A. (1997) Testing for a unit root in the presence of a variance shift, Economics Letters 57, 245-253. Hamori, S. and Tokihisa, A. (2002) Some international evidence on the
seasonality of stock prices, International Journal of Business and Economics 1, 79-86. Harvey, C. R. (1991) The world price of covariance risk, Journal of Finance, 46, 111-157. Higgins, M. L. and
Bera, A. K. (1992) A class of nonlinear ARCH models, International Economic Review, 33, 137-158 Hong, Y. (2001) A test for volatility spillover with application to exchange rates, Journal of
Econometrics 103, 183-224. Hylleberg, S., Engle, R. F., Granger, C. W. J. and Yoo, B. S. (1990) Seasonal integration and cointegration, Journal of Econometrics 44, 215-238. Jarque C. M. and Bera A.
K. (1987) Tests for Normality of Observations and Regression Residuals, International Statistical Review 55, 163-172. Johansen, S. (1988) Statistical analysis of cointegration vectors, Journal of
Economic Dynamics and Control 12, 231-254. Johansen, S. and Juselius, K. (1990) Maximum likelihood estimation and inference on cointegration-with applications to the demand for money, Oxford Bulletin
of Economics and Statistics 52, 169-210. Jorion, P. (1990) The exchange-rate exposure of U.S. multinationals, Journal 0/ Business 63, 331-345. Karolyi, G. A. and Stulz, R. M. (1996) Why do markets
move together? An investigation of U.S.-Japan stock return comovements, Journal of Finance 51, 951-986. Kasa, K. (1992) Common stochastic trends in international stock markets, Journal of Monetary
Economics 29,95-124.
Kim, C. J. and Nelson, C. R. (1999) State Space Models with Regime Switching: classical and Gibbs-Sampling Approaches with Applications, The MIT Press. Kon, S. J. (1984) Models of stock returns: a
comparison 39, 147-165. Lamoureux, C. G. and Lastrapes, W. D. (1990) Persistence in variance, structural change, and the GARCH model, Journal of Business (; Economic Statistics 8, 225-234. Lee, B.-S.
(1992) Causal relationships among stock returns, interest rates, real activity, and inflation, Journal of Finance 41, 1591-1603. Liljeblom, E. and Stenius, M. (1997) Macroeconomic volatility and
stock market volatility: empirical evidence on Finish data, Applied Financial Economics 1, 419426. Liitkepohl, H. (1982) Non-causality due to omitted variables, Journal of Econometrics 19, 367-378.
Ljung, G. and Box, G. (1978) On a measure of lack of fit in time series models, Biometrika 66, 265-270. Malliaris, A. G. and Urrutia, J. L. (1991) An empirical investigation among real, monetary and
financial variables, Economics Letters, 31, 151-58. Mathur, I. and Subrahmanyam, V. (1990) Interdependencies among the Nordic and U.S. stock markets, Scandinavian Journal of Economics 92, 587-597.
Nakagawa, S. and Osawa, N. (2000) Financial market and macroeconomic volatility: relationship and some puzzles, Working Paper 00·9, Research and Statistics Department, Bank of Japan. Nelson, D. B.
(1991) Conditional heteroskedasticity in asset returns: a new approach, Econometrica 59, 347-370. Nelson, D. B. and Cao, C. Q. (1992) Inequality constraints in the univariate GARCH model, Journal of
Business & Economic Statistics 10, 229-235. Nieh, C.-C. and Lee C.-F. (2001) Dynamic relationship between stock prices and exchange rates for G-7 countries, Quarterly Review of Economics and Finance
41, 477-490. Ong, L. L. and Izan, H. Y. (1999) Stocks and currencies: are they related? Applied Financial Economics 9, 523-532. Osterwald-Lenum, M. (1992) A note with quantiles of the asymptotic
distribution of the maximum likelihood cointegration rank test statistics, Oxford Bulletin of Economics and Statistics 54, 461-472. Park, K. and Ratti, R. A. (2000) Real activity, inflation, stock
returns, and monetary policy, Financial Review 35, 59-78. Perman, R. (1991) Cointegration: an introduction to the literature, Journal of Economic Studies 18, 449-459. Perron, P. (1987) The great
crash, the oil price shock, and the unit root hypothesis, Econometrica 51, 1361-1401. Phillips, P. C. B. and Perron, P. (1988) Testing for a unit root in time series regression, Biometrika 15,
335-346. Pierce, D. A. and Haugh, L. D. (1977), Causality in temporal systems: characterizations and a survey, Journal of Econometrics 5, 265-93. Roll, R. (1988) R 2 , Journal of Finance 43, 541-566.
Ross, S. A. (1989) Information and volatility: the no-arbitrage Martingale approach to timing and resolution irrelevancy. Journal of Finance 44, 1-17. Said, S. E. and Dickey, D. A. (1985) Hypothesis
testing in ARIMA (P, 1,q) models, Journal of the American Statistical Association 80, 369-374.
Schwarz, G. (1978) Estimating the dimension of a model, Annals of Statistics 6, 461-464.
Schwert, G. W. (1989) Why does stock market volatility change over time? Journal of Finance 44, 1115-1153. Schwert, G. W. (1990) Stock returns and real activity: a century of evidence Journal of
Finance 45, 1237-1257. Sentana, E. (1995) Quadratic ARCH models, Review of Economic Studies 62, 639661.
Shiller, R. (1989) Market Volatility, The MIT Press. Sims, C. (1980) Macroeconomics and reality, Econometrica 48, 1-48. Shan, J. and Pappas, N. (2000) The relative impacts of Japanese and US interest
rates on local interest rates in Australia and Singapore: a Granger causality test, Applied Financial Economics 10, 291-298. Smith, C. E. (1992) Stock markets and the exchange rate: a multi-country
approach, Journal of Macroeconomics 14,607-629. Solnik, B. (1987) Using financial prices to test exchange rate models: a note, Journal of Finance 42, 141-149. Susmel, R. and Engle, R. F. (1994)
Hourly volatility spillover between national equity markets, Journal of International Money and Finance 13, 3-25. Taylor, S. J. (1986) Modeling Financial Time Series, John Wiley &: Sons. Toda, H. Y.
and Yamamoto, T. (1995) Statistical inference in vector autoregressions with possibly near integrated processes, Journal of Econometrics 66,225-250. Theodossiou, P. and Lee, U. (1993) Mean and
volatility spillovers across major national stock markets: further empirical evidence, Journal of Financial Research 16, 337·350. Tokihisa, A. and Hamori, S. (2001) Seasonal integration for daily
data, Econometric Reviews 20, 187-200. Watanabe, T. (2000) Volatility Hendou Model, Asakura-Shoten (written in Japanese). Zakoian, J-M. (1994) Threshold heteroskedastic models, Journal of Economic
Dynamics and Control 18, 931-955.
absolute error model, 35, 37 AR-EGARCH model, 43, 69, 70, 95, 100, 105, 109 ARCH model, 33, 37 ARCH-type models, 37 Bollerslev-Wooldridge robust standard errors, 43, 69, 70, 95, 100, 105, 109
Bollerslev-Woold ridge standard errors, 42 causality in mean, 78, 79, 114-117 causality in variance, 78, 79, 114-117 causality test, 20, 23 CCF approach, 2 Cheung-Ng Test, 44 Cheung-Ng test, 38
cointegrating vector, 22 cointegration, 8 cointegraticn test, 22 cross-correlation, 49-54, 73-76, 96, 97, 101, 102, 106, 107, 110, 111 EGARCH model, 36, 37 equilibrium error, 14 error-correction
representation, 14 error-correction term, 14 Fama.-Schwert findings, 84, 87, 122 feedback,81 forward-looking behavior, 84 GARCH model, 34, 37 GJR model, 36, 37
kurtosis, 12, 13, 33, 65, 90 LA-VAR,23 Lagrange multiplier test, 20, 21 Ljung-Box test, 43, 44, 69, 70, 94, 95, 100, 105, 109 long-run equilibrium, 14 maximum eigen-value test, 20, 22 mean, 12, 13,
65, 90 NGARCH model, 38 omission-of-variables problem, 27 QGARCH model, 38 SBIC, 21, 42, 94 skewness, 12, 13,65, 90 squares of standardized residuals, 49-54, 73-76, 96, 97, 101, 102, 106, 107, 110,
111 standard deviation, 12, 13, 65, 90 standardized residuals, 49-54, 73-76, 96, 97, 101, 102, 106, 107, 110, 111 summary of literature, 9, 85, 86 summary statistics, 13, 65, 90 trace test, 20, 22
two-step procedure, 39 unit root test, 15, 66, 67, 91
integrated of order one, 8
VAR, 14,23 VECM,14 volatility clustering, 32
Jarque-Bera test, 13, 65, 90 Jensen's inequality, 34, 58
Wald test, 25 | {"url":"https://vdoc.pub/documents/an-empirical-investigation-of-stock-markets-the-ccf-approach-smps08eh8f80","timestamp":"2024-11-06T01:42:32Z","content_type":"text/html","content_length":"284914","record_id":"<urn:uuid:0c7641ff-6af9-45ea-8bef-629026d815ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00802.warc.gz"} |
Background Checks with Lastname
Search over 300 million background checks which include full name, current address, phone number, address history, marriage records, divorce records, criminal records, criminal background checks,
court records, police records, bankruptcy records, property records, arrest records, driving offenses records, felonies, misdemeanors, sex offender records, convictions, incarcerations, inmate
records, and other court documents.
Michael Hardaway, Christopher Hardaway, Jason Hardaway, David Hardaway, James Hardaway, John Hardaway, Robert Hardaway, Brian Hardaway, William Hardaway, Matthew Hardaway, Joseph Hardaway, Daniel
Hardaway, Kevin Hardaway, Eric Hardaway, Jeffrey Hardaway, Richard Hardaway, Scott Hardaway, Mark Hardaway, Steven Hardaway, Thomas Hardaway, Timothy Hardaway, Anthony Hardaway, Charles Hardaway,
Joshua Hardaway, Ryan Hardaway, Jeremy Hardaway, Paul Hardaway, Andrew Hardaway, Gregory Hardaway, Chad Hardaway, Kenneth Hardaway, Jonathan Hardaway, Stephen Hardaway, Shawn Hardaway, Aaron
Hardaway, Adam Hardaway, Patrick Hardaway, Justin Hardaway, Sean Hardaway, Edward Hardaway, Todd Hardaway, Donald Hardaway, Ronald Hardaway, Benjamin Hardaway, Keith Hardaway, Bryan Hardaway, Gary
Hardaway, Jose Hardaway, Nathan Hardaway, Douglas Hardaway, Brandon Hardaway, Nicholas Hardaway, George Hardaway, Travis Hardaway, Peter Hardaway, Craig Hardaway, Bradley Hardaway, Larry Hardaway,
Dennis Hardaway, Shane Hardaway, Raymond Hardaway, Troy Hardaway, Jerry Hardaway, Samuel Hardaway, Frank Hardaway, Jesse Hardaway, Jeffery Hardaway, Juan Hardaway, Terry Hardaway, Corey Hardaway,
Phillip Hardaway, Marcus Hardaway, Derek Hardaway, Rodney Hardaway, Joel Hardaway, Carlos Hardaway, Randy Hardaway, Jacob Hardaway, Jamie Hardaway, Tony Hardaway, Russell Hardaway, Brent Hardaway,
Billy Hardaway, Antonio Hardaway, Derrick Hardaway, Kyle Hardaway, Erik Hardaway, Johnny Hardaway, Marc Hardaway, Carl Hardaway, Philip Hardaway, Roger Hardaway, Bobby Hardaway, Brett Hardaway, Danny
Hardaway, Curtis Hardaway, Jon Hardaway, Vincent Hardaway, Cory Hardaway, Jimmy Hardaway, Victor Hardaway, Lawrence Hardaway, Dustin Hardaway, Gerald Hardaway, Walter Hardaway, Alexander Hardaway,
Joe Hardaway, Christian Hardaway, Chris Hardaway, Alan Hardaway, Shannon Hardaway, Wayne Hardaway, Jared Hardaway, Gabriel Hardaway, Martin Hardaway, Jay Hardaway, Willie Hardaway, Luis Hardaway,
Micheal Hardaway, Henry Hardaway, Wesley Hardaway, Randall Hardaway, Brad Hardaway, Darren Hardaway, Roy Hardaway, Albert Hardaway, Arthur Hardaway, Ricky Hardaway, Lance Hardaway, Allen Hardaway,
Lee Hardaway, Bruce Hardaway, Andre Hardaway, Mario Hardaway, Frederick Hardaway, Louis Hardaway, Darrell Hardaway, Damon Hardaway, Shaun Hardaway, Nathaniel Hardaway, Zachary Hardaway, Casey
Hardaway, Adrian Hardaway, Jesus Hardaway, Jeremiah Hardaway, Jack Hardaway, Ronnie Hardaway, Dale Hardaway, Tyrone Hardaway, Manuel Hardaway, Ricardo Hardaway, Harold Hardaway, Kelly Hardaway, Barry
Hardaway, Reginald Hardaway, Ian Hardaway, Glenn Hardaway, Ernest Hardaway, Steve Hardaway, Seth Hardaway, Eugene Hardaway, Clinton Hardaway, Miguel Hardaway, Tommy Hardaway, Eddie Hardaway, Leonard
Hardaway, Maurice Hardaway, Roberto Hardaway, Dwayne Hardaway, Jerome Hardaway, Ralph Hardaway, Marvin Hardaway, Francisco Hardaway, Jorge Hardaway, Neil Hardaway, Alex Hardaway, Dean Hardaway,
Kristopher Hardaway, Calvin Hardaway, Kurt Hardaway, Theodore Hardaway, Ruben Hardaway, Jermaine Hardaway, Tracy Hardaway, Edwin Hardaway, Stanley Hardaway, Melvin Hardaway, Howard Hardaway, Mitchell
Hardaway, Duane Hardaway, Trevor Hardaway, Jeff Hardaway, Geoffrey Hardaway, Hector Hardaway, Terrence Hardaway, Terrance Hardaway, Oscar Hardaway, Jaime Hardaway, Clifford Hardaway, Harry Hardaway,
Kirk Hardaway, Tyler Hardaway, Jody Hardaway, Greg Hardaway, Earl Hardaway, Angel Hardaway, Daryl Hardaway, Mathew Hardaway, Heath Hardaway, Clayton Hardaway, Clarence Hardaway, Karl Hardaway, Raul
Hardaway, Alfred Hardaway, Isaac Hardaway, Javier Hardaway, Wade Hardaway, Mike Hardaway, Luke Hardaway, Ramon Hardaway, Joey Hardaway, Warren Hardaway, Ray Hardaway, Francis Hardaway, Norman
Hardaway, Toby Hardaway, Leon Hardaway, Glen Hardaway, Lonnie Hardaway, Rafael Hardaway, Cody Hardaway, Pedro Hardaway, Byron Hardaway, Fred Hardaway, Franklin Hardaway, Omar Hardaway, Fernando
Hardaway, Alejandro Hardaway, Bernard Hardaway, Clint Hardaway, Darin Hardaway, Gilbert Hardaway, Andy Hardaway, Lucas Hardaway, Alvin Hardaway, Cedric Hardaway, Eduardo Hardaway, Roderick Hardaway,
Armando Hardaway, Don Hardaway, Leroy Hardaway, Darryl Hardaway, Dana Hardaway, Jessie Hardaway, Cameron Hardaway, Ross Hardaway, Kelvin Hardaway, Marco Hardaway, Evan Hardaway, Johnathan Hardaway,
Clifton Hardaway, Fredrick Hardaway, Vernon Hardaway, Dominic Hardaway, Lewis Hardaway, Colin Hardaway, Donnie Hardaway, Damian Hardaway, Rene Hardaway, Stacy Hardaway, Herbert Hardaway, Sergio
Hardaway, Alberto Hardaway, Grant Hardaway, Leslie Hardaway, Jonathon Hardaway, Erick Hardaway, Marlon Hardaway, Julio Hardaway, Ivan Hardaway, Dwight Hardaway, Micah Hardaway, Darrin Hardaway,
Brendan Hardaway, Spencer Hardaway, Orlando Hardaway, Nelson Hardaway, Lloyd Hardaway, Julian Hardaway, Edgar Hardaway, Marshall Hardaway, Kent Hardaway, Jarrod Hardaway, Lamont Hardaway, Kerry
Hardaway, Trent Hardaway, Garrett Hardaway, Gordon Hardaway, Alfredo Hardaway, Damien Hardaway, Charlie Hardaway, Stuart Hardaway, Demetrius Hardaway, Devin Hardaway, Guy Hardaway, Gene Hardaway,
Arturo Hardaway, Rick Hardaway, Bryant Hardaway, Enrique Hardaway, Cesar Hardaway, Scotty Hardaway, Jim Hardaway, Neal Hardaway, Jimmie Hardaway, Johnnie Hardaway, Jackie Hardaway, Allan Hardaway,
Antoine Hardaway, Ben Hardaway, Marty Hardaway, Bradford Hardaway, Chadwick Hardaway, Floyd Hardaway, Jayson Hardaway, Noah Hardaway, Gerardo Hardaway, Dan Hardaway, Bret Hardaway, Felix Hardaway,
Preston Hardaway, Roland Hardaway, Gregg Hardaway, Angelo Hardaway, Ted Hardaway, Jordan Hardaway, Blake Hardaway, Freddie Hardaway, Tyson Hardaway, Israel Hardaway, Lorenzo Hardaway, Leo Hardaway,
Perry Hardaway, Tim Hardaway, Ethan Hardaway, Rickey Hardaway, Bill Hardaway, Salvador Hardaway, Noel Hardaway, Abraham Hardaway, Lester Hardaway, Ernesto Hardaway, Dewayne Hardaway, Rudy Hardaway,
Nick Hardaway, Tom Hardaway, Carlton Hardaway, Austin Hardaway, Darnell Hardaway, Andres Hardaway, Kenny Hardaway, Terence Hardaway, Herman Hardaway, Milton Hardaway, Jamal Hardaway, Frankie
Hardaway, Robbie Hardaway, Marcos Hardaway, Brady Hardaway, Robin Hardaway, Matt Hardaway, Courtney Hardaway, Sam Hardaway, Quincy Hardaway, Terrell Hardaway, Clyde Hardaway, Clay Hardaway, Quentin
Hardaway, Bryce Hardaway, Ron Hardaway, Stacey Hardaway, Max Hardaway, Ashley Hardaway, Cecil Hardaway, Drew Hardaway, Dion Hardaway, Lamar Hardaway, Chester Hardaway, Alfonso Hardaway, Brock
Hardaway, Rodolfo Hardaway, Dylan Hardaway, Randolph Hardaway, Josh Hardaway, Gerard Hardaway, Loren Hardaway, Pablo Hardaway, Caleb Hardaway, Cornelius Hardaway, Salvatore Hardaway, Kendrick
Hardaway, Timmy Hardaway, Harvey Hardaway, Sammy Hardaway, Dexter Hardaway, Nicolas Hardaway, Malcolm Hardaway, Sidney Hardaway, Levi Hardaway, Bryon Hardaway, Julius Hardaway, Donovan Hardaway,
Oliver Hardaway, Wendell Hardaway, Arnold Hardaway, Kris Hardaway, Rocky Hardaway, Dallas Hardaway, Rolando Hardaway, Everett Hardaway, Garry Hardaway, Rusty Hardaway, Myron Hardaway, Scottie
Hardaway, Gilberto Hardaway, Eli Hardaway, Cary Hardaway, Sheldon Hardaway, Simon Hardaway, Otis Hardaway, Abel Hardaway, Alonzo Hardaway, Scot Hardaway, Jamey Hardaway, Kendall Hardaway, Elijah
Hardaway, Rex Hardaway, Randal Hardaway, Jake Hardaway, Guillermo Hardaway, Roman Hardaway, Morgan Hardaway, Benny Hardaway, Felipe Hardaway, Sherman Hardaway, Derick Hardaway, Ismael Hardaway,
Claude Hardaway, Hugh Hardaway, Mickey Hardaway, Stephan Hardaway, Darrick Hardaway, Damion Hardaway, Virgil Hardaway, Darius Hardaway, Beau Hardaway, Ira Hardaway, Rogelio Hardaway, Devon Hardaway,
Desmond Hardaway, Owen Hardaway, Teddy Hardaway, Gustavo Hardaway, Bart Hardaway, Earnest Hardaway, Kareem Hardaway, Curt Hardaway, Lyle Hardaway, Elias Hardaway, Forrest Hardaway, Erin Hardaway,
Thaddeus Hardaway, Stewart Hardaway, Dante Hardaway, Dominick Hardaway, Xavier Hardaway, Wallace Hardaway, Ramiro Hardaway, Guadalupe Hardaway, Alton Hardaway, Pete Hardaway, Morris Hardaway, Jess
Hardaway, Monte Hardaway, Moses Hardaway, Carey Hardaway, Donny Hardaway, Colby Hardaway, Saul Hardaway, Trenton Hardaway, Marion Hardaway, Rory Hardaway, Tracey Hardaway, Darrel Hardaway, Gavin
Hardaway, Reynaldo Hardaway, Erich Hardaway, Bennie Hardaway, Fabian Hardaway, Luther Hardaway, Tomas Hardaway, Blaine Hardaway, Jerald Hardaway, Dave Hardaway, Shad Hardaway, Edmund Hardaway, Reuben
Hardaway, Clark Hardaway, Kory Hardaway, Jarrett Hardaway, Sonny Hardaway, Donnell Hardaway, Demond Hardaway, Robby Hardaway, Aron Hardaway, Leonardo Hardaway, Jamison Hardaway, Lionel Hardaway,
Kurtis Hardaway, Deon Hardaway, Elliott Hardaway, Stefan Hardaway, Emanuel Hardaway, Nickolas Hardaway, Efrain Hardaway, Kristian Hardaway, Archie Hardaway, Joesph Hardaway, Graham Hardaway, Lynn
Hardaway, Jerrod Hardaway, Wilson Hardaway, Esteban Hardaway, Reggie Hardaway, Mason Hardaway, Sylvester Hardaway, Leland Hardaway, Hugo Hardaway, Willard Hardaway, Vance Hardaway, Ken Hardaway,
Miles Hardaway, Taylor Hardaway, Jeffry Hardaway, Will Hardaway, Jean Hardaway, Rodrick Hardaway, Willis Hardaway, Antwan Hardaway, Hans Hardaway, Rudolph Hardaway, Daren Hardaway, Jefferson
Hardaway, Roosevelt Hardaway, Amos Hardaway, Emmanuel Hardaway, Elmer Hardaway, Grady Hardaway, Joaquin Hardaway, Tommie Hardaway, Jeromy Hardaway, Dorian Hardaway, Dane Hardaway, Tobias Hardaway,
Anton Hardaway, Santiago Hardaway, Sterling Hardaway, Freddy Hardaway, Shon Hardaway, Rico Hardaway, Jarvis Hardaway, Brant Hardaway, Monty Hardaway, Collin Hardaway, Nolan Hardaway, Alexis Hardaway,
Quinton Hardaway, Bob Hardaway, Stevie Hardaway, Avery Hardaway, Aric Hardaway, Emilio Hardaway, Shayne Hardaway, Carlo Hardaway, Conrad Hardaway, Jed Hardaway, Denny Hardaway, Elliot Hardaway,
Laurence Hardaway, Delbert Hardaway, Jennifer Hardaway, Doug Hardaway, Landon Hardaway, Moises Hardaway, Zachariah Hardaway, Kristofer Hardaway, Buddy Hardaway, Ignacio Hardaway, Elvis Hardaway,
Chuck Hardaway, Torrey Hardaway, Jonas Hardaway, Humberto Hardaway, Aubrey Hardaway, Elton Hardaway, Chance Hardaway, Marlin Hardaway, Vicente Hardaway, Solomon Hardaway, Horace Hardaway, Kim
Hardaway, Tory Hardaway, Royce Hardaway, Wilbert Hardaway, Dirk Hardaway, Thad Hardaway, Ervin Hardaway, Louie Hardaway, Jonah Hardaway, Raphael Hardaway, Wilfredo Hardaway, Santos Hardaway, Shelby
Hardaway, Noe Hardaway, Hubert Hardaway, Mack Hardaway, Van Hardaway, Branden Hardaway, Ali Hardaway, Marcel Hardaway, Jamel Hardaway, Jamar Hardaway, Jasen Hardaway, Rodger Hardaway, Pierre
Hardaway, Blair Hardaway, Harley Hardaway, Johnathon Hardaway, Winston Hardaway, Ellis Hardaway, Marquis Hardaway, Galen Hardaway, Dewey Hardaway, Reid Hardaway, Bert Hardaway, Brain Hardaway, Bradly
Hardaway, Dusty Hardaway, Darian Hardaway, Wyatt Hardaway, Alphonso Hardaway, Vaughn Hardaway, Brenton Hardaway, Waylon Hardaway, Benito Hardaway, Kirby Hardaway, Jeramy Hardaway, Jarod Hardaway,
Isaiah Hardaway, Lane Hardaway, Rufus Hardaway, Domingo Hardaway, Jackson Hardaway, Jerod Hardaway, Edmond Hardaway, Tad Hardaway, Ernie Hardaway, Quinn Hardaway, Percy Hardaway, Andrea Hardaway,
Lowell Hardaway, Cornell Hardaway, Rhett Hardaway, Shelton Hardaway, Rickie Hardaway, Korey Hardaway, Cleveland Hardaway, Deandre Hardaway, Nathanael Hardaway, Arron Hardaway, Theron Hardaway, Boyd
Hardaway, Logan Hardaway, Ariel Hardaway, Brendon Hardaway, Adan Hardaway, Donte Hardaway, Doyle Hardaway, Reed Hardaway, Rodrigo Hardaway, Josue Hardaway, Cole Hardaway, Rob Hardaway, Zane Hardaway,
Jeremie Hardaway, Kenyatta Hardaway, Barrett Hardaway, Rocco Hardaway, Sebastian Hardaway, Michel Hardaway, Denis Hardaway, Brooks Hardaway, Danial Hardaway, Garth Hardaway, Heriberto Hardaway, Nakia
Hardaway, Giovanni Hardaway, Mauricio Hardaway, Agustin Hardaway, Cedrick Hardaway, Tyron Hardaway, Brice Hardaway, Darwin Hardaway, Carter Hardaway, Kip Hardaway, Garland Hardaway, Malik Hardaway,
Elbert Hardaway, Gerry Hardaway, Deshawn Hardaway, Ronny Hardaway, Issac Hardaway, Jeramie Hardaway, Emmett Hardaway, Adolfo Hardaway, Daron Hardaway, Rashad Hardaway, Lincoln Hardaway, Homer
Hardaway, Jacques Hardaway, Nicky Hardaway, Dino Hardaway, Markus Hardaway, Ahmad Hardaway, Kristoffer Hardaway, Wilbur Hardaway, Antione Hardaway, Jan Hardaway, Ezra Hardaway, Antony Hardaway, Gino
Hardaway, Mikel Hardaway, Ari Hardaway, Tremayne Hardaway, Judson Hardaway, Garrick Hardaway, Kasey Hardaway, Kraig Hardaway, Rigoberto Hardaway, Diego Hardaway, Edwardo Hardaway, Jarred Hardaway,
Chet Hardaway, Hunter Hardaway, Jude Hardaway, Billie Hardaway, Elvin Hardaway, Alvaro Hardaway, Lenny Hardaway, Irvin Hardaway, Jasper Hardaway, Judd Hardaway, Carson Hardaway, Kenyon Hardaway,
Keven Hardaway, Sammie Hardaway, Keenan Hardaway, Darron Hardaway, Russel Hardaway, Leif Hardaway, Tyree Hardaway, Woodrow Hardaway, Chase Hardaway, Tod Hardaway, Richie Hardaway, Randell Hardaway,
Vito Hardaway, Deron Hardaway, Gregorio Hardaway, Federico Hardaway, Weston Hardaway, Davis Hardaway, Torrance Hardaway, Ulysses Hardaway, Trey Hardaway, Jeremey Hardaway, Riley Hardaway, Vince
Hardaway, Nigel Hardaway, Chauncey Hardaway, Davin Hardaway, Leonel Hardaway, Tobin Hardaway, Michelle Hardaway, Bernardo Hardaway, Dedrick Hardaway, Titus Hardaway, Osvaldo Hardaway, Wilfred
Hardaway, Errol Hardaway, Jade Hardaway, Carmen Hardaway, Cliff Hardaway, Kenya Hardaway, Jammie Hardaway, Kelley Hardaway, Harrison Hardaway, Lonny Hardaway, Abdul Hardaway, Coy Hardaway, Denver
Hardaway, Robb Hardaway, Telly Hardaway, Carroll Hardaway, August Hardaway, Tristan Hardaway, Fidel Hardaway, Octavio Hardaway, Dameon Hardaway, Josef Hardaway, Eloy Hardaway, Dax Hardaway, Barton
Hardaway, Eddy Hardaway, Cyrus Hardaway, Broderick Hardaway, Zachery Hardaway, Hiram Hardaway, Raymundo Hardaway, Giuseppe Hardaway, Terrill Hardaway, Burton Hardaway, Hank Hardaway, Sedrick
Hardaway, Jamil Hardaway, Germaine Hardaway, Myles Hardaway, Maxwell Hardaway, Harlan Hardaway, Norris Hardaway, Emil Hardaway, Kennith Hardaway, Deric Hardaway, Bobbie Hardaway, Levar Hardaway,
Francesco Hardaway, Hassan Hardaway, Jerrold Hardaway, Jayme Hardaway, Josiah Hardaway, Junior Hardaway, Laron Hardaway, Duncan Hardaway, Kenton Hardaway, Sandy Hardaway, Brennan Hardaway, Jamaal
Hardaway, Brook Hardaway, Coby Hardaway, Augustine Hardaway, Merle Hardaway, Abram Hardaway, Rod Hardaway, Jefferey Hardaway, Frederic Hardaway, Eldon Hardaway, Nathanial Hardaway, Silas Hardaway,
Gonzalo Hardaway, Greggory Hardaway, Brannon Hardaway, Demarcus Hardaway, Erwin Hardaway, Shea Hardaway, Lon Hardaway, Alec Hardaway, Antwon Hardaway, Melissa Hardaway, Phil Hardaway, Adrain
Hardaway, Hal Hardaway, Mitchel Hardaway, Antonia Hardaway, Irving Hardaway, Derik Hardaway, Tavares Hardaway, Edgardo Hardaway, Gus Hardaway, Paris Hardaway, Kimberly Hardaway, Jevon Hardaway,
Carmelo Hardaway, Britt Hardaway, Efren Hardaway, Moshe Hardaway, Shanon Hardaway, Dominique Hardaway, Cordell Hardaway, Kermit Hardaway, Whitney Hardaway, Lashawn Hardaway, Jovan Hardaway, Chadd
Hardaway, Dedric Hardaway, Chadrick Hardaway, Lars Hardaway, Kelsey Hardaway, Marcellus Hardaway, Dejuan Hardaway, Tanner Hardaway, Braden Hardaway, Cortez Hardaway, Ezekiel Hardaway, Christoper
Hardaway, Donell Hardaway, Delvin Hardaway, Timmothy Hardaway, Linwood Hardaway, Lisa Hardaway, Alfonzo Hardaway, Tate Hardaway, Douglass Hardaway, Quintin Hardaway, Bennett Hardaway, Pat Hardaway,
Cruz Hardaway, Garret Hardaway, Aldo Hardaway, Brandy Hardaway, Jereme Hardaway, Arnulfo Hardaway, Taurus Hardaway, Dereck Hardaway, Ned Hardaway, Trever Hardaway, Kirt Hardaway, Liam Hardaway,
Angela Hardaway, Marques Hardaway, Amy Hardaway, Roscoe Hardaway, Aurelio Hardaway, Genaro Hardaway, Nestor Hardaway, Dwain Hardaway, German Hardaway, Tarik Hardaway, Cristopher Hardaway, Forest
Hardaway, Barney Hardaway, Chaim Hardaway, Eliseo Hardaway, Thurman Hardaway, Donta Hardaway, Che Hardaway, Mariano Hardaway, Lanny Hardaway, Donavan Hardaway, Jameson Hardaway, Andreas Hardaway,
Isidro Hardaway, Anderson Hardaway, Jedediah Hardaway, Rahsaan Hardaway, Baron Hardaway, Emery Hardaway, Michale Hardaway, Jered Hardaway, Charley Hardaway, Bronson Hardaway, Odell Hardaway, Armand
Hardaway, Shay Hardaway, Grover Hardaway, Carmine Hardaway, Norberto Hardaway, Deangelo Hardaway, Brandt Hardaway, Chandler Hardaway, Otto Hardaway, Dannie Hardaway, Torrence Hardaway, Buck Hardaway,
Darby Hardaway, Lindsey Hardaway, Trinity Hardaway, Cale Hardaway, Franco Hardaway, Damond Hardaway, Jerad Hardaway, Reinaldo Hardaway, Wiley Hardaway, Ivory Hardaway, Everette Hardaway, Leopoldo
Hardaway, Andra Hardaway, Dario Hardaway, Dalton Hardaway, Ward Hardaway, Brien Hardaway, Paulo Hardaway, Lazaro Hardaway, Emory Hardaway, Westley Hardaway, Marcelino Hardaway, Emerson Hardaway, Mary
Hardaway, Kwame Hardaway, Toney Hardaway, Marcelo Hardaway, Rasheed Hardaway, Derrell Hardaway, Jermey Hardaway, Russ Hardaway, Vincenzo Hardaway, Raheem Hardaway, Asa Hardaway, Lemuel Hardaway,
Murray Hardaway, Pasquale Hardaway, Burt Hardaway, Hollis Hardaway, Valentin Hardaway, Bernie Hardaway, Jamin Hardaway, Lenard Hardaway, Roel Hardaway, Weldon Hardaway, Ahmed Hardaway, Cortney
Hardaway, Tito Hardaway, Claudio Hardaway, Kristin Hardaway, Lorne Hardaway, Enrico Hardaway, Marlo Hardaway, Loyd Hardaway, Amir Hardaway, Arnoldo Hardaway, Harris Hardaway, Cristobal Hardaway, Mac
Hardaway, Corbin Hardaway, Cullen Hardaway, Shedrick Hardaway, Maria Hardaway, Jermain Hardaway, Ramsey Hardaway, Torey Hardaway, Nicole Hardaway, Corry Hardaway, Anson Hardaway, Heather Hardaway,
Parker Hardaway, Elizabeth Hardaway, Tye Hardaway, Lesley Hardaway, Lamonte Hardaway, Sanford Hardaway, Karim Hardaway, Stephanie Hardaway, Toriano Hardaway, Bruno Hardaway, Estevan Hardaway, Jerel
Hardaway, Orville Hardaway, Brion Hardaway, Demetrios Hardaway, Anibal Hardaway, Reginal Hardaway, Lavar Hardaway, Rosendo Hardaway, Romeo Hardaway, Simeon Hardaway, Elisha Hardaway, Stan Hardaway,
Vidal Hardaway, Enoch Hardaway, Prince Hardaway, Eugenio Hardaway, Leigh Hardaway, Mervin Hardaway, Bartholomew Hardaway, Delmar Hardaway, Khalid Hardaway, Kristen Hardaway, Johnpaul Hardaway,
Carnell Hardaway, Domenic Hardaway, Adalberto Hardaway, Jaron Hardaway, Luciano Hardaway, Kai Hardaway, Stanford Hardaway, Uriah Hardaway, Demarco Hardaway, Jabari Hardaway, Zackary Hardaway, Ritchie
Hardaway, Ezequiel Hardaway, Basil Hardaway, Brenden Hardaway, Garett Hardaway, Major Hardaway, Jerold Hardaway, Wes Hardaway, Yancy Hardaway, Benjamen Hardaway, Fredric Hardaway, Jerrell Hardaway,
Benji Hardaway, Barron Hardaway, Chip Hardaway, Michele Hardaway, Britton Hardaway, Milo Hardaway, Samir Hardaway, Charlton Hardaway, Royal Hardaway, Rubin Hardaway, Stoney Hardaway, Kieth Hardaway,
Javon Hardaway, Derwin Hardaway, Duke Hardaway, Antone Hardaway, Daryle Hardaway, Demetric Hardaway, Layne Hardaway, Reyes Hardaway, Walker Hardaway, Fletcher Hardaway, Freeman Hardaway, Jarret
Hardaway, Delano Hardaway, Oren Hardaway, Marcell Hardaway, Trinidad Hardaway, Luigi Hardaway, Rustin Hardaway, Shan Hardaway, Sharif Hardaway, Isiah Hardaway, Levon Hardaway, Brody Hardaway, Coleman
Hardaway, Ollie Hardaway, Thor Hardaway, Darell Hardaway, Desi Hardaway, Horacio Hardaway, Kiley Hardaway, Mauro Hardaway, Vinson Hardaway, Dimitrios Hardaway, Kevan Hardaway, Merlin Hardaway, Monroe
Hardaway, Rebecca Hardaway, Rashawn Hardaway, Tyrell Hardaway, Zackery Hardaway, Samson Hardaway, Darcy Hardaway, Donavon Hardaway, Jessica Hardaway, Napoleon Hardaway, Muhammad Hardaway, Trevis
Hardaway, Berry Hardaway, Antwain Hardaway, Eliezer Hardaway, Kendell Hardaway, Lamarr Hardaway, Micky Hardaway, Demian Hardaway, Seneca Hardaway, Tucker Hardaway, Cheyenne Hardaway, Davon Hardaway,
Norbert Hardaway, Regan Hardaway, Adolph Hardaway, Jerimiah Hardaway, Benton Hardaway, Jeb Hardaway, Rian Hardaway, Corwin Hardaway, Jameel Hardaway, Tyrus Hardaway, Alexandro Hardaway, Benson
Hardaway, Darien Hardaway, Augustus Hardaway, Erie Hardaway, Gil Hardaway, Raymon Hardaway, Abelardo Hardaway, Terance Hardaway, Tremaine Hardaway, Cristian Hardaway, Jedidiah Hardaway, Christina
Hardaway, Homero Hardaway, Rueben Hardaway, Rey Hardaway, Jace Hardaway, Caesar Hardaway, Hakim Hardaway, Lydell Hardaway, Marcello Hardaway, Parrish Hardaway, Isaias Hardaway, Johnie Hardaway, Tavis
Hardaway, Eliot Hardaway, Fermin Hardaway, Nikolas Hardaway, Orion Hardaway, Alden Hardaway, Clement Hardaway, Lucio Hardaway, Rondell Hardaway, Ronnell Hardaway, Demetris Hardaway, Juston Hardaway,
Len Hardaway, Cade Hardaway, Detrick Hardaway, Sherwin Hardaway, Truman Hardaway, Reymundo Hardaway, Danielle Hardaway, Domenico Hardaway, Octavius Hardaway, Anwar Hardaway, Christofer Hardaway,
Dwaine Hardaway, Leander Hardaway, Reese Hardaway, Houston Hardaway, Conor Hardaway, Dillon Hardaway, Mitch Hardaway, Demario Hardaway, Artis Hardaway, Boris Hardaway, Samual Hardaway, Dustan
Hardaway, Gorge Hardaway, Jory Hardaway, Stanton Hardaway, Amit Hardaway, Julie Hardaway, Kelby Hardaway, Kody Hardaway, Christos Hardaway, Fritz Hardaway, Lyndon Hardaway, Constantine Hardaway,
Erasmo Hardaway, Jodie Hardaway, Brooke Hardaway, Conan Hardaway, Kane Hardaway, Merrill Hardaway, Bryson Hardaway, Isreal Hardaway, Woody Hardaway, Tadd Hardaway, Emiliano Hardaway, Cyril Hardaway,
Davey Hardaway, Gabe Hardaway, Pernell Hardaway, Curtiss Hardaway, Jacky Hardaway, Nicholaus Hardaway, Hasan Hardaway, Davy Hardaway, Oswaldo Hardaway, Tammy Hardaway, Felton Hardaway, Jarrad
Hardaway, Rosario Hardaway, Yusef Hardaway, Anthoney Hardaway, Dimitri Hardaway, Donn Hardaway, Graig Hardaway, Glendon Hardaway, Alain Hardaway, Alonso Hardaway, Dwane Hardaway, Laura Hardaway,
Millard Hardaway, Milan Hardaway, Everardo Hardaway, Taj Hardaway, Kendal Hardaway, Prentice Hardaway, Hilario Hardaway, Hayden Hardaway, Lukas Hardaway, Lorin Hardaway, Stevan Hardaway, Benedict
Hardaway, Ferdinand Hardaway, Renard Hardaway, Geoff Hardaway, Raleigh Hardaway, Amanda Hardaway, Jules Hardaway, Marshal Hardaway, Ephraim Hardaway, Patricia Hardaway, Sarah Hardaway, Kalvin
Hardaway, Sky Hardaway, Vladimir Hardaway, Kipp Hardaway, Rashid Hardaway, Burke Hardaway, Derrek Hardaway, Margarito Hardaway, Rojelio Hardaway, Von Hardaway, Wayland Hardaway, Mohammad Hardaway,
Sandro Hardaway, Renato Hardaway, Shiloh Hardaway, Torry Hardaway, Kenji Hardaway, Del Hardaway, Elgin Hardaway, Booker Hardaway, Chistopher Hardaway, Lavell Hardaway, Mohammed Hardaway, Refugio
Hardaway, Damen Hardaway, Nicola Hardaway, Daryn Hardaway, Elwood Hardaway, Kenric Hardaway, Andrae Hardaway, Christine Hardaway, Clemente Hardaway, Corby Hardaway, Edmundo Hardaway, Channing
Hardaway, Maynard Hardaway, Roderic Hardaway, Wilton Hardaway, Kedrick Hardaway, Kieran Hardaway, Lucius Hardaway, Cliffton Hardaway, Geraldo Hardaway, Adrien Hardaway, Hershel Hardaway, Reco
Hardaway, Dudley Hardaway, Jame Hardaway, Michal Hardaway, Omari Hardaway, Christophe Hardaway, Delton Hardaway, Lindsay Hardaway, Philippe Hardaway, Faron Hardaway, Brandan Hardaway, Williams
Hardaway, Adonis Hardaway, Jeanpaul Hardaway, Mckinley Hardaway, Bertram Hardaway, Randel Hardaway, Audie Hardaway, Fransisco Hardaway, Gideon Hardaway, Lafayette Hardaway, Renaldo Hardaway, Winfred
Hardaway, Lacy Hardaway, Ravi Hardaway, Denton Hardaway, Bjorn Hardaway, Demetrice Hardaway, Duwayne Hardaway, Lavon Hardaway, Porfirio Hardaway, Eldridge Hardaway, Hosea Hardaway, Lupe Hardaway,
Corbett Hardaway, Grayson Hardaway, Sanjay Hardaway, Emile Hardaway, Emmitt Hardaway, Olin Hardaway, Ramone Hardaway, Yusuf Hardaway, Leandro Hardaway, Amado Hardaway, Leighton Hardaway, Malachi
Hardaway, Stephon Hardaway, Wilford Hardaway, Keon Hardaway, Timmie Hardaway, Errick Hardaway, Jarad Hardaway, Kaleb Hardaway, Dayton Hardaway, Jelani Hardaway, Rance Hardaway, Corrie Hardaway,
Jerrad Hardaway, Yancey Hardaway, Jonpaul Hardaway, Theo Hardaway, Amador Hardaway, Jamaine Hardaway, Lorenza Hardaway, Valentino Hardaway, Dee Hardaway, Eleazar Hardaway, Jarett Hardaway, Justen
Hardaway, Lauren Hardaway, Tyronne Hardaway, Willy Hardaway, Elroy Hardaway, Esequiel Hardaway, Ike Hardaway, Renee Hardaway, Karen Hardaway, Herschel Hardaway, Konstantinos Hardaway, Arlo Hardaway,
Buford Hardaway, Rasheen Hardaway, Reagan Hardaway, Tobey Hardaway, Haywood Hardaway, Khristopher Hardaway, Patricio Hardaway, Zack Hardaway, Jamon Hardaway, Khary Hardaway, Augustin Hardaway,
Benjiman Hardaway, Ishmael Hardaway, Wally Hardaway, Faustino Hardaway, Hamilton Hardaway, Jami Hardaway, Vincente Hardaway, Griffin Hardaway, Jeramiah Hardaway, Manny Hardaway, Santino Hardaway,
Vern Hardaway, Richardo Hardaway, Franklyn Hardaway, Wilmer Hardaway, Christpher Hardaway, Derric Hardaway, Drake Hardaway, Lino Hardaway, Nathen Hardaway, Ryon Hardaway, Vernell Hardaway, Johann
Hardaway, Khalil Hardaway, Cris Hardaway, Foster Hardaway, Gale Hardaway, Teodoro Hardaway, Creighton Hardaway, Farrell Hardaway, Jai Hardaway, Chirstopher Hardaway, Lavelle Hardaway, Mikal Hardaway,
Orin Hardaway, Ceasar Hardaway, Pietro Hardaway, Sedric Hardaway, Sydney Hardaway, Val Hardaway, Dawayne Hardaway, Odis Hardaway, Wardell Hardaway, Murphy Hardaway, Nevin Hardaway, Quenton Hardaway,
Santo Hardaway, Slade Hardaway, Dawn Hardaway, Marko Hardaway, Ricco Hardaway, Rogers Hardaway, Abe Hardaway, Jerimy Hardaway, Jessy Hardaway, Montrell Hardaway, Cassidy Hardaway, Hilton Hardaway,
Christen Hardaway, Kamal Hardaway, Leron Hardaway, Miquel Hardaway, Yosef Hardaway, Judah Hardaway, Malcom Hardaway, Rayford Hardaway, Arlen Hardaway, Gaetano Hardaway, Montez Hardaway, Kareen
Hardaway, Dajuan Hardaway, Marlow Hardaway, Uriel Hardaway, Glynn Hardaway, Bud Hardaway, Eben Hardaway, Geronimo Hardaway, Deshaun Hardaway, Hernan Hardaway, Zebulon Hardaway, Deshon Hardaway,
Gareth Hardaway, Paolo Hardaway, Edison Hardaway, Gaylon Hardaway, Shamus Hardaway, Carleton Hardaway, Florentino Hardaway, Osbaldo Hardaway, Brodie Hardaway, Jaret Hardaway, Johathan Hardaway, Tariq
Hardaway, Seamus Hardaway, Devan Hardaway, Richmond Hardaway, Cynthia Hardaway, Cleo Hardaway, Huey Hardaway, Talmadge Hardaway, Dain Hardaway, Dennie Hardaway, Paxton Hardaway, Peyton Hardaway,
Darrius Hardaway, Jodi Hardaway, Ronell Hardaway, Jeremi Hardaway, Quinten Hardaway, Sandra Hardaway, Waymon Hardaway, Earle Hardaway, Franz Hardaway, Christoher Hardaway, Micahel Hardaway, Sal
Hardaway, Sharon Hardaway, Keegan Hardaway, Eusebio Hardaway, Geoffery Hardaway, Monica Hardaway, Dontae Hardaway, Jimi Hardaway, Kendric Hardaway, Amin Hardaway, Cleon Hardaway, Kale Hardaway, Tina
Hardaway, Ambrose Hardaway, Armond Hardaway, Ashton Hardaway, Lucien Hardaway, Maximo Hardaway, Reno Hardaway, Susan Hardaway, Chico Hardaway, Jermel Hardaway, Lennie Hardaway, Rajesh Hardaway, Artie
Hardaway, Kerwin Hardaway, Sage Hardaway, Dietrich Hardaway, Eriberto Hardaway, Niles Hardaway, Rolland Hardaway, Butch Hardaway, Courtland Hardaway, King Hardaway, Kit Hardaway, Menachem Hardaway,
Rich Hardaway, Florencio Hardaway, Lovell Hardaway, Jacque Hardaway, Antwone Hardaway, Darick Hardaway, Kennth Hardaway, Asher Hardaway, Efrem Hardaway, Lamarcus Hardaway, Terron Hardaway, Jacinto
Hardaway, Karlos Hardaway, Mohamed Hardaway, Rahman Hardaway, Ammon Hardaway, Davie Hardaway, Diallo Hardaway, Kelton Hardaway, Mathias Hardaway, Sunny Hardaway, Zeke Hardaway, Jaison Hardaway, Nate
Hardaway, Demetrious Hardaway, Maceo Hardaway, Vasilios Hardaway, Demetrio Hardaway, Lyman Hardaway, Angus Hardaway, Arvin Hardaway, Camron Hardaway, Daryll Hardaway, Ehren Hardaway, Les Hardaway,
Nils Hardaway, Akil Hardaway, Markeith Hardaway, Santana Hardaway, Warner Hardaway, Alphonse Hardaway, Arnaldo Hardaway, Art Hardaway, Domenick Hardaway, Prentiss Hardaway, Blas Hardaway, Dwan
Hardaway, Geno Hardaway, Kahlil Hardaway, Feliciano Hardaway, Hayward Hardaway, Ulises Hardaway, Valentine Hardaway, Darold Hardaway, Deven Hardaway, Mordechai Hardaway, Rodriquez Hardaway, Gentry
Hardaway, Jorje Hardaway, Sunil Hardaway, Wilburn Hardaway, Anders Hardaway, Damone Hardaway, Sherrod Hardaway, Dashawn Hardaway, Otha Hardaway, Rahim Hardaway, Javan Hardaway, Wendy Hardaway, Camilo
Hardaway, Jeromie Hardaway, Jaysen Hardaway, Modesto Hardaway, Quincey Hardaway, Tyran Hardaway, Cleve Hardaway, Raynard Hardaway, Tori Hardaway, Antonino Hardaway, Bartley Hardaway, Kristofor
Hardaway, Sheridan Hardaway, Alva Hardaway, Correy Hardaway, Dandre Hardaway, Darek Hardaway, Wylie Hardaway, Anand Hardaway, Decarlos Hardaway, Domonic Hardaway, Rachel Hardaway, Jamieson Hardaway,
Nikolaos Hardaway, Rashaan Hardaway, Roque Hardaway, Shelly Hardaway, Sven Hardaway, Evans Hardaway, Jakob Hardaway, Lateef Hardaway, Noble Hardaway, Artemio Hardaway, Kameron Hardaway, Kavin
Hardaway, Dick Hardaway, Filiberto Hardaway, Henri Hardaway, Joby Hardaway, Montgomery Hardaway, Salomon Hardaway, Vashon Hardaway, Abdullah Hardaway, Cal Hardaway, Shain Hardaway, Sherwood Hardaway,
Dewitt Hardaway, Newton Hardaway, Nicklaus Hardaway, Darvin Hardaway, Jeron Hardaway, Khari Hardaway, Tyrome Hardaway, Benjie Hardaway, Kunta Hardaway, Karsten Hardaway, Patric Hardaway, Reece
Hardaway, Olen Hardaway, Rowdy Hardaway, Terrel Hardaway, Addison Hardaway, Blaise Hardaway, Fidencio Hardaway, Ibrahim Hardaway, Idris Hardaway, Lavern Hardaway, Mickel Hardaway, Shun Hardaway,
Athanasios Hardaway, Danilo Hardaway, Giles Hardaway, Mateo Hardaway, Nino Hardaway, Alonza Hardaway, Cardell Hardaway, Job Hardaway, Mahlon Hardaway, Anil Hardaway, Destry Hardaway, Lamon Hardaway,
Martez Hardaway, Mikael Hardaway, Nickey Hardaway, Tywan Hardaway, Abner Hardaway, Jamell Hardaway, Nels Hardaway, Raoul Hardaway, Braulio Hardaway, Randle Hardaway, Roddy Hardaway, Toni Hardaway,
Lynwood Hardaway, Thurston Hardaway, Chrisopher Hardaway, Eron Hardaway, Fausto Hardaway, Keir Hardaway, Kelcey Hardaway, Marquette Hardaway, Mikeal Hardaway, Anselmo Hardaway, Hipolito Hardaway,
Neville Hardaway, Rodriguez Hardaway, Antron Hardaway, Walton Hardaway, Devlin Hardaway, Lucian Hardaway, Rudolfo Hardaway, Avi Hardaway, Brantley Hardaway, Dathan Hardaway, Doran Hardaway, Erica
Hardaway, Franky Hardaway, Kary Hardaway, Tiffany Hardaway, Yuri Hardaway, Bucky Hardaway, Hardy Hardaway, Justus Hardaway, Cecilio Hardaway, Hoyt Hardaway, Jasson Hardaway, Jerardo Hardaway, Johnson
Hardaway, Kenney Hardaway, Panagiotis Hardaway, Ajay Hardaway, Alvis Hardaway, Conrado Hardaway, Mel Hardaway, Burl Hardaway, Daymon Hardaway, Dorsey Hardaway, Giancarlo Hardaway, Riccardo Hardaway,
Antuan Hardaway, Braxton Hardaway, Candelario Hardaway, Colton Hardaway, Rommel Hardaway, Irwin Hardaway, Reynold Hardaway, Tino Hardaway, Alessandro Hardaway, Corie Hardaway, Dameion Hardaway,
Dwyane Hardaway, Jabbar Hardaway, Lajuan Hardaway, Mose Hardaway, Taiwan Hardaway, Yisroel Hardaway, Danniel Hardaway, Darris Hardaway, Jerred Hardaway, Lori Hardaway, Crispin Hardaway, Maximilian
Hardaway, Skip Hardaway, Yaakov Hardaway, Brodrick Hardaway, Fabio Hardaway, Gerrit Hardaway, Iran Hardaway, Trace Hardaway, Amar Hardaway, Aram Hardaway, Buster Hardaway, Byran Hardaway, Jens
Hardaway, Joseluis Hardaway, Karriem Hardaway, Kenan Hardaway, Tonya Hardaway, Trampas Hardaway, Durand Hardaway, Lou Hardaway, Regis Hardaway, Cain Hardaway, Christ Hardaway, Daivd Hardaway,
Marquise Hardaway, Meredith Hardaway, Dell Hardaway, Matthias Hardaway, Alphonzo Hardaway, Latroy Hardaway, Allison Hardaway, Darion Hardaway, Melton Hardaway, Tara Hardaway, Boyce Hardaway, Cletus
Hardaway, Derron Hardaway, Madison Hardaway, Tait Hardaway, Yehuda Hardaway, Arik Hardaway, Donato Hardaway, Duston Hardaway, Kimani Hardaway, Lucious Hardaway, Nader Hardaway, Baldemar Hardaway,
Bertrand Hardaway, Blane Hardaway, Dujuan Hardaway, Ellery Hardaway, Kennedy Hardaway, Misael Hardaway, Tremain Hardaway, Veronica Hardaway, Christy Hardaway, Columbus Hardaway, Little Hardaway,
Marius Hardaway, Skyler Hardaway, Alexandre Hardaway, Celestino Hardaway, Crystal Hardaway, Jordon Hardaway, Justine Hardaway, Demon Hardaway, Derreck Hardaway, Flavio Hardaway, Kennard Hardaway,
Roby Hardaway, Schuyler Hardaway, Garrison Hardaway, Cipriano Hardaway, Cori Hardaway, Denise Hardaway, Keyon Hardaway, Laverne Hardaway, Obie Hardaway, Tige Hardaway, Gabino Hardaway, Joselito
Hardaway, Karlton Hardaway, Lashon Hardaway, Lucky Hardaway, Neill Hardaway, Shmuel Hardaway, Ara Hardaway, Blain Hardaway, Pascual Hardaway, Rashan Hardaway, Serge Hardaway, Thane Hardaway, Bradlee
Hardaway, Connor Hardaway, Cooper Hardaway, Jonny Hardaway, Merrick Hardaway, Porter Hardaway, Zeb Hardaway, April Hardaway, Edrick Hardaway, Flint Hardaway, Heith Hardaway, Jospeh Hardaway, Luiz
Hardaway, Macario Hardaway, Zach Hardaway, Candido Hardaway, Dangelo Hardaway, Dov Hardaway, Dru Hardaway, Eladio Hardaway, Rasheem Hardaway, Tarek Hardaway, Willian Hardaway, Ezell Hardaway, Konrad
Hardaway, Mick Hardaway, Tarrance Hardaway, Yitzchok Hardaway, Rahul Hardaway, Shelley Hardaway, Theodis Hardaway, Timonthy Hardaway, Connie Hardaway, Derrik Hardaway, Ennis Hardaway, Gail Hardaway,
Gaston Hardaway, Gunnar Hardaway, Justice Hardaway, Romel Hardaway, Bilal Hardaway, Jamy Hardaway, Jomo Hardaway, Keary Hardaway, Nehemiah Hardaway, Rashon Hardaway, Rolf Hardaway, Sameer Hardaway,
Jere Hardaway, Jerone Hardaway, Kennon Hardaway, Rodd Hardaway, Ronaldo Hardaway, Teresa Hardaway, Coley Hardaway, Enos Hardaway, Narciso Hardaway, Robinson Hardaway, Tobby Hardaway, Virgilio
Hardaway, Christiaan Hardaway, Clarke Hardaway, Demetruis Hardaway, Jemal Hardaway, Llewellyn Hardaway, Miller Hardaway, Shilo Hardaway, Tarus Hardaway, Young Hardaway, Arlie Hardaway, Djuan
Hardaway, Marcial Hardaway, Pamela Hardaway, Rayshawn Hardaway, Robyn Hardaway, Tirrell Hardaway, Dontay Hardaway, Joshuah Hardaway, Jovon Hardaway, Lemar Hardaway, Oran Hardaway, Taron Hardaway,
Jaimie Hardaway, Kacey Hardaway, Wendall Hardaway, Armondo Hardaway, Curry Hardaway, Klint Hardaway, Lauro Hardaway, Maximillian Hardaway, Rashard Hardaway, Anastasios Hardaway, Jermiah Hardaway,
Joao Hardaway, Lonnell Hardaway, Matias Hardaway, Rollin Hardaway, Thimothy Hardaway, Toma Hardaway, Jahmal Hardaway, Jerame Hardaway, Skye Hardaway, Alford Hardaway, Baltazar Hardaway, Gianni
Hardaway, Hayes Hardaway, Lamond Hardaway, Linda Hardaway, Neftali Hardaway, Antwoine Hardaway, Arden Hardaway, Dionicio Hardaway, Ely Hardaway, Kacy Hardaway, Syed Hardaway, Tyrel Hardaway, Windell
Hardaway, Allyn Hardaway, Antwane Hardaway, Dayne Hardaway, Donal Hardaway, Maxie Hardaway, Rashaun Hardaway, Sampson Hardaway, Torin Hardaway, Axel Hardaway, Delmer Hardaway, Dickie Hardaway,
Evaristo Hardaway, Francois Hardaway, Radames Hardaway, Sixto Hardaway, Terri Hardaway, Wyman Hardaway, Ashanti Hardaway, Benjaman Hardaway, Christophor Hardaway, Delwin Hardaway, Lemont Hardaway,
Lonzo Hardaway, Mustafa Hardaway, Sasha Hardaway, Travon Hardaway, Darryll Hardaway, Garfield Hardaway, Jarrell Hardaway, Lawson Hardaway, Nikki Hardaway, Terell Hardaway, Ace Hardaway, Donna
Hardaway, Gerome Hardaway, Jermane Hardaway, Joeseph Hardaway, Raynaldo Hardaway, Carrie Hardaway, Demetri Hardaway, Rupert Hardaway, Shadrick Hardaway, Sol Hardaway, Wendel Hardaway, Zechariah
Hardaway, Delon Hardaway, Demitrius Hardaway, Mcarthur Hardaway, Nickie Hardaway, Tavaris Hardaway, Baby Hardaway, Cornelious Hardaway, Deborah Hardaway, Haven Hardaway, Merritt Hardaway, Vijay
Hardaway, Granville Hardaway, Rock Hardaway, Romero Hardaway, Tanya Hardaway, Anastacio Hardaway, Ciro Hardaway, Gennaro Hardaway, Jerrel Hardaway, Justo Hardaway, Maury Hardaway, Stafford Hardaway,
Amon Hardaway, Carol Hardaway, Cornelio Hardaway, Dawson Hardaway, Erskine Hardaway, Tarrence Hardaway, Torrie Hardaway, Trevin Hardaway, Breck Hardaway, Delfino Hardaway, Desean Hardaway, Jerron
Hardaway, Milford Hardaway, Nancy Hardaway, Oswald Hardaway, Stefano Hardaway, Demont Hardaway, Lawerence Hardaway, Parris Hardaway, Godfrey Hardaway, Hakeem Hardaway, Leobardo Hardaway, Loran
Hardaway, Garnett Hardaway, Lawrance Hardaway, Darryn Hardaway, Eamon Hardaway, Geremy Hardaway, Kobie Hardaway, Leeroy Hardaway, Maximino Hardaway, Orrin Hardaway, Wayman Hardaway, Brenda Hardaway,
Celso Hardaway, Daman Hardaway, Denzil Hardaway, Geary Hardaway, Ladon Hardaway, Marland Hardaway, Servando Hardaway, Kamau Hardaway, Migel Hardaway, Ramond Hardaway, Terrace Hardaway, Ubaldo
Hardaway, Chaz Hardaway, Demetrick Hardaway, Freddrick Hardaway, Gray Hardaway, Ivy Hardaway, Shlomo Hardaway, Torre Hardaway, Damein Hardaway, Deryl Hardaway, Dusten Hardaway, Katherine Hardaway,
Kreg Hardaway, London Hardaway, Patrice Hardaway, Payton Hardaway, Ramel Hardaway, Rosalio Hardaway, Serjio Hardaway, Tramaine Hardaway, Adolphus Hardaway, Ameer Hardaway, Antwaun Hardaway, Ashish
Hardaway, Benjamine Hardaway, Charleston Hardaway, Darry Hardaway, Elmo Hardaway, Nicklas Hardaway, Nikolai Hardaway, Ricki Hardaway, Tai Hardaway, Bernardino Hardaway, Devaughn Hardaway, Epifanio
Hardaway, Gaylord Hardaway, Mathieu Hardaway, Romell Hardaway, Arlin Hardaway, Bradd Hardaway, Clair Hardaway, Dawud Hardaway, Jeffory Hardaway, Lyndell Hardaway, Sara Hardaway, Adriel Hardaway,
Algernon Hardaway, Benigno Hardaway, Chancey Hardaway, Elan Hardaway, Jeromey Hardaway, Lionell Hardaway, Shone Hardaway, Donyell Hardaway, Herminio Hardaway, Jerrett Hardaway, Naim Hardaway,
Nikolaus Hardaway, Soren Hardaway, Agapito Hardaway, Carlin Hardaway, Cass Hardaway, Harmon Hardaway, Johan Hardaway, Junius Hardaway, Kenrick Hardaway, Kervin Hardaway, Kori Hardaway, Raj Hardaway,
Shimon Hardaway, Basilio Hardaway, Juventino Hardaway, Kirkland Hardaway, Kwesi Hardaway, Manish Hardaway, Omer Hardaway, Tedd Hardaway, Vic Hardaway, Antonie Hardaway, Chauncy Hardaway, Dagoberto
Hardaway, Darel Hardaway, Jerrid Hardaway, Julien Hardaway, Layton Hardaway, Quan Hardaway, Roddrick Hardaway, Shaw Hardaway, Silvestre Hardaway, Tavon Hardaway, Unknown Hardaway, Daniell Hardaway,
Andrei Hardaway, Hashim Hardaway, Hobert Hardaway, Terris Hardaway, Aundre Hardaway, Avrohom Hardaway, Elden Hardaway, Guido Hardaway, Holly Hardaway, Marice Hardaway, Martinez Hardaway, Daneil
Hardaway, Mattew Hardaway, Shamar Hardaway, Shannan Hardaway, Shomari Hardaway, Daniele Hardaway, Darrian Hardaway, Demetrus Hardaway, Erron Hardaway, Hasani Hardaway, Robbin Hardaway, Sabino
Hardaway, Sloan Hardaway, Zev Hardaway, Cassius Hardaway, Dawan Hardaway, Sekou Hardaway, Tyjuan Hardaway, Bentley Hardaway, Ean Hardaway, Edric Hardaway, Gamaliel Hardaway, Gerold Hardaway, Jacoby
Hardaway, Palmer Hardaway, Randon Hardaway, Sami Hardaway, Spence Hardaway, Willaim Hardaway, Alvino Hardaway, Antonios Hardaway, Derry Hardaway, Germain Hardaway, Jewel Hardaway, Lashaun Hardaway,
Tobie Hardaway, Webster Hardaway, Abran Hardaway, Brigham Hardaway, Leshawn Hardaway, Mica Hardaway, Pascal Hardaway, Americo Hardaway, Danyell Hardaway, Jamall Hardaway, Johnell Hardaway, Lambert
Hardaway, Lindell Hardaway, Raffaele Hardaway, Barret Hardaway, Chadley Hardaway, Frantz Hardaway, Gabrial Hardaway, Geroge Hardaway, Harland Hardaway, Obed Hardaway, Sharod Hardaway, Silverio
Hardaway, Vernard Hardaway, Alfie Hardaway, Darryle Hardaway, Daymond Hardaway, Deane Hardaway, Garey Hardaway, Jarid Hardaway, Oronde Hardaway, Rowland Hardaway, Sandeep Hardaway, Stevenson
Hardaway, Aarron Hardaway, Arin Hardaway, Aryeh Hardaway, Catarino Hardaway, Christoph Hardaway, Collins Hardaway, Elio Hardaway, Gearld Hardaway, Jonnie Hardaway, Mackenzie Hardaway, Markell
Hardaway, Shawne Hardaway, Walt Hardaway, Willam Hardaway, Winfield Hardaway, Arnell Hardaway, Aundra Hardaway, Bryn Hardaway, Conway Hardaway, Dempsey Hardaway, Eligio Hardaway, Alicia Hardaway,
Jairo Hardaway, Kathleen Hardaway, Koby Hardaway, Nikia Hardaway, Pierce Hardaway, Sherard Hardaway, Spiro Hardaway, Winford Hardaway, Alfonza Hardaway, Augusto Hardaway, Christin Hardaway, Dakota
Hardaway, Gannon Hardaway, Lucus Hardaway, Rand Hardaway, Renardo Hardaway, Robet Hardaway, Atiba Hardaway, Dondi Hardaway, Jammy Hardaway, Jarrid Hardaway, Jonthan Hardaway, Marlan Hardaway, Olando
Hardaway, Salim Hardaway, Shaine Hardaway, Arun Hardaway, Broc Hardaway, Cash Hardaway, Cord Hardaway, Ebony Hardaway, Gage Hardaway, Jacobo Hardaway, Loy Hardaway, Orlanda Hardaway, Sammuel
Hardaway, Trampus Hardaway, Chaka Hardaway, Lakeith Hardaway, Mendel Hardaway, Saleem Hardaway, Antoin Hardaway, Bernabe Hardaway, Darnel Hardaway, Ioannis Hardaway, Jill Hardaway, Melanie Hardaway,
Oneal Hardaway, Paulino Hardaway, Theadore Hardaway, Yves Hardaway, Aidan Hardaway, Antjuan Hardaway, Decarlo Hardaway, Fitzgerald Hardaway, Frances Hardaway, Jhon Hardaway, Price Hardaway,
Theophilus Hardaway, Theresa Hardaway, Aden Hardaway, Alaric Hardaway, Armon Hardaway, Johny Hardaway, Kenith Hardaway, Mallory Hardaway, Rondal Hardaway, Samer Hardaway, Coty Hardaway, Gerrod
Hardaway, Maximiliano Hardaway, Rito Hardaway, Robie Hardaway, Skipper Hardaway, Wali Hardaway, Blayne Hardaway, Christophr Hardaway, Mandel Hardaway, Nathon Hardaway, Orenthal Hardaway, Rudolf
Hardaway, Bailey Hardaway, Christropher Hardaway, Cleophus Hardaway, Eulalio Hardaway, Jerard Hardaway, Jethro Hardaway, Jonothan Hardaway, Lannie Hardaway, Shaka Hardaway, Arlan Hardaway, Chas
Hardaway, Conley Hardaway, Donzell Hardaway, Jamarr Hardaway, Karon Hardaway, Landis Hardaway, Marie Hardaway, Marino Hardaway, Michial Hardaway, Tyrice Hardaway, Vivek Hardaway, Arian Hardaway,
Carlyle Hardaway, Ibn Hardaway, Jonmichael Hardaway, Ottis Hardaway, Rafeal Hardaway, Serafin Hardaway, Tuan Hardaway, Arne Hardaway, Eliazar Hardaway, Jaren Hardaway, Kedric Hardaway, Kimo Hardaway,
Kristan Hardaway, Kyron Hardaway, Rayburn Hardaway, Sandor Hardaway, Undra Hardaway, Waymond Hardaway, Avraham Hardaway, Brennen Hardaway, Delaney Hardaway, Evangelos Hardaway, Geoffry Hardaway,
Jassen Hardaway, Jermal Hardaway, Jerrick Hardaway, Spiros Hardaway, Uri Hardaway, Andrey Hardaway, Christain Hardaway, Diamond Hardaway, Edgard Hardaway, Gardner Hardaway, Heidi Hardaway, Jerid
Hardaway, Laramie Hardaway, Philbert Hardaway, Arlyn Hardaway, Armin Hardaway, Bonifacio Hardaway, Kari Hardaway, Kofi Hardaway, Tamara Hardaway, Wayde Hardaway, Anna Hardaway, Cheston Hardaway,
Jarom Hardaway, Jemel Hardaway, Jermy Hardaway, Jeronimo Hardaway, Ontario Hardaway, Roshawn Hardaway, Adams Hardaway, Barbara Hardaway, Darran Hardaway, Horatio Hardaway, Jedd Hardaway, Jeramey
Hardaway, Rajeev Hardaway, Rendell Hardaway, Ansel Hardaway, Apolonio Hardaway, Daric Hardaway, Dewan Hardaway, Krishna Hardaway, Lenwood Hardaway, Lex Hardaway, Marlowe Hardaway, Mickael Hardaway,
Octavious Hardaway, Rashaad Hardaway, Catherine Hardaway, Chesley Hardaway, Natividad Hardaway, Ahren Hardaway, Alvie Hardaway, Ardell Hardaway, Gaylen Hardaway, Iain Hardaway, Jeffrie Hardaway, Levy
Hardaway, Naeem Hardaway, Ramey Hardaway, Rapheal Hardaway, Recardo Hardaway, Welton Hardaway, Wiliam Hardaway, Bakari Hardaway, Diana Hardaway, Durrell Hardaway, Gaspar Hardaway, Jacqueline
Hardaway, Jewell Hardaway, Marvell Hardaway, Noland Hardaway, Tedrick Hardaway, Waleed Hardaway, Aloysius Hardaway, Amber Hardaway, Audrey Hardaway, Curley Hardaway, Farris Hardaway, Gian Hardaway,
Jarmaine Hardaway, Lazarus Hardaway, Parish Hardaway, Stavros Hardaway, Sultan Hardaway, Laquan Hardaway, Londell Hardaway, Parnell Hardaway, Ryland Hardaway, Shem Hardaway, Yolanda Hardaway, Arnie
Hardaway, Brennon Hardaway, Chan Hardaway, Dionisio Hardaway, Frederico Hardaway, Joy Hardaway, Juancarlos Hardaway, Keno Hardaway, Kwasi Hardaway, Lael Hardaway, Laroy Hardaway, Normand Hardaway,
Omarr Hardaway, Tyre Hardaway, Valente Hardaway, Gavino Hardaway, Kade Hardaway, Kolby Hardaway, Linton Hardaway, Malvin Hardaway, Moishe Hardaway, Rhonda Hardaway, Sherry Hardaway, Verlin Hardaway,
Antoino Hardaway, Benjy Hardaway, Dmitri Hardaway, Duan Hardaway, Ford Hardaway, Hezekiah Hardaway, Kalani Hardaway, Kinte Hardaway, Lacey Hardaway, Massimo Hardaway, Ryder Hardaway, Seann Hardaway,
Valerie Hardaway, Waldo Hardaway, Edson Hardaway, Eliud Hardaway, Gustave Hardaway, Harrell Hardaway, Henderson Hardaway, Obadiah Hardaway, Christipher Hardaway, Constantinos Hardaway, Cort Hardaway,
Duran Hardaway, Eldred Hardaway, Jennings Hardaway, Jubal Hardaway, Macarthur Hardaway, Marvis Hardaway, Ric Hardaway, Dimas Hardaway, Emily Hardaway, Golden Hardaway, Jessee Hardaway, Juaquin
Hardaway, Kennie Hardaway, Lerone Hardaway, Paige Hardaway, Shean Hardaway, Termaine Hardaway, Antoni Hardaway, Connell Hardaway, Gustav Hardaway, Meir Hardaway, Reza Hardaway, Rollie Hardaway,
Ronney Hardaway, Sherron Hardaway, Teron Hardaway, Theotis Hardaway, Vittorio Hardaway, Athony Hardaway, Burnell Hardaway, Crawford Hardaway, Esau Hardaway, Gina Hardaway, Jamahl Hardaway, Kenyata
Hardaway, Mickeal Hardaway, Rexford Hardaway, Rockey Hardaway, Thadeus Hardaway, Trevon Hardaway, Barclay Hardaway, Chuckie Hardaway, Early Hardaway, Ernst Hardaway, Fredy Hardaway, Gildardo
Hardaway, Hyrum Hardaway, Kerby Hardaway, Kimball Hardaway, Marek Hardaway, Ransom Hardaway, Shareef Hardaway, Sid Hardaway, Tillman Hardaway, West Hardaway, Wynn Hardaway, Danyel Hardaway, Eduard
Hardaway, Elmore Hardaway, Holland Hardaway, Johannes Hardaway, Langston Hardaway, Long Hardaway, Lyn Hardaway, Montreal Hardaway, Rui Hardaway, Shae Hardaway, Shondell Hardaway, Tor Hardaway, Caine
Hardaway, Destin Hardaway, Dionne Hardaway, Hernando Hardaway, Isabel Hardaway, Lamark Hardaway, Leandre Hardaway, Nicholus Hardaway, Nikola Hardaway, Olaf Hardaway, Orson Hardaway, Presley Hardaway,
Randi Hardaway, Regina Hardaway, Rufino Hardaway, Shariff Hardaway, Shaune Hardaway, Torri Hardaway, Arcadio Hardaway, Arlando Hardaway, Carlus Hardaway, Efraim Hardaway, Emmet Hardaway, Etienne
Hardaway, Garvin Hardaway, Jermine Hardaway, Larnell Hardaway, Michiel Hardaway, Pieter Hardaway, Wilber Hardaway, Christapher Hardaway, Derk Hardaway, Enrigue Hardaway, Isidoro Hardaway, Kriston
Hardaway, Ladell Hardaway, Morton Hardaway, Nash Hardaway, Nickolaus Hardaway, Paula Hardaway, Sheron Hardaway, Wellington Hardaway, Antawn Hardaway, Brandyn Hardaway, Christan Hardaway, Deonte
Hardaway, Elie Hardaway, Hampton Hardaway, Jeremias Hardaway, Marque Hardaway, Weylin Hardaway, Antwine Hardaway, Bonnie Hardaway, Chucky Hardaway, Matthews Hardaway, Mikell Hardaway, Padraic
Hardaway, Tarence Hardaway, Christphor Hardaway, Cindy Hardaway, Daven Hardaway, Dock Hardaway, Enzo Hardaway, Gerrick Hardaway, Jeanpierre Hardaway, Jericho Hardaway, Joan Hardaway, Kenley Hardaway,
Kentrell Hardaway, Sherod Hardaway, Carlous Hardaway, Deandra Hardaway, Demar Hardaway, Dieter Hardaway, Frazier Hardaway, Garon Hardaway, Howell Hardaway, Joh Hardaway, Librado Hardaway, Melinda
Hardaway, Micha Hardaway, Steffan Hardaway, Stevens Hardaway, Victoriano Hardaway, Zak Hardaway, Addam Hardaway, Arick Hardaway, Cheryl Hardaway, Cornel Hardaway, Cosme Hardaway, Damani Hardaway,
Dillard Hardaway, Edsel Hardaway, Garner Hardaway, Gerad Hardaway, Giacomo Hardaway, Ismail Hardaway, Link Hardaway, Rohit Hardaway, Alexandros Hardaway, Brandin Hardaway, Dolan Hardaway, Dustyn
Hardaway, Eran Hardaway, Imran Hardaway, Johnthan Hardaway, Lashun Hardaway, Linus Hardaway, Misty Hardaway, Shaheed Hardaway, Ann Hardaway, Antionio Hardaway, Arie Hardaway, Aris Hardaway, Armen
Hardaway, Clancy Hardaway, Colt Hardaway, Dione Hardaway, Garren Hardaway, Janson Hardaway, Justyn Hardaway, Megan Hardaway, Niel Hardaway, Niels Hardaway, Rockie Hardaway, Thornton Hardaway, Turner
Hardaway, Viet Hardaway, Albino Hardaway, Alma Hardaway, Artemus Hardaway, Cam Hardaway, Claud Hardaway, Daimon Hardaway, Dequan Hardaway, Fransico Hardaway, Ilan Hardaway, Laurent Hardaway, Sigmund
Hardaway, Tal Hardaway, Tray Hardaway, Apollo Hardaway, Barak Hardaway, Brentley Hardaway, Cleavon Hardaway, Delvon Hardaway, Dyron Hardaway, Jaye Hardaway, Jermell Hardaway, Ladd Hardaway, Leamon
Hardaway, Nico Hardaway, Ovidio Hardaway, Peder Hardaway, Raven Hardaway, Teofilo Hardaway, Trae Hardaway, Alon Hardaway, Charle Hardaway, Erek Hardaway, Evin Hardaway, Graeme Hardaway, Jerimie
Hardaway, Juanito Hardaway, Keoni Hardaway, Kwan Hardaway, Lathan Hardaway, Leonidas Hardaway, Marrio Hardaway, Mayer Hardaway, Ozzie Hardaway, Rajiv Hardaway, Rami Hardaway, Shann Hardaway, Teague
Hardaway, Terran Hardaway, Branson Hardaway, Briant Hardaway, Christohper Hardaway, Delmas Hardaway, Demone Hardaway, Duffy Hardaway, Grabiel Hardaway, Hudson Hardaway, Jae Hardaway, Jefrey Hardaway,
Kali Hardaway, Klinton Hardaway, Markel Hardaway, Saverio Hardaway, Steffen Hardaway, Akbar Hardaway, Brandi Hardaway, Desmon Hardaway, Detric Hardaway, Doron Hardaway, Georgie Hardaway, Hilary
Hardaway, Isadore Hardaway, Kawika Hardaway, Keefe Hardaway, Montell Hardaway, Prentis Hardaway, Rishi Hardaway, Zacharia Hardaway, Adriano Hardaway, Arash Hardaway, Chancy Hardaway, Deshun Hardaway,
Donnel Hardaway, Johndavid Hardaway, Jonn Hardaway, Kerrick Hardaway, Ruperto Hardaway, Silvio Hardaway, Smith Hardaway, Alen Hardaway, Damarcus Hardaway, Dixon Hardaway, Isacc Hardaway, Landry
Hardaway, Laurie Hardaway, Lennard Hardaway, Mckenzie Hardaway, Rakesh Hardaway, Selwyn Hardaway, Torris Hardaway, Yohance Hardaway, Zakary Hardaway, Alison Hardaway, Baxter Hardaway, Bienvenido
Hardaway, Christon Hardaway, Cosmo Hardaway, Cuauhtemoc Hardaway, Drue Hardaway, Ferris Hardaway, Jensen Hardaway, Jerime Hardaway, Kathryn Hardaway, Kingsley Hardaway, Prescott Hardaway, Rayfield
Hardaway, Sanders Hardaway, Tarvis Hardaway, Bernell Hardaway, Calvert Hardaway, Channon Hardaway, Clovis Hardaway, Duron Hardaway, Dushawn Hardaway, Elam Hardaway, Elwin Hardaway, Johnmichael
Hardaway, Lawton Hardaway, Linden Hardaway, Meyer Hardaway, Raheen Hardaway, Santonio Hardaway, Shonn Hardaway, Vikram Hardaway, Wesly Hardaway, Yoel Hardaway, Ysidro Hardaway, Asim Hardaway, Authur
Hardaway, Cyle Hardaway, General Hardaway, Page Hardaway, Tab Hardaway, Tamar Hardaway, Thaddaeus Hardaway, Tyshawn Hardaway, Akiva Hardaway, Alexi Hardaway, Dannon Hardaway, Draper Hardaway, Garet
Hardaway, Immanuel Hardaway, Julia Hardaway, Manfred Hardaway, Monique Hardaway, Monta Hardaway, Nile Hardaway, Rome Hardaway, Tywon Hardaway, Verne Hardaway, Amando Hardaway, Askia Hardaway, Cannon
Hardaway, Chavis Hardaway, Davide Hardaway, Deepak Hardaway, Garold Hardaway, Jabar Hardaway, Japheth Hardaway, Klaus Hardaway, Naftali Hardaway, Ora Hardaway, Ronn Hardaway, Shandon Hardaway,
Toussaint Hardaway, Travers Hardaway, Verlon Hardaway, Antwann Hardaway, Barnaby Hardaway, Bretton Hardaway, Cesario Hardaway, Georgios Hardaway, Jajuan Hardaway, Lisandro Hardaway, Maurizio
Hardaway, Shakir Hardaway, Tavarus Hardaway, Waverly Hardaway, Chaun Hardaway, Diane Hardaway, Dontrell Hardaway, Jamiel Hardaway, Jashua Hardaway, Leopold Hardaway, Lynell Hardaway, Montie Hardaway,
Rafe Hardaway, Reginold Hardaway, Romulo Hardaway, Thayne Hardaway, Tierre Hardaway, Wilhelm Hardaway, Akim Hardaway, Baldomero Hardaway, Darrien Hardaway, Durell Hardaway, Eden Hardaway, Gergory
Hardaway, Kerri Hardaway, Lennon Hardaway, Margaret Hardaway, Nikita Hardaway, Noam Hardaway, Rajan Hardaway, Reynard Hardaway, Sachin Hardaway, Seferino Hardaway, Ventura Hardaway, Warrick Hardaway,
Washington Hardaway, Antonius Hardaway, Barrington Hardaway, Billyjoe Hardaway, Christo Hardaway, Derald Hardaway, Dewight Hardaway, Ferrell Hardaway, Filippo Hardaway, Garin Hardaway, Herbie
Hardaway, Joachim Hardaway, Khalif Hardaway, Mikey Hardaway, Severo Hardaway, Tahir Hardaway, Taras Hardaway, Vinay Hardaway, Aharon Hardaway, Antiono Hardaway, Apolinar Hardaway, Arsenio Hardaway,
Furman Hardaway, Gayle Hardaway, Girard Hardaway, Grey Hardaway, Hillary Hardaway, Martel Hardaway, Murry Hardaway, Nabil Hardaway, Neel Hardaway, Ozell Hardaway, Rylan Hardaway, Samantha Hardaway,
Tellis Hardaway, Thompson Hardaway, Varian Hardaway, Antoinne Hardaway, Derrel Hardaway, Erika Hardaway, Faisal Hardaway, Kristina Hardaway, Laird Hardaway, Lancer Hardaway, Marcelle Hardaway, Melvyn
Hardaway, Sabin Hardaway, Tajuan Hardaway, Terrick Hardaway, Thadius Hardaway, Timoteo Hardaway, Treavor Hardaway, Tyrese Hardaway, Victoria Hardaway, Vishal Hardaway, Aran Hardaway, Christoffer
Hardaway, Devron Hardaway, Durwin Hardaway, Elon Hardaway, Hansel Hardaway, Haskell Hardaway, Jerell Hardaway, Jonathen Hardaway, Lanier Hardaway, Marcellous Hardaway, Marquez Hardaway, Regino
Hardaway, Skylar Hardaway, Christobal Hardaway, Claudia Hardaway, Dartagnan Hardaway, Errin Hardaway, Hung Hardaway, Juvenal Hardaway, Kaseem Hardaway, Keri Hardaway, Marwan Hardaway, Matteo
Hardaway, Raymone Hardaway, Rennie Hardaway, Rondale Hardaway, Shadd Hardaway, Yul Hardaway, Arnel Hardaway, Concepcion Hardaway, Daran Hardaway, Flynn Hardaway, Jawara Hardaway, Jeremia Hardaway,
Karry Hardaway, Rodric Hardaway, Romon Hardaway, Aren Hardaway, Bevan Hardaway, Casper Hardaway, Collis Hardaway, Delane Hardaway, Erric Hardaway, Gerhard Hardaway, Hillel Hardaway, Jansen Hardaway,
Jenny Hardaway, Keelan Hardaway, Michell Hardaway, Petros Hardaway, Saeed Hardaway, Shamon Hardaway, Starsky Hardaway, Tamir Hardaway, Umar Hardaway, Verdell Hardaway, Vicent Hardaway, Alpha
Hardaway, Arther Hardaway, Case Hardaway, Cavin Hardaway, Deacon Hardaway, Franciso Hardaway, Freedom Hardaway, Gregary Hardaway, Harlen Hardaway, Ichael Hardaway, Jacy Hardaway, Jeremaine Hardaway,
Jud Hardaway, Judge Hardaway, Monico Hardaway, Raffi Hardaway, Ricci Hardaway, Sacha Hardaway, Adelbert Hardaway, Andree Hardaway, Bron Hardaway, Cisco Hardaway, Creed Hardaway, Criag Hardaway,
Edilberto Hardaway, Gillermo Hardaway, Hobart Hardaway, Justino Hardaway, Kodi Hardaway, Martino Hardaway, Mickie Hardaway, Newell Hardaway, Odie Hardaway, Robbi Hardaway, Thaddius Hardaway, Tonny
Hardaway, Tully Hardaway, Werner Hardaway, Zoltan Hardaway, Carrol Hardaway, Cesare Hardaway, Clarance Hardaway, Clem Hardaway, Davion Hardaway, Dvid Hardaway, Graydon Hardaway, Latrell Hardaway,
Manual Hardaway, Pervis Hardaway, Randolf Hardaway, Sinclair Hardaway, Stephane Hardaway, Torian Hardaway, Tylor Hardaway, Tzvi Hardaway, Umberto Hardaway, Baruch Hardaway, Benard Hardaway, Glenwood
Hardaway, Jaun Hardaway, Jermie Hardaway, Johnney Hardaway, Kenard Hardaway, Kyler Hardaway, Luca Hardaway, Martha Hardaway, Nephi Hardaway, Nicholos Hardaway, Nikhil Hardaway, Olan Hardaway, Sir
Hardaway, Stevon Hardaway, Teon Hardaway, Tryone Hardaway, Vanessa Hardaway, Ames Hardaway, Breon Hardaway, Gregroy Hardaway, Heyward Hardaway, Jebediah Hardaway, Kye Hardaway, Larue Hardaway, Lonell
Hardaway, Nevada Hardaway, Norma Hardaway, Renny Hardaway, Rosa Hardaway, Ana Hardaway, Blue Hardaway, Cresencio Hardaway, Danon Hardaway, Debra Hardaway, Denard Hardaway, Deondre Hardaway, Elzie
Hardaway, Gamal Hardaway, Gerrard Hardaway, Heber Hardaway, Jerremy Hardaway, Jobe Hardaway, Jun Hardaway, Karey Hardaway, Marquel Hardaway, Martell Hardaway, Marvel Hardaway, Ocie Hardaway, Raudel
Hardaway, Romaine Hardaway, Rush Hardaway, Ruston Hardaway, Talib Hardaway, Therman Hardaway, Vann Hardaway, Vaughan Hardaway, Wilberto Hardaway, Albin Hardaway, Antwuan Hardaway, Aramis Hardaway,
Bodie Hardaway, Darrion Hardaway, Diron Hardaway, Issa Hardaway, Izaak Hardaway, Javis Hardaway, Jerami Hardaway, Keane Hardaway, Kjell Hardaway, Laurance Hardaway, Marico Hardaway, Philipp Hardaway,
Placido Hardaway, Remy Hardaway, Rion Hardaway, Sanchez Hardaway, Shahid Hardaway, Valdez Hardaway, Acie Hardaway, Aundrey Hardaway, Buckley Hardaway, Eliyahu Hardaway, Erroll Hardaway, Leotis
Hardaway, Levern Hardaway, Mace Hardaway, Perrin Hardaway, Phillipe Hardaway, Rama Hardaway, Steward Hardaway, Tawan Hardaway, Zvi Hardaway, Carlon Hardaway, Daris Hardaway, Deandrea Hardaway,
Dominque Hardaway, Gill Hardaway, Keola Hardaway, Larone Hardaway, Lew Hardaway, Luz Hardaway, Montel Hardaway, Sylvan Hardaway, Tobi Hardaway, Twan Hardaway, Ashraf Hardaway, Avram Hardaway, Benzion
Hardaway, Casimiro Hardaway, Chane Hardaway, Christie Hardaway, Chritopher Hardaway, Cirilo Hardaway, Constantino Hardaway, Dupree Hardaway, Gregor Hardaway, Jathan Hardaway, Keaton Hardaway, Kostas
Hardaway, Larenzo Hardaway, Parag Hardaway, Rudi Hardaway, Sameul Hardaway, Sanjeev Hardaway, Soloman Hardaway, Tarrell Hardaway, Traci Hardaway, Tracie Hardaway, Watson Hardaway, Zoran Hardaway,
Aries Hardaway, Aristotle Hardaway, Avelino Hardaway, Bram Hardaway, Camden Hardaway, Dagan Hardaway, Damin Hardaway, Danta Hardaway, Eliott Hardaway, Felicia Hardaway, Juwan Hardaway, Kemp Hardaway,
Kenn Hardaway, Kenyatte Hardaway, Michail Hardaway, Orval Hardaway, Salvadore Hardaway, Shirley Hardaway, Boone Hardaway, Dionte Hardaway, Durwood Hardaway, Ellison Hardaway, Garron Hardaway, Gerson
Hardaway, Jerrald Hardaway, Lavaughn Hardaway, Musa Hardaway, Rayvon Hardaway, Richrd Hardaway, Sharrod Hardaway, Stony Hardaway, Vikas Hardaway, Ector Hardaway, Elia Hardaway, Jerret Hardaway, Josua
Hardaway, Marlen Hardaway, Mical Hardaway, Niall Hardaway, Nicolaus Hardaway, Niki Hardaway, Raymund Hardaway, Simone Hardaway, Stormy Hardaway, Suzanne Hardaway, Teddie Hardaway, Tron Hardaway,
Wolfgang Hardaway, Anthonio Hardaway, Berton Hardaway, Beverly Hardaway, Clive Hardaway, Corky Hardaway, Danell Hardaway, Darl Hardaway, Ellsworth Hardaway, Gloria Hardaway, Gregery Hardaway, Ivey
Hardaway, Kam Hardaway, Kenderick Hardaway, Magdaleno Hardaway, Markee Hardaway, San Hardaway, Stacie Hardaway, Valdemar Hardaway, Zac Hardaway, Albaro Hardaway, Berkley Hardaway, Boe Hardaway,
Casimir Hardaway, Delroy Hardaway, Janet Hardaway, Jimmey Hardaway, Kalen Hardaway, Kregg Hardaway, Larkin Hardaway, Lenell Hardaway, Natalie Hardaway, Ruel Hardaway, Sabrina Hardaway, Thom Hardaway,
Yale Hardaway, Adnan Hardaway, Ankur Hardaway, Arnett Hardaway, Bishop Hardaway, Brently Hardaway, Carla Hardaway, Courtenay Hardaway, Dashon Hardaway, Ebon Hardaway, Edd Hardaway, Ezzard Hardaway,
Gifford Hardaway, Jermon Hardaway, Jeryl Hardaway, Jock Hardaway, Kiven Hardaway, Mandrell Hardaway, Nichols Hardaway, Norwood Hardaway, Olivier Hardaway, Ramin Hardaway, Rikki Hardaway, Aleksandar
Hardaway, Burley Hardaway, Kawan Hardaway, Kourtney Hardaway, Lary Hardaway, Laszlo Hardaway, Loyal Hardaway, Marin Hardaway, Richey Hardaway, Rio Hardaway, Rogerio Hardaway, Salem Hardaway, Shalom
Hardaway, Sharron Hardaway, Tamer Hardaway, Torie Hardaway, Antar Hardaway, Barrie Hardaway, Cletis Hardaway, Colter Hardaway, Duaine Hardaway, Elpidio Hardaway, Erice Hardaway, Farid Hardaway,
Fortunato Hardaway, Gerrad Hardaway, Jad Hardaway, Jenaro Hardaway, Jones Hardaway, Kamran Hardaway, Kelli Hardaway, Keneth Hardaway, Kortney Hardaway, Levell Hardaway, Minh Hardaway, Monti Hardaway,
Rodell Hardaway, Ronal Hardaway, Sheddrick Hardaway, Sonia Hardaway, Terryl Hardaway, York Hardaway, Andrian Hardaway, Anne Hardaway, Attila Hardaway, Avis Hardaway, Blaze Hardaway, Chi Hardaway,
Corin Hardaway, Dann Hardaway, Darik Hardaway, Deleon Hardaway, Demarko Hardaway, Dennison Hardaway, Dovid Hardaway, Dutch Hardaway, Emmit Hardaway, Evert Hardaway, Iman Hardaway, Joshue Hardaway,
Lindy Hardaway, Mohamad Hardaway, Purnell Hardaway, Rayshon Hardaway, Tyreese Hardaway, Alistair Hardaway, Arman Hardaway, Blakely Hardaway, Brit Hardaway, Carlis Hardaway, Doniel Hardaway, Earnie
Hardaway, Hershell Hardaway, Jasin Hardaway, Kendale Hardaway, Laine Hardaway, Leandrew Hardaway, Lenin Hardaway, Macon Hardaway, Merrell Hardaway, Mychal Hardaway, Nat Hardaway, Nicanor Hardaway,
Nirav Hardaway, Waldemar Hardaway, Yakov Hardaway, Adrion Hardaway, Bowen Hardaway, Carolyn Hardaway, Creg Hardaway, Farley Hardaway, Favian Hardaway, Ferman Hardaway, Hurley Hardaway, Jairus
Hardaway, Kevyn Hardaway, Korry Hardaway, Mandell Hardaway, Melchor Hardaway, Osualdo Hardaway, Shanti Hardaway, Sheila Hardaway, Sotirios Hardaway, Taft Hardaway, Tijuan Hardaway, Antonyo Hardaway,
Antwoin Hardaway, Arley Hardaway, Carlisle Hardaway, Christerpher Hardaway, Derell Hardaway, Dimitrius Hardaway, Fabrizio Hardaway, Gibran Hardaway, Jasn Hardaway, June Hardaway, Kenta Hardaway,
Kenyetta Hardaway, Lantz Hardaway, Mart Hardaway, Naaman Hardaway, Nasser Hardaway, Norvell Hardaway, Ole Hardaway, Perfecto Hardaway, Philemon Hardaway, Primitivo Hardaway, Quint Hardaway, Rony
Hardaway, Shanta Hardaway, Tighe Hardaway, Trini Hardaway, Antionne Hardaway, Antwaine Hardaway, Bradon Hardaway, Campbell Hardaway, Chao Hardaway, Donel Hardaway, Isa Hardaway, Kendra Hardaway, Kiel
Hardaway, Lejon Hardaway, Lonn Hardaway, Micael Hardaway, Namon Hardaway, Perez Hardaway, Ramses Hardaway, Renwick Hardaway, Rigo Hardaway, Rober Hardaway, Rondall Hardaway, Terral Hardaway, Tobe
Hardaway, Akili Hardaway, Dalen Hardaway, Deanthony Hardaway, Elson Hardaway, Fulton Hardaway, Hillard Hardaway, Jedadiah Hardaway, Jihad Hardaway, Keevin Hardaway, Kile Hardaway, Kito Hardaway,
Kwane Hardaway, Lamount Hardaway, Lanell Hardaway, Leith Hardaway, Marshawn Hardaway, Marwin Hardaway, Raymont Hardaway, Rumaldo Hardaway, Tabari Hardaway, Tex Hardaway, Yechiel Hardaway, Chirag
Hardaway, Crist Hardaway, Damain Hardaway, Damar Hardaway, Deke Hardaway, Earlie Hardaway, Eleuterio Hardaway, Garen Hardaway, Joanthan Hardaway, Josephus Hardaway, Marisol Hardaway, Phillips
Hardaway, Priest Hardaway, Romie Hardaway, Sonya Hardaway, Sung Hardaway, Tandy Hardaway, Traves Hardaway, Augusta Hardaway, Avon Hardaway, Daemon Hardaway, Danie Hardaway, Delwyn Hardaway, Eldrick
Hardaway, Gerod Hardaway, Hannibal Hardaway, Harper Hardaway, Javiel Hardaway, Jawan Hardaway, Kathy Hardaway, Kiran Hardaway, Linn Hardaway, Manolito Hardaway, Remus Hardaway, Salome Hardaway,
Savalas Hardaway, Shamel Hardaway, Sigfredo Hardaway, Tylon Hardaway, Yehoshua Hardaway, Damean Hardaway, Gualberto Hardaway, Guiseppe Hardaway, Huy Hardaway, Jaymes Hardaway, Jaymie Hardaway, Jeshua
Hardaway, Maher Hardaway, Mahmoud Hardaway, Otoniel Hardaway, Radford Hardaway, Rodman Hardaway, Tevis Hardaway, Timohty Hardaway, Tripp Hardaway, Wells Hardaway, Allie Hardaway, Ason Hardaway, Beth
Hardaway, Carlas Hardaway, Cederick Hardaway, Colleen Hardaway, Cortland Hardaway, Deanna Hardaway, Farron Hardaway, Gayland Hardaway, Jaeson Hardaway, Jamerson Hardaway, Kenon Hardaway, Keoki
Hardaway, Kevon Hardaway, Kristy Hardaway, Ladale Hardaway, Mare Hardaway, Maurilio Hardaway, Niko Hardaway, Nikos Hardaway, Orland Hardaway, Remigio Hardaway, Rhys Hardaway, Ruby Hardaway, Simcha
Hardaway, Thatcher Hardaway, Travas Hardaway, Zachry Hardaway, Zenon Hardaway, Amish Hardaway, Anthon Hardaway, Aswad Hardaway, Erickson Hardaway, Fernandez Hardaway, Finley Hardaway, Gorden
Hardaway, Ignatius Hardaway, Illya Hardaway, Jerico Hardaway, Jermayne Hardaway, Kamron Hardaway, Keita Hardaway, Kwabena Hardaway, Lonzell Hardaway, Manley Hardaway, Markanthony Hardaway, Mika
Hardaway, Muhammed Hardaway, Nicholes Hardaway, Percival Hardaway, Quin Hardaway, Radley Hardaway, Ruth Hardaway, Sabas Hardaway, Shaft Hardaway, Shante Hardaway, Sullivan Hardaway, Waylan Hardaway,
Westly Hardaway, Woodie Hardaway, Algie Hardaway, Arvel Hardaway, Benedetto Hardaway, Bradrick Hardaway, Chancellor Hardaway, Chawn Hardaway, Daylon Hardaway, Devonne Hardaway, Galvin Hardaway, Herb
Hardaway, Jacen Hardaway, Jadon Hardaway, Janice Hardaway, Kristi Hardaway, Leeland Hardaway, Merton Hardaway, Ossie Hardaway, Patsy Hardaway, Reynolds Hardaway, Sebastiano Hardaway, Seymour
Hardaway, Shanan Hardaway, Shawndell Hardaway, Terrie Hardaway, Toure Hardaway, Zakee Hardaway, Zebulun Hardaway, Alexei Hardaway, Branon Hardaway, Derrin Hardaway, Deryck Hardaway, Edelmiro
Hardaway, Felice Hardaway, Hamid Hardaway, Higinio Hardaway, Jule Hardaway, Kellie Hardaway, Kern Hardaway, Khaled Hardaway, Marciano Hardaway, Morrell Hardaway, Nazario Hardaway, Oneil Hardaway,
Said Hardaway, Silvester Hardaway, Sotero Hardaway, Yvonne Hardaway, Adron Hardaway, Alando Hardaway, Anothony Hardaway, Brando Hardaway, Brenten Hardaway, Canaan Hardaway, Colon Hardaway, Dany
Hardaway, Fenton Hardaway, Griffith Hardaway, Halbert Hardaway, Johney Hardaway, Karlo Hardaway, Keone Hardaway, Leviticus Hardaway, Maurio Hardaway, Murad Hardaway, Nahum Hardaway, Vester Hardaway,
Anita Hardaway, Archibald Hardaway, Brinton Hardaway, Cayce Hardaway, Craige Hardaway, Damont Hardaway, Heinz Hardaway, Joaquim Hardaway, Lajuane Hardaway, Lakendrick Hardaway, Lanard Hardaway, Lesly
Hardaway, Nam Hardaway, Remi Hardaway, Semaj Hardaway, Shadrach Hardaway, Sholom Hardaway, Silvano Hardaway, Timothey Hardaway, Vernal Hardaway, Vernie Hardaway, Alix Hardaway, Aristeo Hardaway,
Bubba Hardaway, Carlie Hardaway, Ceaser Hardaway, Dallin Hardaway, Darrow Hardaway, Deondra Hardaway, Deonta Hardaway, Engelbert Hardaway, Jabier Hardaway, Jarron Hardaway, Kekoa Hardaway, Lasean
Hardaway, Manolo Hardaway, Marqus Hardaway, Marshell Hardaway, Menno Hardaway, Ram Hardaway, Raynell Hardaway, Rohan Hardaway, Amadeo Hardaway, Arch Hardaway, Arlis Hardaway, Ayinde Hardaway, Barnard
Hardaway, Darriel Hardaway, Darvis Hardaway, Derryl Hardaway, Dewain Hardaway, Dietrick Hardaway, Glyn Hardaway, Jasmine Hardaway, Kenyan Hardaway, Michae Hardaway, Montoya Hardaway, Oracio Hardaway,
Roney Hardaway, Sascha Hardaway, Shermaine Hardaway, Sigifredo Hardaway, Theodor Hardaway, Virginia Hardaway, Waco Hardaway, Yaphet Hardaway, Zebadiah Hardaway, Alika Hardaway, Aundrea Hardaway,
Brendt Hardaway, Carvin Hardaway, Christophere Hardaway, Clevon Hardaway, Drayton Hardaway, Evelio Hardaway, Gerasimos Hardaway, Jemaine Hardaway, Kian Hardaway, Lucan Hardaway, Moise Hardaway, Richy
Hardaway, Tyshon Hardaway, Zebedee Hardaway, Alcides Hardaway, Allister Hardaway, Altonio Hardaway, Bengie Hardaway, Cavan Hardaway, Desiderio Hardaway, Dewon Hardaway, Eulogio Hardaway, Faheem
Hardaway, Georges Hardaway, Hasson Hardaway, Hollie Hardaway, Izell Hardaway, Jamee Hardaway, Jestin Hardaway, Keldrick Hardaway, Kier Hardaway, Lorn Hardaway, Mister Hardaway, Niraj Hardaway, Omega
Hardaway, Osman Hardaway, Ragan Hardaway, Raja Hardaway, Talbert Hardaway, Thanh Hardaway, Triston Hardaway, Vashawn Hardaway, Victorio Hardaway, Aubry Hardaway, Audwin Hardaway, Babatunde Hardaway,
Bejamin Hardaway, Cantrell Hardaway, Carman Hardaway, Chaddrick Hardaway, Claiborne Hardaway, Creston Hardaway, Deshone Hardaway, Dewane Hardaway, Gabriele Hardaway, Hason Hardaway, Jessi Hardaway,
Keenon Hardaway, Keithan Hardaway, Kostantinos Hardaway, Leah Hardaway, Ludwig Hardaway, Maverick Hardaway, Neilson Hardaway, Octaviano Hardaway, Orestes Hardaway, Otho Hardaway, Parry Hardaway,
Percell Hardaway, Rashod Hardaway, Rochelle Hardaway, Rondel Hardaway, Rondey Hardaway, Rowan Hardaway, Sang Hardaway, Shawnta Hardaway, Sulaiman Hardaway, Suresh Hardaway, Taran Hardaway, Tatum
Hardaway, Venson Hardaway, Agustine Hardaway, Babak Hardaway, Biagio Hardaway, Cedar Hardaway, Christorpher Hardaway, Cloyd Hardaway, Corinthian Hardaway, Curran Hardaway, Dani Hardaway, Dannell
Hardaway, Dayle Hardaway, Dermot Hardaway, Encarnacion Hardaway, Hendrick Hardaway, Imani Hardaway, Jackey Hardaway, Keion Hardaway, Kennis Hardaway, Langdon Hardaway, Lawayne Hardaway, Leticia
Hardaway, Levan Hardaway, Liberty Hardaway, Mardy Hardaway, Nicolo Hardaway, Nolen Hardaway, Octavian Hardaway, Purvis Hardaway, Reuven Hardaway, Ronrico Hardaway, Tharon Hardaway, Turhan Hardaway,
Tyren Hardaway, Wess Hardaway, Aman Hardaway, Amiel Hardaway, Apostolos Hardaway, Clayborn Hardaway, Devone Hardaway, Genesis Hardaway, Giorgio Hardaway, Jaleel Hardaway, Jarmar Hardaway, Jasan
Hardaway, Jermar Hardaway, Jojo Hardaway, Jondavid Hardaway, Martyn Hardaway, Melecio Hardaway, Michelangelo Hardaway, Mikkel Hardaway, Montague Hardaway, Nicki Hardaway, Quenten Hardaway, Ramie
Hardaway, Rudolpho Hardaway, Shenandoah Hardaway, Tilden Hardaway, Tosh Hardaway, Zedrick Hardaway, Able Hardaway, Aldon Hardaway, Alejo Hardaway, Atif Hardaway, Cason Hardaway, Daiel Hardaway,
Davian Hardaway, Donaciano Hardaway, Dywane Hardaway, Eliberto Hardaway, Ephriam Hardaway, Galo Hardaway, Giulio Hardaway, Hanif Hardaway, Harlon Hardaway, Hermon Hardaway, Homar Hardaway, Jhonny
Hardaway, Jonnathan Hardaway, Kara Hardaway, Keath Hardaway, Kennieth Hardaway, Kirsten Hardaway, Kriss Hardaway, Lennis Hardaway, Leonides Hardaway, Levelle Hardaway, Marlyn Hardaway, Merced
Hardaway, Montrel Hardaway, Naveen Hardaway, Newman Hardaway, Osiris Hardaway, Raymondo Hardaway, Renzo Hardaway, Royden Hardaway, Shawnn Hardaway, Shuan Hardaway, Thurmond Hardaway, Treg Hardaway,
Tremell Hardaway, Trenten Hardaway, Aldric Hardaway, Alexandra Hardaway, Alston Hardaway, Anish Hardaway, Anuj Hardaway, Arif Hardaway, Ashby Hardaway, Ashok Hardaway, Chavez Hardaway, Cinque
Hardaway, Derris Hardaway, Donovon Hardaway, Doroteo Hardaway, Fontaine Hardaway, Hadley Hardaway, Jeanclaude Hardaway, Josha Hardaway, Kenyada Hardaway, Laban Hardaway, Neeraj Hardaway, Nilesh
Hardaway, Oral Hardaway, Quran Hardaway, Raheim Hardaway, Rockford Hardaway, Rondy Hardaway, Sesar Hardaway, Thayer Hardaway, Vincient Hardaway, Amilcar Hardaway, Antion Hardaway, Antrone Hardaway,
Ardis Hardaway, Arvil Hardaway, Avid Hardaway, Branton Hardaway, Carmon Hardaway, Claudell Hardaway, Deion Hardaway, Dejon Hardaway, Delonte Hardaway, Delvecchio Hardaway, Dickson Hardaway, Elwyn
Hardaway, Fredie Hardaway, Garnet Hardaway, Gunther Hardaway, Jawanza Hardaway, Jermone Hardaway, Katrina Hardaway, Krista Hardaway, Landy Hardaway, Levester Hardaway, Levin Hardaway, Rahiem
Hardaway, Rashun Hardaway, Rinaldo Hardaway, Ronson Hardaway, Terril Hardaway, Ambrosio Hardaway, Arvid Hardaway, Atlee Hardaway, Calogero Hardaway, Cobey Hardaway, Donathan Hardaway, Donelle
Hardaway, Dorman Hardaway, Dung Hardaway, Garcia Hardaway, Jase Hardaway, Jeanette Hardaway, Kolin Hardaway, Kurtiss Hardaway, Lorenz Hardaway, Mathis Hardaway, Natale Hardaway, Patton Hardaway,
Petar Hardaway, Richar Hardaway, Thadd Hardaway, Timithy Hardaway, Tomislav Hardaway, Tommaso Hardaway, Tyris Hardaway, Uvaldo Hardaway, Anothy Hardaway, Aristides Hardaway, Arno Hardaway, Deshannon
Hardaway, Eamonn Hardaway, Hiawatha Hardaway, Jamol Hardaway, Jasun Hardaway, Joedy Hardaway, Lenn Hardaway, Lord Hardaway, Manoj Hardaway, Merl Hardaway, Michaelangelo Hardaway, Navin Hardaway, Per
Hardaway, Peterson Hardaway, Sherri Hardaway, Simmie Hardaway, Socrates Hardaway, Tramell Hardaway, Trevino Hardaway, Winslow Hardaway, Andrell Hardaway, Arlington Hardaway, Austen Hardaway, Aziz
Hardaway, Benuel Hardaway, Cristofer Hardaway, Delrico Hardaway, Dereke Hardaway, Edmon Hardaway, Finis Hardaway, Frederich Hardaway, Hussein Hardaway, Jerman Hardaway, Johnathen Hardaway, Joni
Hardaway, Kariem Hardaway, Kasim Hardaway, Klayton Hardaway, Lafe Hardaway, Lamario Hardaway, Manning Hardaway, Markos Hardaway, Natasha Hardaway, Nicholis Hardaway, Nickolaos Hardaway, Nima
Hardaway, Nitin Hardaway, Piero Hardaway, Robertson Hardaway, Sabastian Hardaway, Sandon Hardaway, Shandy Hardaway, Sylvia Hardaway, Tacuma Hardaway, Tarance Hardaway, Tarl Hardaway, Traver Hardaway,
Ander Hardaway, Bartolo Hardaway, Brentt Hardaway, Chace Hardaway, Chay Hardaway, Clete Hardaway, Colbert Hardaway, Domonick Hardaway, Dondre Hardaway, Emanuele Hardaway, Enoc Hardaway, Hercules
Hardaway, Inocencio Hardaway, Ivery Hardaway, Jaja Hardaway, Jalal Hardaway, Jamelle Hardaway, Jin Hardaway, Jumaane Hardaway, Kendel Hardaway, Kurk Hardaway, Lendell Hardaway, Linc Hardaway, Louise
Hardaway, Luc Hardaway, Mackie Hardaway, Mehul Hardaway, Myrick Hardaway, Nasir Hardaway, Omero Hardaway, Osiel Hardaway, Rhyan Hardaway, Roni Hardaway, Roshan Hardaway, Sander Hardaway, Thai
Hardaway, Toy Hardaway, Willia Hardaway, Zacharias Hardaway, Zenas Hardaway, Abelino Hardaway, Ameen Hardaway, Bridget Hardaway, Claudius Hardaway, Creig Hardaway, Danton Hardaway, Danuel Hardaway,
Durward Hardaway, Duval Hardaway, Fredrico Hardaway, Gibson Hardaway, Hoang Hardaway, Huston Hardaway, Ildefonso Hardaway, Jamaar Hardaway, Jerrard Hardaway, Jsaon Hardaway, Korie Hardaway, Lang
Hardaway, Latron Hardaway, Lawernce Hardaway, Lief Hardaway, Manu Hardaway, Mervyn Hardaway, Mingo Hardaway, Nadeem Hardaway, Nunzio Hardaway, Odin Hardaway, Pinchas Hardaway, Quoc Hardaway, Ramy
Hardaway, Randol Hardaway, Reginaldo Hardaway, Tam Hardaway, Theodoros Hardaway, Vinnie Hardaway, Arland Hardaway, Atul Hardaway, Baudelio Hardaway, Bernhard Hardaway, Chadric Hardaway, Charon
Hardaway, Damaris Hardaway, Derin Hardaway, Donavin Hardaway, Feliz Hardaway, Ferdinando Hardaway, Fortino Hardaway, Geron Hardaway, Gershon Hardaway, Hanson Hardaway, Helen Hardaway, Hodari
Hardaway, Jamen Hardaway, Jayce Hardaway, Kalonji Hardaway, Kayle Hardaway, Lamorris Hardaway, Lennox Hardaway, Mannix Hardaway, Minor Hardaway, Nathaneal Hardaway, Olegario Hardaway, Pantelis
Hardaway, Princeton Hardaway, Quince Hardaway, Rashied Hardaway, Rose Hardaway, Shaye Hardaway, Son Hardaway, Stephone Hardaway, Trenell Hardaway, Ulric Hardaway, Arion Hardaway, Burgess Hardaway,
Chadwin Hardaway, Collier Hardaway, Derico Hardaway, Egan Hardaway, Foy Hardaway, Jacobi Hardaway, Jemar Hardaway, Kay Hardaway, Lake Hardaway, Markey Hardaway, Micaiah Hardaway, Nasario Hardaway,
Oakley Hardaway, Remo Hardaway, Richad Hardaway, Ringo Hardaway, Romano Hardaway, Thierry Hardaway, Vergil Hardaway, Vinton Hardaway, Ajamu Hardaway, Andrzej Hardaway, Annette Hardaway, Asif
Hardaway, Aundray Hardaway, Benn Hardaway, Bernice Hardaway, Byrant Hardaway, Clemon Hardaway, Cleotha Hardaway, Cully Hardaway, Darious Hardaway, Diondre Hardaway, Donivan Hardaway, Fard Hardaway,
Gianfranco Hardaway, Gyasi Hardaway, Hosie Hardaway, Inez Hardaway, Jaden Hardaway, Julious Hardaway, Junious Hardaway, Ketan Hardaway, Kieron Hardaway, Loring Hardaway, Noal Hardaway, Pepe Hardaway,
Pharoah Hardaway, Pilar Hardaway, Raquel Hardaway, Remon Hardaway, Sederick Hardaway, Severiano Hardaway, Shawna Hardaway, Shawnee Hardaway, Sim Hardaway, Spurgeon Hardaway, Stace Hardaway, Taber
Hardaway, Tarron Hardaway, Thorin Hardaway, Wil Hardaway, Zaire Hardaway, Andray Hardaway, Berlin Hardaway, Betty Hardaway, Biran Hardaway, Chaney Hardaway, Chon Hardaway, Dearl Hardaway, Demetria
Hardaway, Dewaine Hardaway, Edan Hardaway, Ediberto Hardaway, Froilan Hardaway, Henrique Hardaway, Jamille Hardaway, Jessey Hardaway, Kaine Hardaway, Kendricks Hardaway, Kynan Hardaway, Laray
Hardaway, Laren Hardaway, Mandy Hardaway, Marilyn Hardaway, Neale Hardaway, Nyle Hardaway, Okey Hardaway, Ramesh Hardaway, Ricard Hardaway, Saladin Hardaway, Sherrill Hardaway, Shonta Hardaway, Sione
Hardaway, Steaven Hardaway, Stefon Hardaway, Taji Hardaway, Toris Hardaway, Travell Hardaway, Urbano Hardaway, Wolf Hardaway, Alvah Hardaway, Ante Hardaway, Arlon Hardaway, Auther Hardaway, Bran
Hardaway, Chevy Hardaway, Cormac Hardaway, Damione Hardaway, Delando Hardaway, Deno Hardaway, Dow Hardaway, Earvin Hardaway, Erinn Hardaway, Florian Hardaway, Georg Hardaway, Gerren Hardaway, Henery
Hardaway, Isac Hardaway, Jamond Hardaway, Jaramie Hardaway, Kinta Hardaway, Kylan Hardaway, Lashan Hardaway, Mckay Hardaway, Molly Hardaway, Ngai Hardaway, Nolberto Hardaway, Quang Hardaway, Solon
Hardaway, Tobiah Hardaway, Tricia Hardaway, Wai Hardaway, Willem Hardaway, Yaron Hardaway, Agostino Hardaway, Alfonse Hardaway, Antonious Hardaway, Antwion Hardaway, Bryen Hardaway, Camille Hardaway,
Carsten Hardaway, Cheo Hardaway, Chevis Hardaway, Colen Hardaway, Corneilus Hardaway, Davidson Hardaway, Dawon Hardaway, Deldrick Hardaway, Dia Hardaway, Dimitris Hardaway, Donnelle Hardaway, Dory
Hardaway, Ewell Hardaway, Hai Hardaway, Heron Hardaway, Hope Hardaway, Imari Hardaway, Jaremy Hardaway, Lindon Hardaway, Loreto Hardaway, Nichole Hardaway, Nieves Hardaway, Phelan Hardaway, Randale
Hardaway, Rik Hardaway, Saturnino Hardaway, Schawn Hardaway, Sterlin Hardaway, Talmage Hardaway, Tavoris Hardaway, Toribio Hardaway, Trebor Hardaway, Vick Hardaway, Worth Hardaway, Zaid Hardaway,
Adin Hardaway, Alim Hardaway, Artez Hardaway, Bashir Hardaway, Dameian Hardaway, Demorris Hardaway, Deryk Hardaway, Estill Hardaway, Gautam Hardaway, Hyman Hardaway, Isom Hardaway, Jeral Hardaway,
Juanita Hardaway, Kalman Hardaway, Kee Hardaway, Keithen Hardaway, Kipling Hardaway, Lesean Hardaway, Love Hardaway, Nadir Hardaway, Norton Hardaway, Philander Hardaway, Sebastien Hardaway, Sherif
Hardaway, Tabitha Hardaway, Webb Hardaway, Akin Hardaway, Blu Hardaway, Calbert Hardaway, Cassandra Hardaway, Cephus Hardaway, Dalvin Hardaway, Daxton Hardaway, Delfin Hardaway, Drexel Hardaway,
Elijio Hardaway, Fareed Hardaway, Geffrey Hardaway, Jabin Hardaway, Jodey Hardaway, Jomar Hardaway, Judy Hardaway, Jujuan Hardaway, Klay Hardaway, Kyriakos Hardaway, Laderrick Hardaway, Landen
Hardaway, Latoya Hardaway, Lebron Hardaway, Mat Hardaway, Rashi Hardaway, Roberta Hardaway, Rodderick Hardaway, Shant Hardaway, Summer Hardaway, Viktor Hardaway, Abayomi Hardaway, Adrienne Hardaway,
Akeem Hardaway, Amaury Hardaway, Andrez Hardaway, Antuane Hardaway, Barnabas Hardaway, Corneluis Hardaway, Delray Hardaway, Demarlo Hardaway, Deshan Hardaway, Dev Hardaway, Diarra Hardaway, Duriel
Hardaway, Emigdio Hardaway, Eon Hardaway, Evelyn Hardaway, Hasaan Hardaway, Janmichael Hardaway, Jeston Hardaway, Jobie Hardaway, Kalon Hardaway, Kumar Hardaway, Lemond Hardaway, Raimundo Hardaway,
Rainier Hardaway, Robbert Hardaway, Rulon Hardaway, Skeeter Hardaway, Starling Hardaway, Sun Hardaway, Tammie Hardaway, Terrelle Hardaway, Tiwan Hardaway, Toddrick Hardaway, Varick Hardaway, Zavier
Hardaway, Zion Hardaway, Aniceto Hardaway, Asmar Hardaway, Boaz Hardaway, Cayetano Hardaway, Dakarai Hardaway, Deadrick Hardaway, Debbie Hardaway, Dominik Hardaway, Dugan Hardaway, Emad Hardaway,
Glennon Hardaway, Haralambos Hardaway, Harlin Hardaway, Hilliard Hardaway, Hristopher Hardaway, Humphrey Hardaway, Javar Hardaway, Javin Hardaway, Jeffre Hardaway, Josuha Hardaway, Kashif Hardaway,
Keino Hardaway, Kemper Hardaway, Kendrix Hardaway, Keshawn Hardaway, Lamone Hardaway, Lashone Hardaway, Lekeith Hardaway, Markham Hardaway, Maruice Hardaway, Miguelangel Hardaway, Nichalos Hardaway,
Olanda Hardaway, Rodrigues Hardaway, Tarig Hardaway, Tarius Hardaway, Tonnie Hardaway, Velton Hardaway, Walden Hardaway, Yasin Hardaway, Alexie Hardaway, Autry Hardaway, Burnett Hardaway, Burnis
Hardaway, Cephas Hardaway, Crandall Hardaway, Curvin Hardaway, Dashaun Hardaway, Donyel Hardaway, Emir Hardaway, Eston Hardaway, Farrel Hardaway, Giovani Hardaway, Jamaul Hardaway, Jarmel Hardaway,
Jarold Hardaway, Jeorge Hardaway, Joshwa Hardaway, Jousha Hardaway, Joyce Hardaway, Justan Hardaway, Kem Hardaway, Lavance Hardaway, Ledell Hardaway, Manuelito Hardaway, Michaell Hardaway, Mischa
Hardaway, Osborne Hardaway, Park Hardaway, Partick Hardaway, Rasaan Hardaway, Ravon Hardaway, Romone Hardaway, Stefen Hardaway, Veron Hardaway, Vondell Hardaway, Airrion Hardaway, Aki Hardaway,
Alegandro Hardaway, Author Hardaway, Autumn Hardaway, Brannan Hardaway, Caliph Hardaway, Carles Hardaway, Cartez Hardaway, Codey Hardaway, Dammon Hardaway, Daunte Hardaway, Demonte Hardaway, Devell
Hardaway, Domonique Hardaway, Donyale Hardaway, Dornell Hardaway, Duwan Hardaway, Edvardo Hardaway, Emeterio Hardaway, Gerado Hardaway, Gordan Hardaway, Jahi Hardaway, Jair Hardaway, Jasmin Hardaway,
Jaydee Hardaway, Judas Hardaway, Kennan Hardaway, Lehman Hardaway, Macy Hardaway, Mayo Hardaway, Mazen Hardaway, Mikhail Hardaway, Osama Hardaway, Rainer Hardaway, Raman Hardaway, Rocio Hardaway,
Ryen Hardaway, Shayn Hardaway, Tee Hardaway, Temple Hardaway, Timathy Hardaway, Vinh Hardaway, Adarryl Hardaway, Adrin Hardaway, Alastair Hardaway, Anh Hardaway, Braun Hardaway, Burrell Hardaway,
Dalon Hardaway, Demetrias Hardaway, Derrill Hardaway, Donnis Hardaway, Dwon Hardaway, Fredick Hardaway, Gasper Hardaway, Glade Hardaway, Hulon Hardaway, Jonatha Hardaway, Jorma Hardaway, Kaipo
Hardaway, Karel Hardaway, Kejuan Hardaway, Kristine Hardaway, Larrie Hardaway, Majid Hardaway, Mar Hardaway, Mareo Hardaway, Nahshon Hardaway, Ondre Hardaway, Rasool Hardaway, Rayn Hardaway, Reade
Hardaway, Roshon Hardaway, Rozell Hardaway, Ryann Hardaway, Salbador Hardaway, Seon Hardaway, Sharone Hardaway, Shown Hardaway, Taggart Hardaway, Tarell Hardaway, Tymon Hardaway, Akira Hardaway,
Alice Hardaway, Arend Hardaway, Baretta Hardaway, Billyjack Hardaway, Breton Hardaway, Cleophas Hardaway, Dejan Hardaway, Delante Hardaway, Delmus Hardaway, Derren Hardaway, Doni Hardaway, Ezekial
Hardaway, Fernand Hardaway, Gable Hardaway, Gilford Hardaway, Irfan Hardaway, Jamarcus Hardaway, Jamason Hardaway, Jermond Hardaway, Jozef Hardaway, Kal Hardaway, Kalin Hardaway, Kayne Hardaway,
Kosta Hardaway, Lancelot Hardaway, Lois Hardaway, Marlos Hardaway, Penny Hardaway, Rodriques Hardaway, Rogerick Hardaway, Roldan Hardaway, Ronel Hardaway, Sha Hardaway, Shana Hardaway, Taris
Hardaway, Torres Hardaway, Trapper Hardaway, Venancio Hardaway, Vinod Hardaway, Whit Hardaway, Allah Hardaway, Angelos Hardaway, Antawan Hardaway, Bobbi Hardaway, Declan Hardaway, Eliasar Hardaway,
Emanual Hardaway, Ericson Hardaway, Erol Hardaway, Faris Hardaway, Filemon Hardaway, Guthrie Hardaway, Helder Hardaway, Jahn Hardaway, Jamine Hardaway, Jeoffrey Hardaway, Jervis Hardaway, Ketih
Hardaway, Kimble Hardaway, Kole Hardaway, Krzysztof Hardaway, Lorena Hardaway, Martice Hardaway, Orie Hardaway, Oziel Hardaway, Ralphael Hardaway, Randie Hardaway, Seldon Hardaway, Shanna Hardaway,
Shawndale Hardaway, Shawnte Hardaway, Smokey Hardaway, Teri Hardaway, Tyrrell Hardaway, Wilburt Hardaway, Winton Hardaway, Yonatan Hardaway, Brand Hardaway, Briton Hardaway, Budd Hardaway, Charlene
Hardaway, Chetan Hardaway, Chivas Hardaway, Chriss Hardaway, Daryel Hardaway, Davi Hardaway, Dewarren Hardaway, Elston Hardaway, Garo Hardaway, Hilberto Hardaway, Hughie Hardaway, Jarrel Hardaway,
Juma Hardaway, Kala Hardaway, Kamel Hardaway, Lasalle Hardaway, Leoncio Hardaway, Linnie Hardaway, Markis Hardaway, Melville Hardaway, Nekia Hardaway, Obert Hardaway, Otilio Hardaway, Quintus
Hardaway, Romualdo Hardaway, Romulus Hardaway, Sanjuan Hardaway, Shadi Hardaway, Shonte Hardaway, Stelios Hardaway, Tavaras Hardaway, Tennyson Hardaway, Timon Hardaway, Torrell Hardaway, Tramel
Hardaway, Tyrece Hardaway, Vertis Hardaway, Vipul Hardaway, Wenceslao Hardaway, Zuri Hardaway, Amer Hardaway, Aquil Hardaway, Ardie Hardaway, Brayden Hardaway, Cederic Hardaway, Claudie Hardaway,
Colm Hardaway, Cristen Hardaway, Crosby Hardaway, Darlene Hardaway, Darrol Hardaway, Darroll Hardaway, Dequincy Hardaway, Donold Hardaway, Eddrick Hardaway, Elder Hardaway, Gaines Hardaway, Giovanny
Hardaway, Hermilo Hardaway, Ivo Hardaway, Jermale Hardaway, Jibri Hardaway, Joshawa Hardaway, Kendrell Hardaway, Kerin Hardaway, Kert Hardaway, Kojo Hardaway, Koy Hardaway, Lawerance Hardaway, Letroy
Hardaway, Lumumba Hardaway, Machael Hardaway, Marshon Hardaway, Napolean Hardaway, Nimrod Hardaway, Phong Hardaway, Ramondo Hardaway, Ricahrd Hardaway, Sanjiv Hardaway, Scotti Hardaway, Shaunn
Hardaway, Shedric Hardaway, Sundance Hardaway, Tally Hardaway, Tiant Hardaway, Tysen Hardaway, Ulyses Hardaway, Vivian Hardaway, Ajani Hardaway, Anthany Hardaway, Antowan Hardaway, Brack Hardaway,
Chason Hardaway, Damany Hardaway, Danney Hardaway, Demitri Hardaway, Diangelo Hardaway, Dontaye Hardaway, Dragan Hardaway, Einar Hardaway, Gaspare Hardaway, Gerrell Hardaway, Gery Hardaway, Heston
Hardaway, Isai Hardaway, Jamale Hardaway, Jamone Hardaway, Jane Hardaway, Jerri Hardaway, Karla Hardaway, Kelan Hardaway, Kerrie Hardaway, Kobi Hardaway, Krist Hardaway, Lalo Hardaway, Latif
Hardaway, Leman Hardaway, Levert Hardaway, Lorenso Hardaway, Marcy Hardaway, Markese Hardaway, Merlyn Hardaway, Milburn Hardaway, Nicolai Hardaway, Norval Hardaway, Oris Hardaway, Ranjit Hardaway,
Rashidi Hardaway, Richardson Hardaway, Rohn Hardaway, Roshaun Hardaway, Shepherd Hardaway, Sonnie Hardaway, Stephens Hardaway, Stirling Hardaway, Sundiata Hardaway, Troyce Hardaway, Vander Hardaway,
Waddell Hardaway, Yarnell Hardaway, Yitzchak Hardaway, Zerrick Hardaway, Adair Hardaway, Amani Hardaway, Arvind Hardaway, Ascencion Hardaway, Athan Hardaway, Biff Hardaway, Burnice Hardaway, Carlito
Hardaway, Carlitos Hardaway, Carlson Hardaway, Carry Hardaway, Christepher Hardaway, Clemmie Hardaway, Corbet Hardaway, Cristoval Hardaway, Damaso Hardaway, Dickey Hardaway, Doral Hardaway, Eliu
Hardaway, Erving Hardaway, Favio Hardaway, Helmut Hardaway, Hisham Hardaway, Ignazio Hardaway, Jamail Hardaway, Jarius Hardaway, Jermanie Hardaway, Jonte Hardaway, Juluis Hardaway, Jushua Hardaway,
Keron Hardaway, Kevis Hardaway, Kollin Hardaway, Labron Hardaway, Lavan Hardaway, Lavoris Hardaway, Lazar Hardaway, Nafis Hardaway, Nathanel Hardaway, Othello Hardaway, Quantrell Hardaway, Ramzi
Hardaway, Rossi Hardaway, Rye Hardaway, Siegfried Hardaway, Stanly Hardaway, Takashi Hardaway, Taro Hardaway, Tarvares Hardaway, Tehran Hardaway, Therron Hardaway, Thoms Hardaway, Tico Hardaway, Timm
Hardaway, Tylan Hardaway, Wheeler Hardaway, Zalman Hardaway, Aleksander Hardaway, Alok Hardaway, Alvan Hardaway, Angelica Hardaway, Ato Hardaway, Aureliano Hardaway, Ayman Hardaway, Berkeley
Hardaway, Bracken Hardaway, Bren Hardaway, Brittany Hardaway, Caley Hardaway, Camara Hardaway, Christifer Hardaway, Clenton Hardaway, Corley Hardaway, Deante Hardaway, Denorris Hardaway, Donielle
Hardaway, Eleftherios Hardaway, Eleno Hardaway, Eliah Hardaway, Enrrique Hardaway, Frederik Hardaway, French Hardaway, Hani Hardaway, Herby Hardaway, Howie Hardaway, Joab Hardaway, Kylie Hardaway,
Leshon Hardaway, Maribel Hardaway, Marquese Hardaway, Martine Hardaway, Maurico Hardaway, Melford Hardaway, Mikie Hardaway, Myreon Hardaway, Nabeel Hardaway, Nabor Hardaway, Orlandus Hardaway, Rafi
Hardaway, Rahn Hardaway, Ramont Hardaway, Romain Hardaway, Ronold Hardaway, Rosco Hardaway, Shabazz Hardaway, Shandell Hardaway, Shaughn Hardaway, Sumner Hardaway, Tamika Hardaway, Tarry Hardaway,
Tedric Hardaway, Tellas Hardaway, Theordore Hardaway, Tolbert Hardaway, Wanda Hardaway, Winthrop Hardaway, Wyndell Hardaway, Abrahm Hardaway, Arthuro Hardaway, Attilio Hardaway, Baraka Hardaway,
Chanse Hardaway, Codie Hardaway, Conner Hardaway, Demarkus Hardaway, Demetres Hardaway, Estel Hardaway, Eva Hardaway, Greyson Hardaway, Hazen Hardaway, Ivor Hardaway, Janie Hardaway, Jarian Hardaway,
Jene Hardaway, Joseh Hardaway, Kajuan Hardaway, Kamil Hardaway, Kean Hardaway, Larson Hardaway, Laymon Hardaway, Marian Hardaway, Mosi Hardaway, Nathaneil Hardaway, Octavis Hardaway, Regginald
Hardaway, Schon Hardaway, Secundino Hardaway, Slyvester Hardaway, Socorro Hardaway, Sundeep Hardaway, Tao Hardaway, Tasha Hardaway, Tivon Hardaway, Twain Hardaway, Verl Hardaway, Yousef Hardaway,
Adem Hardaway, Akash Hardaway, Archer Hardaway, Barnett Hardaway, Binh Hardaway, Darrly Hardaway, Detrich Hardaway, Diandre Hardaway, Dinesh Hardaway, Donya Hardaway, Eirik Hardaway, Fuquan Hardaway,
Garrin Hardaway, Goerge Hardaway, Griff Hardaway, Haneef Hardaway, Jamian Hardaway, Janos Hardaway, Jervon Hardaway, Josie Hardaway, Jumar Hardaway, Keithon Hardaway, Kelii Hardaway, Konstantine
Hardaway, Latonya Hardaway, Laurel Hardaway, Liborio Hardaway, Lyon Hardaway, Markie Hardaway, Mikhael Hardaway, Montana Hardaway, Mordecai Hardaway, Nana Hardaway, Onesimo Hardaway, Reymond
Hardaway, Ronda Hardaway, Rosevelt Hardaway, Sandip Hardaway, Shade Hardaway, Stanislaus Hardaway, Trung Hardaway, Zeferino Hardaway, Abu Hardaway, Adaryll Hardaway, Adolf Hardaway, Allon Hardaway,
Almon Hardaway, Alphonza Hardaway, Andras Hardaway, Angelito Hardaway, Aurelius Hardaway, Banjamin Hardaway, Bartlett Hardaway, Caprice Hardaway, Cheskel Hardaway, Corrado Hardaway, Damieon Hardaway,
Dariel Hardaway, Darral Hardaway, Demetreus Hardaway, Fleming Hardaway, Fotios Hardaway, Graylin Hardaway, Hari Hardaway, Hilbert Hardaway, Holden Hardaway, Iram Hardaway, Jalon Hardaway, Jams
Hardaway, Jaquan Hardaway, Jerimey Hardaway, Johnthomas Hardaway, Kain Hardaway, Lonney Hardaway, Lukus Hardaway, Marquet Hardaway, Melody Hardaway, Michall Hardaway, Nicodemus Hardaway, Octavia
Hardaway, Petro Hardaway, Prashant Hardaway, Primo Hardaway, Rahmel Hardaway, Rita Hardaway, Shepard Hardaway, Siddhartha Hardaway, Talbot Hardaway, Tomy Hardaway, Tyon Hardaway, Tyrie Hardaway,
Victory Hardaway, Aaren Hardaway, Ade Hardaway, Adel Hardaway, Akida Hardaway, Anjel Hardaway, Arrick Hardaway, Becky Hardaway, Bekim Hardaway, Camillo Hardaway, Cerrone Hardaway, Chett Hardaway,
Cicero Hardaway, Collie Hardaway, Cregg Hardaway, Danel Hardaway, Demitrios Hardaway, Desiree Hardaway, Dontez Hardaway, Doris Hardaway, Dorrell Hardaway, Edouard Hardaway, Emmette Hardaway, Everton
Hardaway, Eward Hardaway, Geofrey Hardaway, Gregrey Hardaway, Greig Hardaway, Gust Hardaway, Jaycee Hardaway, Jes Hardaway, Jocob Hardaway, Judith Hardaway, Karlin Hardaway, Kaveh Hardaway, Keil
Hardaway, Kino Hardaway, Kwaku Hardaway, Lamel Hardaway, Lea Hardaway, Leodis Hardaway, Loni Hardaway, Lonie Hardaway, Miriam Hardaway, Nnamdi Hardaway, Olufemi Hardaway, Patrik Hardaway, Raun
Hardaway, Refujio Hardaway, Rossie Hardaway, Rydell Hardaway, Severin Hardaway, Shadeed Hardaway, Shannen Hardaway, Shellie Hardaway, Sheri Hardaway, Sumit Hardaway, Tamiko Hardaway, Tien Hardaway,
Tirso Hardaway, Tonie Hardaway, Trino Hardaway, Tyra Hardaway, Wright Hardaway, Zalmen Hardaway, Abdon Hardaway, Abrahan Hardaway, Adrean Hardaway, Aeron Hardaway, Ajene Hardaway, Atthew Hardaway,
Augie Hardaway, Carmel Hardaway, Cezar Hardaway, Chioke Hardaway, Chung Hardaway, Delonta Hardaway, Derrich Hardaway, Deyon Hardaway, Dicky Hardaway, Elihu Hardaway, Eustacio Hardaway, Fernado
Hardaway, Filbert Hardaway, Free Hardaway, Goran Hardaway, Habib Hardaway, Haile Hardaway, Han Hardaway, Heraclio Hardaway, Jarin Hardaway, Javaris Hardaway, Jeramine Hardaway, Joanna Hardaway,
Johnel Hardaway, Joie Hardaway, Juddson Hardaway, Kason Hardaway, Leverne Hardaway, Liston Hardaway, Marck Hardaway, Marcoantonio Hardaway, Margarita Hardaway, Messiah Hardaway, Nicholaos Hardaway,
Onofre Hardaway, Pacer Hardaway, Parke Hardaway, Sion Hardaway, Spyros Hardaway, Starr Hardaway, Storm Hardaway, Tami Hardaway, Theon Hardaway, Tramayne Hardaway, Tyronn Hardaway, Tywone Hardaway,
Undray Hardaway, Wray Hardaway, Yon Hardaway, Zeno Hardaway, Abimael Hardaway, Adell Hardaway, Adren Hardaway, Agron Hardaway, Brick Hardaway, Bridger Hardaway, Buddie Hardaway, Cosimo Hardaway,
Cristin Hardaway, Deitrich Hardaway, Demetre Hardaway, Dolphus Hardaway, Dorell Hardaway, Dylon Hardaway, Fadi Hardaway, Fredrich Hardaway, Gerron Hardaway, Jacobe Hardaway, Jarell Hardaway, Karson
Hardaway, Kegan Hardaway, Latasha Hardaway, Laureano Hardaway, Lemonte Hardaway, Lev Hardaway, Marcellino Hardaway, Marsha Hardaway, Maureen Hardaway, Ori Hardaway, Prudencio Hardaway, Rockwell
Hardaway, Selby Hardaway, Shah Hardaway, Stefanos Hardaway, Tarris Hardaway, Taryll Hardaway, Telley Hardaway, Teryl Hardaway, Thoams Hardaway, Tiko Hardaway, Tilford Hardaway, Toraino Hardaway, Tyce
Hardaway, Undrea Hardaway, Walid Hardaway, Williard Hardaway, Yasir Hardaway, Yvette Hardaway, Abiel Hardaway, Aimee Hardaway, Ainsley Hardaway, Akram Hardaway, Antino Hardaway, Bengy Hardaway,
Benjimen Hardaway, Bohdan Hardaway, Canyon Hardaway, Chandra Hardaway, Cheron Hardaway, Christhoper Hardaway, Cordney Hardaway, Covey Hardaway, Danyale Hardaway, Daquan Hardaway, Darnelle Hardaway,
Dong Hardaway, Donjuan Hardaway, Elieser Hardaway, Feliberto Hardaway, Fernie Hardaway, Gaylan Hardaway, Gerritt Hardaway, Glennis Hardaway, Goldie Hardaway, Hermes Hardaway, Hernandez Hardaway,
Herny Hardaway, Hervey Hardaway, Italo Hardaway, Jeno Hardaway, Joell Hardaway, Kaiser Hardaway, Kalem Hardaway, Kijana Hardaway, Knute Hardaway, Korby Hardaway, Leaf Hardaway, Lejuan Hardaway, Lin
Hardaway, Mahdi Hardaway, Mindy Hardaway, Mithcell Hardaway, Mylon Hardaway, Nichlas Hardaway, Nickolus Hardaway, Quanah Hardaway, Rachael Hardaway, Rafiq Hardaway, Ranier Hardaway, Rashaud Hardaway,
Rebel Hardaway, Reuel Hardaway, Rodgerick Hardaway, Ryun Hardaway, Samad Hardaway, Steffon Hardaway, Tan Hardaway, Tonio Hardaway, Trevell Hardaway, Truett Hardaway, Woodley Hardaway, Ziad Hardaway,
Alisha Hardaway, Amory Hardaway, Ashford Hardaway, Bard Hardaway, Brenan Hardaway, Burney Hardaway, Carnel Hardaway, Cevin Hardaway, Clare Hardaway, Corban Hardaway, Dashan Hardaway, Deddrick
Hardaway, Dennise Hardaway, Dimetrius Hardaway, Diogenes Hardaway, Dodd Hardaway, Dominico Hardaway, Dorion Hardaway, Eberardo Hardaway, Ethen Hardaway, Eyal Hardaway, Fedrick Hardaway, Gonsalo
Hardaway, Gorje Hardaway, Hart Hardaway, Irby Hardaway, Jarmal Hardaway, Jaworski Hardaway, Johnston Hardaway, Joshuwa Hardaway, Json Hardaway, Kenzie Hardaway, Kristie Hardaway, Labaron Hardaway,
Lan Hardaway, Leroi Hardaway, Lysander Hardaway, Marten Hardaway, Matthieu Hardaway, Meghan Hardaway, Misha Hardaway, Norvel Hardaway, Ohn Hardaway, Olivia Hardaway, Omoro Hardaway, Powell Hardaway,
Qunicy Hardaway, Shaheen Hardaway, Sirron Hardaway, Suraj Hardaway, Teran Hardaway, Theopolis Hardaway, Tion Hardaway, Toddy Hardaway, Toronto Hardaway, Tyrin Hardaway, Urban Hardaway, Vencent
Hardaway, Vida Hardaway, Virgle Hardaway, Weslee Hardaway, Wilfrido Hardaway, Ahron Hardaway, Alanzo Hardaway, Amol Hardaway, Asad Hardaway, Avel Hardaway, Bao Hardaway, Benancio Hardaway, Britten
Hardaway, Cassie Hardaway, Choya Hardaway, Christien Hardaway, Chrles Hardaway, Cochise Hardaway, Criss Hardaway, Culley Hardaway, Danyelle Hardaway, Deforest Hardaway, Deland Hardaway, Deshane
Hardaway, Dynell Hardaway, Eluterio Hardaway, Emeka Hardaway, Eyad Hardaway, Fabricio Hardaway, Graviel Hardaway, Gunner Hardaway, Harun Hardaway, Hiroshi Hardaway, Ihsan Hardaway, Irineo Hardaway,
Jacon Hardaway, Jamile Hardaway, Jawad Hardaway, Jeramia Hardaway, Joann Hardaway, Johnn Hardaway, Josph Hardaway, Kaleo Hardaway, Katie Hardaway, Kelwin Hardaway, Kwanza Hardaway, Laddie Hardaway,
Lanorris Hardaway, Larron Hardaway, Maron Hardaway, Mehdi Hardaway, Monolito Hardaway, Mont Hardaway, Munir Hardaway, Murrell Hardaway, Nachman Hardaway, Natan Hardaway, Nguyen Hardaway, Nickalus
Hardaway, Nikitas Hardaway, Orlin Hardaway, Phillippe Hardaway, Rasean Hardaway, Rino Hardaway, Robi Hardaway, Rodgers Hardaway, Rolan Hardaway, Roswell Hardaway, Sagar Hardaway, Schaun Hardaway,
Segundo Hardaway, Senaca Hardaway, Severino Hardaway, Sloane Hardaway, Spenser Hardaway, Talvin Hardaway, Teddrick Hardaway, Theran Hardaway, Tiron Hardaway, Tshombe Hardaway, Zephaniah Hardaway,
Abbas Hardaway, Adil Hardaway, Alireza Hardaway, Ancil Hardaway, Antoan Hardaway, Atanacio Hardaway, Benedicto Hardaway, Benjimin Hardaway, Blandon Hardaway, Bradey Hardaway, Brannen Hardaway,
Chalmer Hardaway, Charleton Hardaway, Chrsitopher Hardaway, Cleatus Hardaway, Corwyn Hardaway, Costas Hardaway, Cranston Hardaway, Damario Hardaway, Delmon Hardaway, Dermaine Hardaway, Duc Hardaway,
Duff Hardaway, Farhad Hardaway, Gabor Hardaway, Garreth Hardaway, Heshimu Hardaway, Ines Hardaway, Ivar Hardaway, Jahan Hardaway, Jamael Hardaway, Janes Hardaway, Jaycen Hardaway, Jett Hardaway,
Johnhenry Hardaway, Jona Hardaway, Josey Hardaway, Khristian Hardaway, Kilian Hardaway, Lander Hardaway, Laval Hardaway, Lebaron Hardaway, Leocadio Hardaway, Lonzie Hardaway, Mackey Hardaway, Marios
Hardaway, Maxim Hardaway, Naquan Hardaway, Nickalaus Hardaway, Nicolaos Hardaway, Nkosi Hardaway, Ozie Hardaway, Ponciano Hardaway, Praveen Hardaway, Ramar Hardaway, Rizwan Hardaway, Ronen Hardaway,
Saad Hardaway, Shalin Hardaway, Shalon Hardaway, Shaunte Hardaway, Steele Hardaway, Sten Hardaway, Terrin Hardaway, Tiran Hardaway, Valdis Hardaway, Virgel Hardaway, Yamil Hardaway, Adriana Hardaway,
Ajit Hardaway, Alger Hardaway, Amil Hardaway, Ananda Hardaway, Angie Hardaway, Anup Hardaway, Arjun Hardaway, Ark Hardaway, Bayard Hardaway, Bolivar Hardaway, Burnie Hardaway, Chae Hardaway, Dallan
Hardaway, Danile Hardaway, Deward Hardaway, Didier Hardaway, Dusan Hardaway, Elaine Hardaway, Franchot Hardaway, Gavriel Hardaway, Heliodoro Hardaway, Igor Hardaway, Irene Hardaway, Iven Hardaway,
Jamien Hardaway, Jaquay Hardaway, Kaylon Hardaway, Larico Hardaway, Lenord Hardaway, Lenton Hardaway, Lynden Hardaway, Mahesh Hardaway, Malo Hardaway, Mercedes Hardaway, Nicklos Hardaway, Raiford
Hardaway, Rashed Hardaway, Rayshaun Hardaway, Reubin Hardaway, Senica Hardaway, Shaan Hardaway, Shermon Hardaway, Sherrell Hardaway, Sierra Hardaway, Tamarcus Hardaway, Tavius Hardaway, Thurmon
Hardaway, Tonia Hardaway, Torino Hardaway, Urian Hardaway, Verner Hardaway, Zacarias Hardaway, Abdel Hardaway, Aniel Hardaway, Arjuna Hardaway, Ashon Hardaway, Candy Hardaway, Caroline Hardaway,
Chukwuemeka Hardaway, Cid Hardaway, Court Hardaway, Cristino Hardaway, Darshan Hardaway, Delshawn Hardaway, Demetrie Hardaway, Deondray Hardaway, Deone Hardaway, Diante Hardaway, Dolores Hardaway,
Donahue Hardaway, Eligah Hardaway, Essex Hardaway, Eutimio Hardaway, Gustavus Hardaway, Hebert Hardaway, Hyun Hardaway, Janssen Hardaway, Jaxon Hardaway, Jerre Hardaway, Joanne Hardaway, Jorden
Hardaway, Joseantonio Hardaway, Kainoa Hardaway, Kalif Hardaway, Karin Hardaway, Katina Hardaway, Keisha Hardaway, Khalfani Hardaway, Leanthony Hardaway, Michaelpaul Hardaway, Murl Hardaway, Nazareth
Hardaway, Pearl Hardaway, Purcell Hardaway, Quay Hardaway, Raylon Hardaway, Ren Hardaway, Renell Hardaway, Reshard Hardaway, Rodel Hardaway, Shaen Hardaway, Talon Hardaway, Torsten Hardaway,
Victorino Hardaway, Vonzell Hardaway, Yong Hardaway, Zaki Hardaway, Abdiel Hardaway, Adon Hardaway, Alferd Hardaway, Amjad Hardaway, Arlester Hardaway, Bassam Hardaway, Bela Hardaway, Belton
Hardaway, Branko Hardaway, Burk Hardaway, Calixto Hardaway, Cassey Hardaway, Cayle Hardaway, Cedrie Hardaway, Cuong Hardaway, Damu Hardaway, Danelle Hardaway, Darcey Hardaway, Dayna Hardaway, Domanic
Hardaway, Dougles Hardaway, Elex Hardaway, Elrico Hardaway, Estaban Hardaway, Ferlando Hardaway, Filipe Hardaway, Geovanni Hardaway, Glenford Hardaway, Gumaro Hardaway, Gustabo Hardaway, Haley
Hardaway, Herberto Hardaway, Jabez Hardaway, Jaman Hardaway, Joshau Hardaway, Jullian Hardaway, Kairi Hardaway, Karnell Hardaway, Kin Hardaway, Kinsey Hardaway, Kipper Hardaway, Laquincy Hardaway,
Laterrance Hardaway, Laurice Hardaway, Leshaun Hardaway, Marcia Hardaway, Marcio Hardaway, Melquiades Hardaway, Miklos Hardaway, Ming Hardaway, Mylo Hardaway, Omid Hardaway, Oskar Hardaway, Paco
Hardaway, Rafel Hardaway, Ralf Hardaway, Ravindra Hardaway, Rivers Hardaway, Rody Hardaway, Saint Hardaway, Santee Hardaway, Shai Hardaway, Shelvin Hardaway, Shyam Hardaway, Silvino Hardaway, Squire
Hardaway, Star Hardaway, Steen Hardaway, Stein Hardaway, Tae Hardaway, Taryn Hardaway, Tavare Hardaway, Thedore Hardaway, Yehudah Hardaway, Zacchaeus Hardaway, Zachory Hardaway, Zan Hardaway, Adewale
Hardaway, Aleksandr Hardaway, Alf Hardaway, Alvester Hardaway, Amiri Hardaway, Araceli Hardaway, Arcenio Hardaway, Arvell Hardaway, Arvis Hardaway, Atlas Hardaway, Binyomin Hardaway, Chasen Hardaway,
Cheyne Hardaway, Coray Hardaway, Dak Hardaway, Davone Hardaway, Egbert Hardaway, Ferlin Hardaway, Gilles Hardaway, Glendell Hardaway, Granger Hardaway, Jafar Hardaway, Jalil Hardaway, Jamain
Hardaway, Jarrette Hardaway, Jeret Hardaway, Jocelyn Hardaway, Joenathan Hardaway, Jurgen Hardaway, Keene Hardaway, Keller Hardaway, Kenneith Hardaway, Kinney Hardaway, Labarron Hardaway, Levie
Hardaway, Linford Hardaway, Marcas Hardaway, Mickle Hardaway, Morrison Hardaway, Oba Hardaway, Qasim Hardaway, Rahsan Hardaway, Rajah Hardaway, Raphel Hardaway, Rehan Hardaway, Rodregus Hardaway,
Ronan Hardaway, Salman Hardaway, Seddrick Hardaway, Shauna Hardaway, Simpson Hardaway, Sunshine Hardaway, Terrall Hardaway, Tiawan Hardaway, Waylen Hardaway, Weslie Hardaway, Abdur Hardaway, Alric
Hardaway, Arien Hardaway, Arren Hardaway, Bari Hardaway, Briar Hardaway, Burdette Hardaway, Carver Hardaway, Charvis Hardaway, Contrell Hardaway, Costa Hardaway, Dara Hardaway, Darly Hardaway, Darwyn
Hardaway, Davied Hardaway, Dayon Hardaway, Delmont Hardaway, Dena Hardaway, Devlon Hardaway, Dimitry Hardaway, Duwane Hardaway, Esteven Hardaway, Fergus Hardaway, Harles Hardaway, Hutch Hardaway, Ion
Hardaway, Isidore Hardaway, Jamus Hardaway, Jerett Hardaway, Jerrit Hardaway, Jontue Hardaway, Lemarcus Hardaway, Lequan Hardaway, Leslee Hardaway, Linell Hardaway, Lugene Hardaway, Makoto Hardaway,
Malek Hardaway, Marquell Hardaway, Osei Hardaway, Peggy Hardaway, Ralphie Hardaway, Ramona Hardaway, Rickard Hardaway, Ronnald Hardaway, Sabian Hardaway, Saleh Hardaway, Salih Hardaway, Samule
Hardaway, Sarkis Hardaway, Sharrieff Hardaway, Snehal Hardaway, Sony Hardaway, Timotheus Hardaway, Tood Hardaway, Tyreece Hardaway, Wei Hardaway, Zackariah Hardaway, Aleem Hardaway, Antwione
Hardaway, Ayodele Hardaway, Barkley Hardaway, Bartt Hardaway, Bethany Hardaway, Bing Hardaway, Braheem Hardaway, Brek Hardaway, Caron Hardaway, Cathy Hardaway, Cecilia Hardaway, Cristina Hardaway,
Dainel Hardaway, Dal Hardaway, Darrek Hardaway, Darrelle Hardaway, Dashun Hardaway, Delrick Hardaway, Demitrus Hardaway, Dishon Hardaway, Donnald Hardaway, Ehab Hardaway, Erico Hardaway, Erskin
Hardaway, Faraji Hardaway, Gillis Hardaway, Hamed Hardaway, Hermann Hardaway, Hondo Hardaway, Jamis Hardaway, Jeric Hardaway, Jermyn Hardaway, Jessup Hardaway, Jonatan Hardaway, Joshaua Hardaway,
Jotham Hardaway, Khaalis Hardaway, Kion Hardaway, Lam Hardaway, Larence Hardaway, Lavonne Hardaway, Marti Hardaway, Mclean Hardaway, Mercury Hardaway, Miki Hardaway, Norvin Hardaway, Onnie Hardaway,
Paulmichael Hardaway, Pavan Hardaway, Paz Hardaway, Phuong Hardaway, Race Hardaway, Ralston Hardaway, Ramadan Hardaway, Rashee Hardaway, Roan Hardaway, Sally Hardaway, Shanga Hardaway, Slater
Hardaway, Tarvaris Hardaway, Tevin Hardaway, Travus Hardaway, Viviano Hardaway, Zerick Hardaway, Abdula Hardaway, Achilles Hardaway, Adarryll Hardaway, Ananias Hardaway, Antinio Hardaway, Antrell
Hardaway, Anzio Hardaway, Aristidis Hardaway, Ashante Hardaway, Atom Hardaway, Audley Hardaway, Azim Hardaway, Bandy Hardaway, Bary Hardaway, Britain Hardaway, Bryne Hardaway, Calen Hardaway,
Ceddrick Hardaway, Chalmers Hardaway, Chanc Hardaway, Chap Hardaway, Charly Hardaway, Charron Hardaway, Cornellius Hardaway, Darth Hardaway, Dayan Hardaway, Deran Hardaway, Dewand Hardaway, Fran
Hardaway, Girolamo Hardaway, Harvie Hardaway, Haskel Hardaway, Hoy Hardaway, Islam Hardaway, Ivon Hardaway, Jafari Hardaway, Jamario Hardaway, Jehu Hardaway, Josemanuel Hardaway, Jung Hardaway, Kabir
Hardaway, Kael Hardaway, Khan Hardaway, Khayree Hardaway, Kimber Hardaway, Koran Hardaway, Lamonta Hardaway, Lavalle Hardaway, Lavel Hardaway, Lenardo Hardaway, Lydia Hardaway, Mandrill Hardaway,
Mannie Hardaway, Marguis Hardaway, Marisa Hardaway, Marissa Hardaway, Marshaun Hardaway, Medardo Hardaway, Naomi Hardaway, Naser Hardaway, Natanael Hardaway, Oji Hardaway, Orvil Hardaway, Pasqual
Hardaway, Previn Hardaway, Reo Hardaway, Ruddy Hardaway, Shahram Hardaway, Sharad Hardaway, Shaunta Hardaway, Shuron Hardaway, Teo Hardaway, Thalmus Hardaway, Theodoric Hardaway, Tibor Hardaway, Tres
Hardaway, Ulrich Hardaway, Winter Hardaway, Yasser Hardaway, Ygnacio Hardaway, Abasi Hardaway, Alwin Hardaway, Alwyn Hardaway, Amalio Hardaway, Arty Hardaway, Baker Hardaway, Bishara Hardaway, Briane
Hardaway, Catalino Hardaway, Darrall Hardaway, Dartanyon Hardaway, Deano Hardaway, Demico Hardaway, Deontae Hardaway, Deveron Hardaway, Devonn Hardaway, Donnovan Hardaway, Durant Hardaway, Duy
Hardaway, Evander Hardaway, Ever Hardaway, Ewing Hardaway, Fouad Hardaway, Froylan Hardaway, Gabriela Hardaway, Garnell Hardaway, Gerlad Hardaway, Ginger Hardaway, Glendale Hardaway, Grafton
Hardaway, Haig Hardaway, Hale Hardaway, Iver Hardaway, Ivin Hardaway, Jarel Hardaway, Javen Hardaway, Jeran Hardaway, Jereld Hardaway, Jermichael Hardaway, Jimel Hardaway, Karreem Hardaway, Keiron
Hardaway, Kester Hardaway, Laronn Hardaway, Lavor Hardaway, Lehi Hardaway, Lemon Hardaway, Lourdes Hardaway, Marqui Hardaway, Marsh Hardaway, Marx Hardaway, Montego Hardaway, Murice Hardaway, Mychael
Hardaway, Nashawn Hardaway, Nichalas Hardaway, Nickalas Hardaway, Nuri Hardaway, Obinna Hardaway, Onaje Hardaway, Patrich Hardaway, Pavel Hardaway, Priscilla Hardaway, Rakeem Hardaway, Rande
Hardaway, Rashann Hardaway, Rayan Hardaway, Raynold Hardaway, Ricko Hardaway, Riki Hardaway, Rooney Hardaway, Saulo Hardaway, Schad Hardaway, Seiji Hardaway, Shahn Hardaway, Shantel Hardaway, Shanton
Hardaway, Sisto Hardaway, Taiwo Hardaway, Travion Hardaway, Tristen Hardaway, Truong Hardaway, Tywann Hardaway, Tywayne Hardaway, Uhuru Hardaway, Welby Hardaway, Wesam Hardaway, Alexius Hardaway,
Ammar Hardaway, Andrews Hardaway, Atticus Hardaway, Audy Hardaway, Belinda Hardaway, Bennet Hardaway, Bomani Hardaway, Bulmaro Hardaway, Charlies Hardaway, Chato Hardaway, Chelsea Hardaway, Chipper
Hardaway, Clearence Hardaway, Cleotis Hardaway, Cleven Hardaway, Corydon Hardaway, Dadrian Hardaway, Damiano Hardaway, Darrold Hardaway, Davell Hardaway, Derone Hardaway, Derral Hardaway, Derwood
Hardaway, Django Hardaway, Doc Hardaway, Dontell Hardaway, Doren Hardaway, Elza Hardaway, Esmeralda Hardaway, Estil Hardaway, Finn Hardaway, Fredrik Hardaway, Garrie Hardaway, Greogory Hardaway,
Harlow Hardaway, Iris Hardaway, Jascha Hardaway, Jediah Hardaway, Jef Hardaway, Jefferie Hardaway, Jerimah Hardaway, Jerrie Hardaway, Joal Hardaway, Johsua Hardaway, Joon Hardaway, Joson Hardaway,
Kamar Hardaway, Kellen Hardaway, Kelvis Hardaway, Ketrick Hardaway, Kimberley Hardaway, Kingston Hardaway, Meldon Hardaway, Mendell Hardaway, Michaeljohn Hardaway, Moss Hardaway, Nuno Hardaway,
Onofrio Hardaway, Phineas Hardaway, Ramell Hardaway, Raydell Hardaway, Regie Hardaway, Ronnel Hardaway, Sadat Hardaway, Sajid Hardaway, Shanard Hardaway, Shaul Hardaway, Shondale Hardaway, Shyrone
Hardaway, Simuel Hardaway, Sjon Hardaway, Stephenson Hardaway, Tadeusz Hardaway, Takeshi Hardaway, Tilman Hardaway, Tomasz Hardaway, Travares Hardaway, Truitt Hardaway, Ulrick Hardaway, Vandy
Hardaway, Yvon Hardaway, Zebulan Hardaway, Adolpho Hardaway, Alek Hardaway, Amery Hardaway, Andrej Hardaway, Aniello Hardaway, Anurag Hardaway, Audra Hardaway, Boruch Hardaway, Burr Hardaway, Cable
Hardaway, Cheney Hardaway, Cimarron Hardaway, Clell Hardaway, Clinten Hardaway, Daimen Hardaway, Dameyon Hardaway, Dason Hardaway, Deanglo Hardaway, Delaine Hardaway, Demeco Hardaway, Denim Hardaway,
Derryck Hardaway, Domnick Hardaway, Doy Hardaway, Dwyne Hardaway, Edwards Hardaway, Elisa Hardaway, Esmond Hardaway, Gawain Hardaway, Green Hardaway, Hallie Hardaway, Hameed Hardaway, Heinrich
Hardaway, Jeri Hardaway, Kaj Hardaway, Kalan Hardaway, Kash Hardaway, Kente Hardaway, Koji Hardaway, Kunte Hardaway, Laurin Hardaway, Livingston Hardaway, Meliton Hardaway, Michaeal Hardaway, Myran
Hardaway, Nephtali Hardaway, Nicasio Hardaway, Nyles Hardaway, Obediah Hardaway, Pride Hardaway, Primus Hardaway, Rae Hardaway, Rahmaan Hardaway, Rhoderick Hardaway, Rip Hardaway, River Hardaway,
Roberts Hardaway, Rodricus Hardaway, Rommie Hardaway, Romney Hardaway, Roverto Hardaway, Samar Hardaway, Sergei Hardaway, Shadrack Hardaway, Shavon Hardaway, Stevin Hardaway, Tarone Hardaway, Tashawn
Hardaway, Tora Hardaway, Tri Hardaway, Vinny Hardaway, Yesenia Hardaway, Yvan Hardaway, Zubin Hardaway, Absalom Hardaway, Aldrick Hardaway, Alif Hardaway, Amad Hardaway, Anacleto Hardaway, Andru
Hardaway, Argelio Hardaway, Arrington Hardaway, Ashely Hardaway, Auston Hardaway, Baird Hardaway, Beryl Hardaway, Celester Hardaway, Chadron Hardaway, Chante Hardaway, Dack Hardaway, Darain Hardaway,
Darvell Hardaway, Daune Hardaway, Daylan Hardaway, Dedan Hardaway, Deitrick Hardaway, Dejaun Hardaway, Denzel Hardaway, Deverick Hardaway, Eris Hardaway, Flor Hardaway, Frankey Hardaway, Gaberial
Hardaway, Gianluca Hardaway, Hesham Hardaway, Iban Hardaway, Ilya Hardaway, Isham Hardaway, Issiah Hardaway, Jacey Hardaway, Jahmel Hardaway, Jaimey Hardaway, Jamane Hardaway, Jammey Hardaway, Jareb
Hardaway, Jarek Hardaway, Jemery Hardaway, Jeremaih Hardaway, Jernard Hardaway, Jerramy Hardaway, Jese Hardaway, Johnaton Hardaway, Jontae Hardaway, Kayode Hardaway, Kaz Hardaway, Kedar Hardaway,
Kell Hardaway, Kewan Hardaway, Kiyoshi Hardaway, Ladarryl Hardaway, Lamart Hardaway, Lataurus Hardaway, Lavarr Hardaway, Lavert Hardaway, Lawan Hardaway, Loney Hardaway, Lonnel Hardaway, Luan
Hardaway, Macio Hardaway, Mansa Hardaway, Miachel Hardaway, Neftaly Hardaway, Nichlos Hardaway, Nicklous Hardaway, Nik Hardaway, Orlander Hardaway, Quadir Hardaway, Raeford Hardaway, Rajohn Hardaway,
Ranson Hardaway, Rasul Hardaway, Renald Hardaway, Reynald Hardaway, Roben Hardaway, Rutherford Hardaway, Salathiel Hardaway, Savoy Hardaway, Sharief Hardaway, Sherrick Hardaway, Siddharth Hardaway,
Staci Hardaway, Steed Hardaway, Tauheed Hardaway, Thaddus Hardaway, Toland Hardaway, Toran Hardaway, Torriano Hardaway, Turon Hardaway, Vashaun Hardaway, Zebediah Hardaway, Aasim Hardaway, Abron
Hardaway, Ahman Hardaway, Aiden Hardaway, Alexsander Hardaway, Alias Hardaway, Aly Hardaway, Ameet Hardaway, Anothny Hardaway, Ansley Hardaway, Ballard Hardaway, Bedford Hardaway, Beecher Hardaway,
Benaiah Hardaway, Clent Hardaway, Colvin Hardaway, Dakari Hardaway, Darric Hardaway, Daved Hardaway, Demichael Hardaway, Deray Hardaway, Deren Hardaway, Dionisios Hardaway, Dionta Hardaway, Dorothy
Hardaway, Dorrian Hardaway, Dorris Hardaway, Dyke Hardaway, Elmon Hardaway, Erle Hardaway, Eunice Hardaway, Eustace Hardaway, Evon Hardaway, Fay Hardaway, Federick Hardaway, Feras Hardaway, Gabrielle
Hardaway, Gaven Hardaway, Gay Hardaway, Gjon Hardaway, Grier Hardaway, Guillaume Hardaway, Hansen Hardaway, Harrel Hardaway, Hazel Hardaway, Heathe Hardaway, Herschell Hardaway, Ingram Hardaway,
Isauro Hardaway, Jaimy Hardaway, Jaymar Hardaway, Jerin Hardaway, Jermall Hardaway, Jevin Hardaway, Jillian Hardaway, Johnmark Hardaway, Jovaughn Hardaway, Kamuela Hardaway, Kendon Hardaway, Khoa
Hardaway, Lamberto Hardaway, Latham Hardaway, Luster Hardaway, Malon Hardaway, Micholas Hardaway, Mitul Hardaway, Montrice Hardaway, Moroni Hardaway, Mylan Hardaway, Nason Hardaway, Nic Hardaway,
Nowell Hardaway, Oryan Hardaway, Owens Hardaway, Phill Hardaway, Raed Hardaway, Raji Hardaway, Ramzy Hardaway, Rebekah Hardaway, Reeve Hardaway, Regenald Hardaway, Roddie Hardaway, Roshun Hardaway,
Salvator Hardaway, Seanpaul Hardaway, Shanne Hardaway, Shawntay Hardaway, Sheldrick Hardaway, Sheppard Hardaway, Snapper Hardaway, Tarick Hardaway, Terelle Hardaway, Tery Hardaway, Thelonious
Hardaway, Trampis Hardaway, Tushar Hardaway, Vernice Hardaway, Whalen Hardaway, Willim Hardaway, Alban Hardaway, Alexy Hardaway, Almalik Hardaway, Amedeo Hardaway, Antoney Hardaway, Asia Hardaway,
Aurthur Hardaway, Avelardo Hardaway, Barth Hardaway, Cartrell Hardaway, Chaderick Hardaway, Champ Hardaway, Chano Hardaway, Charity Hardaway, Christhopher Hardaway, Christino Hardaway, Clavin
Hardaway, Cuyler Hardaway, Dalbert Hardaway, Darcel Hardaway, Darcell Hardaway, Darrie Hardaway, Denney Hardaway, Denson Hardaway, Dezmond Hardaway, Donyea Hardaway, Dorien Hardaway, Emidio Hardaway,
Etan Hardaway, Evagelos Hardaway, Everado Hardaway, Farren Hardaway, Fate Hardaway, Filomeno Hardaway, Fonzie Hardaway, Franko Hardaway, Garrod Hardaway, Graciano Hardaway, Gustin Hardaway, Isrrael
Hardaway, Jac Hardaway, Jakie Hardaway, Jamas Hardaway, Jamiyl Hardaway, Jaylon Hardaway, Jeanmarc Hardaway, Jebadiah Hardaway, Jeffey Hardaway, Jiles Hardaway, Johnnell Hardaway, Josip Hardaway,
Juanmanuel Hardaway, Kemo Hardaway, Kemuel Hardaway, Kindrick Hardaway, Kirtis Hardaway, Kyley Hardaway, Lacorey Hardaway, Lansing Hardaway, Lawyer Hardaway, Lenzie Hardaway, Lessie Hardaway, Lindley
Hardaway, Lynette Hardaway, Lyonel Hardaway, Magnus Hardaway, Marki Hardaway, Marrell Hardaway, Mccoy Hardaway, Mcdonald Hardaway, Meade Hardaway, Mickell Hardaway, Mikah Hardaway, Miranda Hardaway,
Mitchael Hardaway, Mitcheal Hardaway, Montee Hardaway, Morio Hardaway, Murat Hardaway, Nashon Hardaway, Nova Hardaway, Olusegun Hardaway, Ondra Hardaway, Orian Hardaway, Orren Hardaway, Reggy
Hardaway, Rodrico Hardaway, Ronzell Hardaway, Runako Hardaway, Ryszard Hardaway, Shady Hardaway, Shango Hardaway, Shawan Hardaway, Terez Hardaway, Thang Hardaway, Tomothy Hardaway, Toron Hardaway,
Torrin Hardaway, Toya Hardaway, Trayon Hardaway, Trellis Hardaway, Tremel Hardaway, Tryon Hardaway, Tyrek Hardaway, Wessley Hardaway, Yan Hardaway, Zed Hardaway, Ako Hardaway, Aldrich Hardaway,
Alesandro Hardaway, Alrick Hardaway, Ayo Hardaway, Bobbyjoe Hardaway, Broadus Hardaway, Brockton Hardaway, Callie Hardaway, Ched Hardaway, Conn Hardaway, Cristo Hardaway, Dallen Hardaway, Dannel
Hardaway, Dasan Hardaway, Datron Hardaway, Dawaun Hardaway, Delance Hardaway, Delawrence Hardaway, Demarrio Hardaway, Denell Hardaway, Derak Hardaway, Derrie Hardaway, Deundra Hardaway, Dorin
Hardaway, Dyon Hardaway, Elizardo Hardaway, Filimon Hardaway, Fitzroy Hardaway, Gari Hardaway, Glover Hardaway, Gretchen Hardaway, Hartley Hardaway, Holt Hardaway, Jabe Hardaway, Jaclyn Hardaway,
Jadrien Hardaway, Jamah Hardaway, Jarman Hardaway, Jarreau Hardaway, Jearld Hardaway, Jerimi Hardaway, Jerris Hardaway, Joeph Hardaway, Jorel Hardaway, Joshus Hardaway, Jovany Hardaway, Kaven
Hardaway, Kawaski Hardaway, Kiron Hardaway, Korrey Hardaway, Kort Hardaway, Lavale Hardaway, Lenorris Hardaway, Lizandro Hardaway, Lynard Hardaway, Manson Hardaway, Marky Hardaway, Marlando Hardaway,
Maxime Hardaway, Mcgarrett Hardaway, Merwin Hardaway, Miko Hardaway, Niklas Hardaway, Nkrumah Hardaway, Orman Hardaway, Osmar Hardaway, Paresh Hardaway, Ponce Hardaway, Prem Hardaway, Rachelle
Hardaway, Rajendra Hardaway, Rashone Hardaway, Rawn Hardaway, Raynor Hardaway, Rechard Hardaway, Rogan Hardaway, Rollo Hardaway, Romando Hardaway, Royd Hardaway, Salah Hardaway, Scoey Hardaway,
Selvin Hardaway, Swain Hardaway, Tarun Hardaway, Tawn Hardaway, Teal Hardaway, Telesforo Hardaway, Thaddeaus Hardaway, Timoth Hardaway, Toren Hardaway, Vicki Hardaway, Wael Hardaway, Whitfield
Hardaway, Winifred Hardaway, Wyeth Hardaway, Abdu Hardaway, Abdulla Hardaway, Akio Hardaway, Aleck Hardaway, Antwian Hardaway, Aquiles Hardaway, Armad Hardaway, Averill Hardaway, Bijan Hardaway,
Billyjo Hardaway, Bladimir Hardaway, Bray Hardaway, Brigido Hardaway, Candice Hardaway, Changa Hardaway, Chanon Hardaway, Charlotte Hardaway, Colie Hardaway, Colonel Hardaway, Crespin Hardaway,
Dabney Hardaway, Dace Hardaway, Damaine Hardaway, Danna Hardaway, Danya Hardaway, Darelle Hardaway, Darrnell Hardaway, Deaundre Hardaway, Dijon Hardaway, Dina Hardaway, Dontel Hardaway, Dracy
Hardaway, Duayne Hardaway, Easton Hardaway, Eban Hardaway, Elric Hardaway, Ervey Hardaway, Festus Hardaway, Filip Hardaway, Furnell Hardaway, Gardiner Hardaway, Gleen Hardaway, Herve Hardaway, Hogan
Hardaway, Hossein Hardaway, Ikaika Hardaway, Imad Hardaway, Jacek Hardaway, Jamaica Hardaway, Jamichael Hardaway, Jammal Hardaway, Jarmon Hardaway, Jerren Hardaway, Joshu Hardaway, Juanjose Hardaway,
Kanoa Hardaway, Kia Hardaway, Kiernan Hardaway, Kirtus Hardaway, Lacedric Hardaway, Legrand Hardaway, Lekendrick Hardaway, Lemoyne Hardaway, Lige Hardaway, Loranzo Hardaway, Loyce Hardaway, Masai
Hardaway, Master Hardaway, Michah Hardaway, Min Hardaway, Mir Hardaway, Mohan Hardaway, Montrail Hardaway, Montray Hardaway, Nacoma Hardaway, Nakoa Hardaway, Navid Hardaway, Nole Hardaway, Opie
Hardaway, Orpheus Hardaway, Payam Hardaway, Pearce Hardaway, Ramal Hardaway, Recco Hardaway, Renn Hardaway, Rigel Hardaway, Roe Hardaway, Rolondo Hardaway, Romaldo Hardaway, Severn Hardaway, Shakim
Hardaway, Shulem Hardaway, Silviano Hardaway, Strider Hardaway, Tayari Hardaway, Tiburcio Hardaway, Tilmon Hardaway, Tomar Hardaway, Tran Hardaway, Travaris Hardaway, Trayvon Hardaway, Trev Hardaway,
Trevar Hardaway, Tyreek Hardaway, Windle Hardaway, Yaw Hardaway, Yechezkel Hardaway, Zeth Hardaway, Zollie Hardaway, Acey Hardaway, Adisa Hardaway, Arlanda Hardaway, Artavius Hardaway, Avin Hardaway,
Beauford Hardaway, Bernadette Hardaway, Birch Hardaway, Brittan Hardaway, Brown Hardaway, Cabe Hardaway, Calton Hardaway, Caribe Hardaway, Carrington Hardaway, Chadwich Hardaway, Chrstopher Hardaway,
Chun Hardaway, Crescencio Hardaway, Damyon Hardaway, Dantrell Hardaway, Dariusz Hardaway, Dartanion Hardaway, Delio Hardaway, Demarius Hardaway, Deniz Hardaway, Dennard Hardaway, Domingos Hardaway,
Dominie Hardaway, Doyal Hardaway, Dwanye Hardaway, Earon Hardaway, Efstathios Hardaway, Elkin Hardaway, Ellen Hardaway, Fabain Hardaway, Frans Hardaway, Gopal Hardaway, Haleem Hardaway, Iverson
Hardaway, Japeth Hardaway, Johanna Hardaway, Jorgen Hardaway, Jourdan Hardaway, Kassim Hardaway, Keidrick Hardaway, Keif Hardaway, Knox Hardaway, Lacharles Hardaway, Ladarius Hardaway, Ladarrell
Hardaway, Leonce Hardaway, Lipa Hardaway, Lorren Hardaway, Luqman Hardaway, Mcihael Hardaway, Mercer Hardaway, Micharl Hardaway, Mikle Hardaway, Montre Hardaway, Morad Hardaway, Nichael Hardaway, Phi
Hardaway, Quantez Hardaway, Quency Hardaway, Quinlan Hardaway, Quintrell Hardaway, Redell Hardaway, Regnald Hardaway, Renauld Hardaway, Ronelle Hardaway, Rudell Hardaway, Ruffin Hardaway, Salil
Hardaway, Seanmichael Hardaway, Shaheem Hardaway, Sharieff Hardaway, Shawndel Hardaway, Stamatis Hardaway, Tadashi Hardaway, Tiras Hardaway, Toan Hardaway, Toben Hardaway, Tyrik Hardaway, Tyshaun
Hardaway, Ulisses Hardaway, Visente Hardaway, Wadell Hardaway, Wilfrid Hardaway, Won Hardaway, Aashish Hardaway, Adama Hardaway, Aja Hardaway, Alcindor Hardaway, Alejandra Hardaway, Alyn Hardaway,
Ami Hardaway, Andrel Hardaway, Antonine Hardaway, Artemas Hardaway, Artimus Hardaway, Ary Hardaway, Atiim Hardaway, Avinash Hardaway, Brittain Hardaway, Callan Hardaway, Cari Hardaway, Claire
Hardaway, Conard Hardaway, Danan Hardaway, Darol Hardaway, Day Hardaway, Delma Hardaway, Demetricus Hardaway, Dewone Hardaway, Dougals Hardaway, Elester Hardaway, Espiridion Hardaway, Estanislado
Hardaway, Esther Hardaway, Fabien Hardaway, Famous Hardaway, Fredderick Hardaway, Friedrich Hardaway, Garrette Hardaway, Greogry Hardaway, Gwendolyn Hardaway, Hardin Hardaway, Hendrik Hardaway,
Hendrix Hardaway, Jabulani Hardaway, Jacin Hardaway, Jacquelyn Hardaway, Jantzen Hardaway, Jarren Hardaway, Jasyn Hardaway, Jayde Hardaway, Jeris Hardaway, Jery Hardaway, Jolyon Hardaway, Karem
Hardaway, Key Hardaway, Kibwe Hardaway, Kyon Hardaway, Lamin Hardaway, Lasaro Hardaway, Ledon Hardaway, Malcomb Hardaway, Mando Hardaway, Marcelus Hardaway, Margus Hardaway, Marzell Hardaway, Mearl
Hardaway, Miron Hardaway, Mitesh Hardaway, Montay Hardaway, Montrey Hardaway, Morey Hardaway, Morrie Hardaway, Mozell Hardaway, Narada Hardaway, Naveed Hardaway, Nenad Hardaway, Nicolaas Hardaway,
Nilton Hardaway, Ojay Hardaway, Olga Hardaway, Oman Hardaway, Pancho Hardaway, Pepper Hardaway, Poul Hardaway, Rajon Hardaway, Rawley Hardaway, Rayon Hardaway, Read Hardaway, Rees Hardaway, Reeves
Hardaway, Romond Hardaway, Rondrick Hardaway, Sancho Hardaway, Santi Hardaway, Selso Hardaway, Sevan Hardaway, Shondel Hardaway, Tavarius Hardaway, Thelbert Hardaway, Theodoro Hardaway, Theus
Hardaway, Thien Hardaway, Timoty Hardaway, Timur Hardaway, Travius Hardaway, Trisha Hardaway, Tyquan Hardaway, Usbaldo Hardaway, Varnell Hardaway, Videl Hardaway, Willi Hardaway, Zacariah Hardaway,
Zain Hardaway, Aamir Hardaway, Abdallah Hardaway, Almando Hardaway, Almond Hardaway, Alpheus Hardaway, Alvarez Hardaway, Alverto Hardaway, Andrick Hardaway, Anthonie Hardaway, Anthoy Hardaway,
Antiona Hardaway, Antwand Hardaway, Arrie Hardaway, Artur Hardaway, Baldwin Hardaway, Bardo Hardaway, Carols Hardaway, Chau Hardaway, Clayborne Hardaway, Conell Hardaway, Confesor Hardaway, Cottrell
Hardaway, Craven Hardaway, Curits Hardaway, Damiel Hardaway, Deandrae Hardaway, Derrius Hardaway, Dirrick Hardaway, Dora Hardaway, Dorain Hardaway, Dushaun Hardaway, Eliodoro Hardaway, Elvie
Hardaway, Faraz Hardaway, Fernanda Hardaway, Ferron Hardaway, Frankin Hardaway, Fraser Hardaway, Giuliano Hardaway, Godwin Hardaway, Hakan Hardaway, Hershal Hardaway, Jacobie Hardaway, Jada Hardaway,
Jaimeson Hardaway, Jamill Hardaway, Jarrard Hardaway, Jeromi Hardaway, Jerral Hardaway, Jilberto Hardaway, Jonel Hardaway, Joseangel Hardaway, Josedejesus Hardaway, Joshva Hardaway, Jovani Hardaway,
Juana Hardaway, Juvencio Hardaway, Karol Hardaway, Keanon Hardaway, Keli Hardaway, Kelven Hardaway, Kemal Hardaway, Khanh Hardaway, Kiah Hardaway, Kimmy Hardaway, Kitt Hardaway, Laith Hardaway,
Latravis Hardaway, Lebarron Hardaway, Lenon Hardaway, Leondre Hardaway, Lue Hardaway, Luka Hardaway, Lyndal Hardaway, Macedonio Hardaway, Mahdee Hardaway, Marcellas Hardaway, Mardell Hardaway, Monnie
Hardaway, Naron Hardaway, Nehal Hardaway, Nicholai Hardaway, Nicol Hardaway, Nimesh Hardaway, Orlan Hardaway, Pal Hardaway, Paras Hardaway, Pasha Hardaway, Pio Hardaway, Piotr Hardaway, Rahmon
Hardaway, Reilly Hardaway, Rommell Hardaway, Romondo Hardaway, Ronie Hardaway, Rontae Hardaway, Roth Hardaway, Ruven Hardaway, Senecca Hardaway, Sequoia Hardaway, Shadwick Hardaway, Shaquan Hardaway,
Shaundell Hardaway, Shaya Hardaway, Shloma Hardaway, Shonnon Hardaway, Shunta Hardaway, Silver Hardaway, Stanislaw Hardaway, Sumeet Hardaway, Sylvain Hardaway, Tajiddin Hardaway, Tejas Hardaway,
Tereso Hardaway, Tilton Hardaway, Tjay Hardaway, Tramon Hardaway, Troi Hardaway, Troye Hardaway, Tyrelle Hardaway, Usher Hardaway, Valeriano Hardaway, Verlyn Hardaway, Vickie Hardaway, Wallis
Hardaway, Yahya Hardaway, Yair Hardaway, Zacharie Hardaway, Zakery Hardaway, Zoe Hardaway, Abdual Hardaway, Adeyemi Hardaway, Alisa Hardaway, Alois Hardaway, Alondo Hardaway, Alprentice Hardaway,
Altariq Hardaway, Amaro Hardaway, Amen Hardaway, Andria Hardaway, Anthoni Hardaway, Apurva Hardaway, Arrow Hardaway, Aryn Hardaway, Atilano Hardaway, Auburn Hardaway, Auturo Hardaway, Aven Hardaway,
Averil Hardaway, Bernerd Hardaway, Bethel Hardaway, Biju Hardaway, Bogdan Hardaway, Boston Hardaway, Bree Hardaway, Camaron Hardaway, Candace Hardaway, Cara Hardaway, Cato Hardaway, Ceferino
Hardaway, Chantry Hardaway, Chappell Hardaway, Charels Hardaway, Ciaran Hardaway, Codi Hardaway, Colan Hardaway, Crayton Hardaway, Dakar Hardaway, Dallis Hardaway, Danen Hardaway, Daoud Hardaway,
Davien Hardaway, Deny Hardaway, Derius Hardaway, Derrico Hardaway, Devere Hardaway, Dmarcus Hardaway, Dwright Hardaway, Dyami Hardaway, Eileen Hardaway, Eitan Hardaway, Elgie Hardaway, Elimelech
Hardaway, Ellie Hardaway, Ellwood Hardaway, Emmanual Hardaway, Eryk Hardaway, Esdras Hardaway, Estanislao Hardaway, Euell Hardaway, Everard Hardaway, Hagen Hardaway, Happy Hardaway, Harald Hardaway,
Harrington Hardaway, Hays Hardaway, Henrik Hardaway, Hillery Hardaway, Honorio Hardaway, Hristos Hardaway, Ilario Hardaway, Indalecio Hardaway, Izaac Hardaway, Jahmar Hardaway, Jarome Hardaway,
Jarone Hardaway, Javaughn Hardaway, Jeanne Hardaway, Jeannie Hardaway, Jearl Hardaway, Jenner Hardaway, Jennie Hardaway, Jerit Hardaway, Jet Hardaway, Jonath Hardaway, Jumal Hardaway, Kahil Hardaway,
Kealii Hardaway, Keeley Hardaway, Kehinde Hardaway, Kene Hardaway, Kermitt Hardaway, Keshon Hardaway, Kore Hardaway, Lapatrick Hardaway, Larmar Hardaway, Laroyce Hardaway, Larren Hardaway, Latisha
Hardaway, Latonio Hardaway, Lewayne Hardaway, Malachy Hardaway, Manford Hardaway, Marcanthony Hardaway, Marks Hardaway, Marley Hardaway, Marlone Hardaway, Marquies Hardaway, Mihir Hardaway, Minas
Hardaway, Montae Hardaway, Mortimer Hardaway, Mykel Hardaway, Nadia Hardaway, Nadim Hardaway, Naftoli Hardaway, Nakai Hardaway, Nashid Hardaway, Nello Hardaway, Nichola Hardaway, Nilo Hardaway,
Nthony Hardaway, Orenthial Hardaway, Palani Hardaway, Pericles Hardaway, Quentine Hardaway, Rafiel Hardaway, Rainey Hardaway, Remberto Hardaway, Renan Hardaway, Riyad Hardaway, Robey Hardaway,
Rochell Hardaway, Romaro Hardaway, Roosvelt Hardaway, Satish Hardaway, Scotte Hardaway, Sentell Hardaway, Serapio Hardaway, Shabaka Hardaway, Srinivas Hardaway, Stanely Hardaway, Taraus Hardaway,
Tehron Hardaway, Temujin Hardaway, Todrick Hardaway, Torrian Hardaway, Treven Hardaway, Trinidy Hardaway, Tung Hardaway, Uchenna Hardaway, Vineet Hardaway, Waldon Hardaway, Wilkins Hardaway, Xzavier
Hardaway, Yaser Hardaway, Zahir Hardaway, Zeus Hardaway, Abdurrahim Hardaway, Ahmet Hardaway, Albion Hardaway, Andi Hardaway, Andri Hardaway, Angello Hardaway, Angle Hardaway, Aquilla Hardaway, Arlee
Hardaway, Arliss Hardaway, Asante Hardaway, Ashwin Hardaway, Benford Hardaway, Bernd Hardaway, Bevin Hardaway, Blanton Hardaway, Brandie Hardaway, Brint Hardaway, Carle Hardaway, Carvel Hardaway,
Carvell Hardaway, Chang Hardaway, Charls Hardaway, Christa Hardaway, Deidrick Hardaway, Derrion Hardaway, Dervin Hardaway, Devine Hardaway, Dixie Hardaway, Dom Hardaway, Dwone Hardaway, Edman
Hardaway, Efthimios Hardaway, Eian Hardaway, Elazar Hardaway, Elridge Hardaway, Emma Hardaway, Eoin Hardaway, Ericka Hardaway, Everick Hardaway, Everitt Hardaway, Franck Hardaway, Fredi Hardaway,
Ganesh Hardaway, Gaurav Hardaway, Geza Hardaway, Gianpaolo Hardaway, Grahm Hardaway, Haden Hardaway, Haitham Hardaway, Hansford Hardaway, Heberto Hardaway, Hieu Hardaway, Jabali Hardaway, Jabir
Hardaway, Jamilah Hardaway, Jarriel Hardaway, Javares Hardaway, Jawaun Hardaway, Jessen Hardaway, Jonanthan Hardaway, Joshoa Hardaway, Joslyn Hardaway, Jovian Hardaway, Jozsef Hardaway, Juanpablo
Hardaway, Justun Hardaway, Kalief Hardaway, Kaseen Hardaway, Kato Hardaway, Keagan Hardaway, Keefer Hardaway, Kishan Hardaway, Kitwana Hardaway, Konata Hardaway, Lajohn Hardaway, Lakeisha Hardaway,
Lamichael Hardaway, Latonia Hardaway, Lelan Hardaway, Lendon Hardaway, Leovardo Hardaway, Leray Hardaway, Linh Hardaway, Lovie Hardaway, Manasseh Hardaway, Marcella Hardaway, Marguette Hardaway,
Marke Hardaway, Marlene Hardaway, Martie Hardaway, Martrell Hardaway, Marus Hardaway, Masaki Hardaway, Mathhew Hardaway, Maurisio Hardaway, Mayur Hardaway, Meco Hardaway, Meguel Hardaway, Meko
Hardaway, Mensah Hardaway, Micki Hardaway, Mikele Hardaway, Milas Hardaway, Mirko Hardaway, Mondell Hardaway, Mondo Hardaway, Mondrell Hardaway, Murdock Hardaway, Nairobi Hardaway, Nathaiel Hardaway,
Neco Hardaway, Neely Hardaway, Nelvin Hardaway, Nichol Hardaway, Nicholous Hardaway, Norm Hardaway, Olander Hardaway, Pace Hardaway, Paschal Hardaway, Patick Hardaway, Pearson Hardaway, Phuc
Hardaway, Ramil Hardaway, Ravinder Hardaway, Rayman Hardaway, Remond Hardaway, Rendall Hardaway, Reshawn Hardaway, Rhian Hardaway, Romy Hardaway, Roozbeh Hardaway, Rutledge Hardaway, Ryker Hardaway,
Satoshi Hardaway, Shafton Hardaway, Shahab Hardaway, Shantell Hardaway, Shontez Hardaway, Standley Hardaway, Taha Hardaway, Tarren Hardaway, Tavio Hardaway, Terah Hardaway, Texas Hardaway, Thabiti
Hardaway, Theodric Hardaway, Timtohy Hardaway, Torence Hardaway, Torrick Hardaway, Towan Hardaway, Travor Hardaway, Tyge Hardaway, Vandell Hardaway, Virginio Hardaway, Wilkie Hardaway, Windsor
Hardaway, Yarrow Hardaway, Yuji Hardaway, Zedric Hardaway, Ziyad Hardaway, Abie Hardaway, Abundio Hardaway, Adi Hardaway, Aisha Hardaway, Akintunde Hardaway, Alfio Hardaway, Anival Hardaway, Antonnio
Hardaway, Asencion Hardaway, Ashly Hardaway, Beaux Hardaway, Bodhi Hardaway, Bowie Hardaway, Brandal Hardaway, Bristol Hardaway, Carrick Hardaway, Chales Hardaway, Chantz Hardaway, Chenier Hardaway,
Chima Hardaway, Chrystopher Hardaway, Cian Hardaway, Clearance Hardaway, Cohen Hardaway, Colman Hardaway, Corneil Hardaway, Dannis Hardaway, Dantonio Hardaway, Danyal Hardaway, Darus Hardaway, Daudi
Hardaway, Dejohn Hardaway, Demeko Hardaway, Dewaun Hardaway, Dondrick Hardaway, Donley Hardaway, Donney Hardaway, Donyelle Hardaway, Dorjan Hardaway, Drummond Hardaway, Dwayn Hardaway, Dywayne
Hardaway, Euclides Hardaway, Farhan Hardaway, Fritzgerald Hardaway, Gabreil Hardaway, Gerrald Hardaway, Greer Hardaway, Halim Hardaway, Haroon Hardaway, Hatim Hardaway, Herald Hardaway, Ismeal
Hardaway, Jamesmichael Hardaway, Jamir Hardaway, Janel Hardaway, Janelle Hardaway, Jerrol Hardaway, Jerryl Hardaway, Jonta Hardaway, Jumoke Hardaway, Karanja Hardaway, Karina Hardaway, Karis
Hardaway, Karrem Hardaway, Kavan Hardaway, Keddrick Hardaway, Kedrin Hardaway, Keifer Hardaway, Keldric Hardaway, Kenichi Hardaway, Kery Hardaway, Khris Hardaway, Kiko Hardaway, Kimothy Hardaway,
Kippy Hardaway, Kivin Hardaway, Kojak Hardaway, Kortez Hardaway, Lambros Hardaway, Lealand Hardaway, Lillian Hardaway, Llewelyn Hardaway, Loron Hardaway, Maneesh Hardaway, Mansoor Hardaway, Mariah
Hardaway, Maricus Hardaway, Marta Hardaway, Marte Hardaway, Martese Hardaway, Maston Hardaway, Maylon Hardaway, Mcclain Hardaway, Mikko Hardaway, Montrez Hardaway, Najee Hardaway, Najeeb Hardaway,
Nakoma Hardaway, Nakuma Hardaway, Oliverio Hardaway, Orlondo Hardaway, Osceola Hardaway, Osric Hardaway, Othoniel Hardaway, Ovid Hardaway, Peterjohn Hardaway, Rad Hardaway, Rogue Hardaway, Rolly
Hardaway, Roshad Hardaway, Sahwn Hardaway, Santosh Hardaway, Shakeem Hardaway, Shanen Hardaway, Shey Hardaway, Sina Hardaway, Sonja Hardaway, Stone Hardaway, Stphen Hardaway, Sue Hardaway, Sylvanus
Hardaway, Tameka Hardaway, Tauris Hardaway, Tawayne Hardaway, Tennille Hardaway, Terrice Hardaway, Thalamus Hardaway, Tore Hardaway, Tranell Hardaway, Travone Hardaway, Trevan Hardaway, Tristram
Hardaway, Tywaun Hardaway, Valton Hardaway, Venice Hardaway, Wagner Hardaway, Walberto Hardaway, Weyman Hardaway, Willliam Hardaway, Wisam Hardaway, Yisrael Hardaway, Zachari Hardaway, Zon Hardaway,
Amal Hardaway, Amie Hardaway, Antwun Hardaway, Audre Hardaway, Bertrum Hardaway, Bond Hardaway, Canon Hardaway, Careem Hardaway, Carly Hardaway, Charmaine Hardaway, Corian Hardaway, Coye Hardaway,
Daisuke Hardaway, Damacio Hardaway, Damarius Hardaway, Damiane Hardaway, Darrik Hardaway, Dartanian Hardaway, Dary Hardaway, Davan Hardaway, Deana Hardaway, Deaundra Hardaway, Delos Hardaway, Develle
Hardaway, Dontray Hardaway, Dorsett Hardaway, Dyrell Hardaway, Gered Hardaway, Gilad Hardaway, Hadrian Hardaway, Haji Hardaway, Harl Hardaway, Hideki Hardaway, Husani Hardaway, Jacson Hardaway,
Jaimes Hardaway, Jama Hardaway, Janathan Hardaway, Jaran Hardaway, Jauron Hardaway, Jeffie Hardaway, Jejuan Hardaway, Jerick Hardaway, Jermery Hardaway, Jerramie Hardaway, Kalei Hardaway, Karam
Hardaway, Keevan Hardaway, Kei Hardaway, Kel Hardaway, Keo Hardaway, Khaleel Hardaway, Klye Hardaway, Korrie Hardaway, Kunal Hardaway, Kyrone Hardaway, Ladaryl Hardaway, Laman Hardaway, Larome
Hardaway, Latwan Hardaway, Lavaris Hardaway, Lavoy Hardaway, Lenoard Hardaway, Levarr Hardaway, Matilde Hardaway, Mazin Hardaway, Nakie Hardaway, Nikolos Hardaway, Oak Hardaway, Orrie Hardaway, Pleas
Hardaway, Quanta Hardaway, Rayshun Hardaway, Rhasaan Hardaway, Ruark Hardaway, Saxon Hardaway, Seith Hardaway, Sham Hardaway, Shamont Hardaway, Shontel Hardaway, Terre Hardaway, Thoma Hardaway,
Thunder Hardaway, Torance Hardaway, Travelle Hardaway, Tulio Hardaway, Tyrance Hardaway, Ustin Hardaway, Verle Hardaway, Vicky Hardaway, Wlliam Hardaway, Xerxes Hardaway, Yero Hardaway, Adetokunbo
Hardaway, Adika Hardaway, Adriane Hardaway, Afshin Hardaway, Alto Hardaway, Alvon Hardaway, Anant Hardaway, Anatole Hardaway, Aston Hardaway, Aul Hardaway, Aza Hardaway, Azriel Hardaway, Bay
Hardaway, Bee Hardaway, Benoit Hardaway, Bradden Hardaway, Brayton Hardaway, Buren Hardaway, Caton Hardaway, Cesareo Hardaway, Chandar Hardaway, Charistopher Hardaway, Chike Hardaway, Chong Hardaway,
Cline Hardaway, Dabid Hardaway, Damascus Hardaway, Delron Hardaway, Delvis Hardaway, Demeterius Hardaway, Demetries Hardaway, Demorio Hardaway, Demtrius Hardaway, Derrian Hardaway, Dilanjan Hardaway,
Donti Hardaway, Doulgas Hardaway, Doyce Hardaway, Eber Hardaway, Elena Hardaway, Ellington Hardaway, Eros Hardaway, Esiquio Hardaway, Esley Hardaway, Ewan Hardaway, Firas Hardaway, Freeland Hardaway,
Frenchie Hardaway, Garlan Hardaway, Gaylin Hardaway, Hamin Hardaway, Hoyle Hardaway, Hubbard Hardaway, Ikechukwu Hardaway, Issam Hardaway, Jameil Hardaway, Janard Hardaway, Jann Hardaway, Jarmain
Hardaway, Jayjay Hardaway, Jemell Hardaway, Jerem Hardaway, Jeremian Hardaway, Jerritt Hardaway, Juda Hardaway, Kahn Hardaway, Kartik Hardaway, Katrell Hardaway, Kavon Hardaway, Kelsie Hardaway,
Kelsy Hardaway, Kenna Hardaway, Kenner Hardaway, Kentaro Hardaway, Kenyotta Hardaway, Keonte Hardaway, Kindell Hardaway, Kodie Hardaway, Kurtus Hardaway, Lacory Hardaway, Latanya Hardaway, Lathaniel
Hardaway, Lauriano Hardaway, Lavares Hardaway, Lemarr Hardaway, Lomont Hardaway, Lyndel Hardaway, Mandeep Hardaway, Mansur Hardaway, Marrion Hardaway, Mathan Hardaway, Maurie Hardaway, Melburn
Hardaway, Mia Hardaway, Micheil Hardaway, Milledge Hardaway, Mourice Hardaway, Mousa Hardaway, Muneer Hardaway, Naphtali Hardaway, Navarro Hardaway, Nehemias Hardaway, Nickalous Hardaway, Nickolous
Hardaway, Ola Hardaway, Oshua Hardaway, Osmond Hardaway, Quinta Hardaway, Quintel Hardaway, Rahshawn Hardaway, Raney Hardaway, Raye Hardaway, Rayne Hardaway, Rockne Hardaway, Romar Hardaway,
Sebastain Hardaway, Shang Hardaway, Sherrard Hardaway, Smauel Hardaway, Stamatios Hardaway, Talal Hardaway, Taurin Hardaway, Terdell Hardaway, Thong Hardaway, Tishawn Hardaway, Tizoc Hardaway, Trad
Hardaway, Trafton Hardaway, Trenity Hardaway, Tyray Hardaway, Ulices Hardaway, Vareck Hardaway, Videll Hardaway, Voltaire Hardaway, Wilmot Hardaway, Zacary Hardaway, Aaran Hardaway, Abdalla Hardaway,
Abdias Hardaway, Abiodun Hardaway, Adlai Hardaway, Ahamad Hardaway, Alexandar Hardaway, Alie Hardaway, Alissa Hardaway, Alterick Hardaway, Amitabh Hardaway, Andros Hardaway, Angelia Hardaway,
Antwonne Hardaway, Arnez Hardaway, Arrion Hardaway, Asaf Hardaway, Auden Hardaway, Bacilio Hardaway, Brynn Hardaway, Byan Hardaway, Can Hardaway, Carlen Hardaway, Carliss Hardaway, Carlose Hardaway,
Cazzie Hardaway, Chadney Hardaway, Chapman Hardaway, Chedrick Hardaway, Cherokee Hardaway, Christophen Hardaway, Clearnce Hardaway, Cleto Hardaway, Cliford Hardaway, Colburn Hardaway, Cordy Hardaway,
Craigory Hardaway, Dalan Hardaway, Dam Hardaway, Damonn Hardaway, Dandrea Hardaway, Danford Hardaway, Danieal Hardaway, Dano Hardaway, Daril Hardaway, Darrence Hardaway, Dartanyan Hardaway, Dawone
Hardaway, Deatrick Hardaway, Delayne Hardaway, Delfred Hardaway, Delshon Hardaway, Demark Hardaway, Demea Hardaway, Demetra Hardaway, Demetress Hardaway, Demeturis Hardaway, Dene Hardaway, Derran
Hardaway, Dewann Hardaway, Dezi Hardaway, Dondrell Hardaway, Donsha Hardaway, Dontai Hardaway, Durelle Hardaway, Ebay Hardaway, Edin Hardaway, Efran Hardaway, Emelio Hardaway, Erasto Hardaway, Erby
Hardaway, Eriks Hardaway, Erine Hardaway, Ester Hardaway, Everet Hardaway, Excell Hardaway, Falando Hardaway, Gaius Hardaway, Garald Hardaway, Garlon Hardaway, Germane Hardaway, Gernard Hardaway,
Glenden Hardaway, Graciela Hardaway, Gram Hardaway, Grayling Hardaway, Gunter Hardaway, Gustaf Hardaway, Gwyn Hardaway, Hamp Hardaway, Hanley Hardaway, Harrold Hardaway, Harvest Hardaway,
Hermenegildo Hardaway, Hooman Hardaway, Husam Hardaway, Idrees Hardaway, Jaaron Hardaway, Jaben Hardaway, Jacent Hardaway, Jacyn Hardaway, Jadrian Hardaway, Jaisen Hardaway, Jamard Hardaway, Jamare
Hardaway, Jarl Hardaway, Jarvin Hardaway, Jary Hardaway, Jashon Hardaway, Jaso Hardaway, Jaton Hardaway, Jawann Hardaway, Jebidiah Hardaway, Jedadia Hardaway, Jenard Hardaway, Jeramaine Hardaway,
Jerrimy Hardaway, Jerrin Hardaway, Jibreel Hardaway, Jigar Hardaway, Joah Hardaway, Jolene Hardaway, Jonell Hardaway, Jordy Hardaway, Josuah Hardaway, Jwan Hardaway, Kantrell Hardaway, Keithrick
Hardaway, Kennet Hardaway, Keshaun Hardaway, Kief Hardaway, Kisha Hardaway, Kiwan Hardaway, Kizzy Hardaway, Koury Hardaway, Kylon Hardaway, Lakisha Hardaway, Lamarco Hardaway, Lavonte Hardaway,
Lemanuel Hardaway, Lennell Hardaway, Letcher Hardaway, Lethaniel Hardaway, Levone Hardaway, Lilton Hardaway, Lindel Hardaway, Linzy Hardaway, Lora Hardaway, Lorie Hardaway, Lovelle Hardaway, Lowry
Hardaway, Lucia Hardaway, Luisa Hardaway, Macheal Hardaway, Magdiel Hardaway, Mahmood Hardaway, Manrique Hardaway, Manus Hardaway, Maricela Hardaway, Marjoe Hardaway, Marquett Hardaway, Mars
Hardaway, Mathews Hardaway, Mattthew Hardaway, Mehmet Hardaway, Mehran Hardaway, Menashe Hardaway, Merril Hardaway, Moe Hardaway, Morgen Hardaway, Natalio Hardaway, Nero Hardaway, Nickolos Hardaway,
Nidal Hardaway, Nishant Hardaway, Nixon Hardaway, Oreste Hardaway, Osie Hardaway, Parks Hardaway, Pax Hardaway, Phillipp Hardaway, Qaadir Hardaway, Quantas Hardaway, Rameen Hardaway, Ranell Hardaway,
Rani Hardaway, Rasheim Hardaway, Rawlin Hardaway, Rena Hardaway, Rett Hardaway, Riko Hardaway, Rimas Hardaway, Roark Hardaway, Roma Hardaway, Romale Hardaway, Rosemary Hardaway, Roshell Hardaway,
Sadiki Hardaway, Saquan Hardaway, Sargon Hardaway, Shantez Hardaway, Shawntel Hardaway, Shelia Hardaway, Sheryl Hardaway, Sholem Hardaway, Sly Hardaway, Sohail Hardaway, Spero Hardaway, Stepehn
Hardaway, Tabor Hardaway, Taki Hardaway, Talley Hardaway, Tamboura Hardaway, Tarrus Hardaway, Tau Hardaway, Taz Hardaway, Tejuan Hardaway, Terik Hardaway, Terone Hardaway, Terreance Hardaway, Terriel
Hardaway, Terriss Hardaway, Theodus Hardaway, Tobian Hardaway, Tolan Hardaway, Tollie Hardaway, Townsend Hardaway, Tramain Hardaway, Trennis Hardaway, Tshaka Hardaway, Tynell Hardaway, Tyone
Hardaway, Tywain Hardaway, Uganda Hardaway, Unique Hardaway, Vahe Hardaway, Vashone Hardaway, Verna Hardaway, Welsey Hardaway, Woodson Hardaway, Wydell Hardaway, Youssef Hardaway, Zebulin Hardaway,
Zeyad Hardaway, Aaryn Hardaway, Ademola Hardaway, Adom Hardaway, Alandus Hardaway, Alcide Hardaway, Aldridge Hardaway, Alfonsa Hardaway, Alfonzia Hardaway, Allyson Hardaway, Alter Hardaway, Amari
Hardaway, Amr Hardaway, Ancel Hardaway, Andie Hardaway, Andrus Hardaway, Antiwan Hardaway, Antoinette Hardaway, Antonis Hardaway, Antwyne Hardaway, Anupam Hardaway, Aquila Hardaway, Arville Hardaway,
Asael Hardaway, Assad Hardaway, Aurohom Hardaway, Autrey Hardaway, Ayal Hardaway, Azikiwe Hardaway, Bashar Hardaway, Basim Hardaway, Beauregard Hardaway, Belal Hardaway, Benyamin Hardaway, Bertis
Hardaway, Bertran Hardaway, Boban Hardaway, Boby Hardaway, Bracy Hardaway, Branndon Hardaway, Burtis Hardaway, Cameo Hardaway, Carley Hardaway, Carnelius Hardaway, Caroll Hardaway, Carolos Hardaway,
Carvis Hardaway, Cashawn Hardaway, Cean Hardaway, Charod Hardaway, Chevelle Hardaway, Chrisotpher Hardaway, Christerfer Hardaway, Christi Hardaway, Clois Hardaway, Cloyce Hardaway, Constancio
Hardaway, Corkey Hardaway, Corvin Hardaway, Corye Hardaway, Costello Hardaway, Coury Hardaway, Crecencio Hardaway, Cuahutemoc Hardaway, Curby Hardaway, Curly Hardaway, Dade Hardaway, Dag Hardaway,
Daimeon Hardaway, Dakin Hardaway, Damiean Hardaway, Dandy Hardaway, Danian Hardaway, Davinder Hardaway, Daylin Hardaway, Deljuan Hardaway, Delone Hardaway, Delta Hardaway, Demaris Hardaway, Demel
Hardaway, Demerick Hardaway, Demetrous Hardaway, Demiko Hardaway, Demorrio Hardaway, Deonne Hardaway, Derian Hardaway, Detron Hardaway, Deva Hardaway, Dillion Hardaway, Divine Hardaway, Dniel
Hardaway, Doanld Hardaway, Donaldo Hardaway, Donshay Hardaway, Dontee Hardaway, Doyne Hardaway, Dubois Hardaway, Dulani Hardaway, Duong Hardaway, Dupre Hardaway, Durante Hardaway, Edith Hardaway,
Edris Hardaway, Edwyn Hardaway, Efrin Hardaway, Ehrin Hardaway, Ehsan Hardaway, Eldwin Hardaway, Elic Hardaway, Elis Hardaway, Elven Hardaway, Eren Hardaway, Errik Hardaway, Erubey Hardaway, Evertt
Hardaway, Exavier Hardaway, Fahad Hardaway, Fares Hardaway, Fatmir Hardaway, Ferando Hardaway, Fonta Hardaway, Fotis Hardaway, Gaither Hardaway, Gar Hardaway, Garvey Hardaway, Gilmer Hardaway,
Godofredo Hardaway, Greggery Hardaway, Gregoire Hardaway, Gregorey Hardaway, Hagan Hardaway, Haim Hardaway, Hanz Hardaway, Hartford Hardaway, Hassen Hardaway, Haynes Hardaway, Herrick Hardaway, Herry
Hardaway, Hoa Hardaway, Hud Hardaway, Hurbert Hardaway, Ilir Hardaway, Ilyas Hardaway, Ivica Hardaway, Jabarr Hardaway, Jamara Hardaway, Jamarco Hardaway, Jana Hardaway, Janis Hardaway, Jarry
Hardaway, Javad Hardaway, Javas Hardaway, Javed Hardaway, Jayden Hardaway, Jebb Hardaway, Jeptha Hardaway, Jerimia Hardaway, Joffrey Hardaway, Johanthan Hardaway, Johari Hardaway, Johnanthony
Hardaway, Johnnathan Hardaway, Johnta Hardaway, Jolly Hardaway, Jonahtan Hardaway, Jong Hardaway, Jonh Hardaway, Josep Hardaway, Joss Hardaway, Juergen Hardaway, Juliocesar Hardaway, Junichi
Hardaway, Jusitn Hardaway, Kalid Hardaway, Kamali Hardaway, Kannon Hardaway, Karlon Hardaway, Kashawn Hardaway, Keilan Hardaway, Kelcy Hardaway, Kellis Hardaway, Kelon Hardaway, Kendrich Hardaway,
Kentrel Hardaway, Kerwyn Hardaway, Kesley Hardaway, Kimon Hardaway, Kindu Hardaway, Kinneth Hardaway, Kondwani Hardaway, Krystal Hardaway, Kyran Hardaway, Lacarlos Hardaway, Laney Hardaway, Lelon
Hardaway, Lem Hardaway, Lena Hardaway, Leno Hardaway, Leshan Hardaway, Lexie Hardaway, Lisle Hardaway, Lonne Hardaway, Lorence Hardaway, Lowery Hardaway, Lundy Hardaway, Lynford Hardaway, Maleek
Hardaway, Maliek Hardaway, Mano Hardaway, Marcellis Hardaway, Marcellius Hardaway, Marci Hardaway, Maris Hardaway, Marl Hardaway, Marston Hardaway, Matthe Hardaway, Matther Hardaway, Maximiano
Hardaway, Mecca Hardaway, Melesio Hardaway, Mic Hardaway, Mican Hardaway, Mickal Hardaway, Mitchum Hardaway, Mohit Hardaway, Myrl Hardaway, Nael Hardaway, Najja Hardaway, Nedal Hardaway, Nektarios
Hardaway, Nemiah Hardaway, Niclas Hardaway, Nilay Hardaway, Nina Hardaway, Noberto Hardaway, Noell Hardaway, Obrian Hardaway, Orbie Hardaway, Ormond Hardaway, Orvin Hardaway, Osborn Hardaway, Pasco
Hardaway, Pele Hardaway, Penn Hardaway, Perryn Hardaway, Philipe Hardaway, Prakash Hardaway, Quarterrio Hardaway, Quiency Hardaway, Rahaman Hardaway, Rahshan Hardaway, Rahson Hardaway, Raimond
Hardaway, Ramsay Hardaway, Ranard Hardaway, Ranulfo Hardaway, Rasha Hardaway, Rashine Hardaway, Rason Hardaway, Ravis Hardaway, Raylan Hardaway, Rayme Hardaway, Raymel Hardaway, Redrick Hardaway,
Regi Hardaway, Renier Hardaway, Rhea Hardaway, Riad Hardaway, Richerd Hardaway, Rishard Hardaway, Rockland Hardaway, Rolfe Hardaway, Roudy Hardaway, Saed Hardaway, Salaam Hardaway, Samie Hardaway,
Savino Hardaway, Selim Hardaway, Senneca Hardaway, Seung Hardaway, Shadric Hardaway, Shaman Hardaway, Sharn Hardaway, Shawen Hardaway, Shiron Hardaway, Shohn Hardaway, Shunn Hardaway, Sims Hardaway,
Skeet Hardaway, Soctt Hardaway, Sohn Hardaway, Standly Hardaway, Starbuck Hardaway, Stratton Hardaway, Sulton Hardaway, Sunday Hardaway, Sylas Hardaway, Taff Hardaway, Tallis Hardaway, Tamal
Hardaway, Tarrant Hardaway, Taurice Hardaway, Teko Hardaway, Telvis Hardaway, Terek Hardaway, Terrius Hardaway, Thermon Hardaway, Thorsten Hardaway, Timothee Hardaway, Tirus Hardaway, Tomi Hardaway,
Torren Hardaway, Travolta Hardaway, Travoris Hardaway, Tyan Hardaway, Vasco Hardaway, Vashion Hardaway, Verron Hardaway, Vinicio Hardaway, Vyron Hardaway, Wilkin Hardaway, Winson Hardaway, Worthy
Hardaway, Wynne Hardaway, Yance Hardaway, Yates Hardaway, Yecheskel Hardaway, Yonah Hardaway, Yuseff Hardaway, Abby Hardaway, Adaryl Hardaway, Addis Hardaway, Afrim Hardaway, Ahmand Hardaway, Ahmon
Hardaway, Alejos Hardaway, Alphonsa Hardaway, Alphonsus Hardaway, Altie Hardaway, Anastasio Hardaway, Andris Hardaway, Ankit Hardaway, Antario Hardaway, Antero Hardaway, Arafat Hardaway, Armard
Hardaway, Armel Hardaway, Arrin Hardaway, Artavious Hardaway, Ashlee Hardaway, Atari Hardaway, Aundrae Hardaway, Avan Hardaway, Avion Hardaway, Azad Hardaway, Azariah Hardaway, Babe Hardaway, Banks
Hardaway, Bentura Hardaway, Bily Hardaway, Blakeley Hardaway, Bonner Hardaway, Bradshaw Hardaway, Braeden Hardaway, Brainard Hardaway, Brianne Hardaway, Brigg Hardaway, Broch Hardaway, Bryian
Hardaway, Buell Hardaway, Burchell Hardaway, Butler Hardaway, Callen Hardaway, Cardale Hardaway, Carlester Hardaway, Carwin Hardaway, Cassell Hardaway, Casy Hardaway, Chancelor Hardaway, Chapin
Hardaway, Chee Hardaway, Chey Hardaway, Chianti Hardaway, Clarenc Hardaway, Clayburn Hardaway, Cleofas Hardaway, Clevester Hardaway, Clydell Hardaway, Constance Hardaway, Coran Hardaway, Correll
Hardaway, Cotton Hardaway, Currie Hardaway, Dae Hardaway, Dai Hardaway, Daisy Hardaway, Dametrius Hardaway, Damico Hardaway, Danal Hardaway, Danien Hardaway, Danis Hardaway, Dar Hardaway, Darivs
Hardaway, Darmon Hardaway, Deejay Hardaway, Delia Hardaway, Demarquis Hardaway, Demetrik Hardaway, Demonta Hardaway, Denys Hardaway, Derec Hardaway, Derl Hardaway, Deshay Hardaway, Destiny Hardaway,
Detroy Hardaway, Devar Hardaway, Dewyane Hardaway, Dierre Hardaway, Dillan Hardaway, Din Hardaway, Dionysios Hardaway, Dishawn Hardaway, Doak Hardaway, Donaldson Hardaway, Donaven Hardaway, Donzel
Hardaway, Doremus Hardaway, Dorn Hardaway, Doss Hardaway, Duglas Hardaway, Durk Hardaway, Dushon Hardaway, Duvon Hardaway, Dwaylon Hardaway, Dywan Hardaway, Edjuan Hardaway, Effrem Hardaway, Eliel
Hardaway, Elizah Hardaway, Elray Hardaway, Elsa Hardaway, Emerick Hardaway, Epigmenio Hardaway, Epimenio Hardaway, Erasmus Hardaway, Erez Hardaway, Erman Hardaway, Esgar Hardaway, Everrett Hardaway,
Fateen Hardaway, Felimon Hardaway, Felis Hardaway, Ferguson Hardaway, Firman Hardaway, Fitz Hardaway, France Hardaway, Freddick Hardaway, Frederi Hardaway, Gadiel Hardaway, Galdino Hardaway, Gant
Hardaway, Gared Hardaway, Garic Hardaway, Garrell Hardaway, Gavan Hardaway, Gemini Hardaway, Genard Hardaway, Geoge Hardaway, Germany Hardaway, Gioacchino Hardaway, Gladys Hardaway, Glennie Hardaway,
Glenroy Hardaway, Gordy Hardaway, Graylon Hardaway, Griselda Hardaway, Gussie Hardaway, Hall Hardaway, Hammad Hardaway, Hamza Hardaway, Handy Hardaway, Hardie Hardaway, Hassel Hardaway, Heiko
Hardaway, Henley Hardaway, Hersh Hardaway, Hewitt Hardaway, Hiran Hardaway, Hoke Hardaway, Holmes Hardaway, Hunt Hardaway, Ikenna Hardaway, Imre Hardaway, Irma Hardaway, Isaul Hardaway, Ishmail
Hardaway, Ishmel Hardaway, Iyad Hardaway, Jaamal Hardaway, Jabaar Hardaway, Jacquin Hardaway, Jameison Hardaway, Jasom Hardaway, Jefery Hardaway, Jeffy Hardaway, Jermanine Hardaway, Jerol Hardaway,
Jesses Hardaway, Jibril Hardaway, Jimy Hardaway, Joffre Hardaway, Johm Hardaway, Johne Hardaway, Johnothan Hardaway, Josias Hardaway, Joslin Hardaway, Jovanny Hardaway, Juel Hardaway, Kaare Hardaway,
Kalib Hardaway, Kalim Hardaway, Kalum Hardaway, Kedron Hardaway, Kerim Hardaway, Kerman Hardaway, Keyan Hardaway, Khai Hardaway, Khareem Hardaway, Kierre Hardaway, Kimmie Hardaway, Kipton Hardaway,
Kord Hardaway, Lajaune Hardaway, Lamor Hardaway, Lancy Hardaway, Lando Hardaway, Landrum Hardaway, Lani Hardaway, Larmont Hardaway, Laurens Hardaway, Lawrenc Hardaway, Lealon Hardaway, Leeman
Hardaway, Legrande Hardaway, Leone Hardaway, Lewie Hardaway, Loretta Hardaway, Lorrin Hardaway, Loukas Hardaway, Luisito Hardaway, Lutalo Hardaway, Machel Hardaway, Macklin Hardaway, Mahir Hardaway,
Manpreet Hardaway, Mansour Hardaway, Marcum Hardaway, Mariusz Hardaway, Markcus Hardaway, Marquest Hardaway, Marven Hardaway, Mattias Hardaway, Melbourne Hardaway, Melisa Hardaway, Mihcael Hardaway,
Mile Hardaway, Monserrate Hardaway, Montique Hardaway, Morice Hardaway, Nataniel Hardaway, Neiman Hardaway, Noa Hardaway, Obdulio Hardaway, Odilon Hardaway, Oma Hardaway, Onathan Hardaway, Orlandis
Hardaway, Osamah Hardaway, Pastor Hardaway, Patterson Hardaway, Pattrick Hardaway, Pavlos Hardaway, Phoenix Hardaway, Pierson Hardaway, Puneet Hardaway, Quartez Hardaway, Quetin Hardaway, Rahsheen
Hardaway, Randee Hardaway, Ras Hardaway, Raymie Hardaway, Raymound Hardaway, Raysean Hardaway, Regginal Hardaway, Reico Hardaway, Reino Hardaway, Reynardo Hardaway, Rodrecus Hardaway, Rodrickus
Hardaway, Rolin Hardaway, Royston Hardaway, Rynell Hardaway, Ryo Hardaway, Sacramento Hardaway, Seaborn Hardaway, Sedgwick Hardaway, Shaddrick Hardaway, Shaffer Hardaway, Shamal Hardaway, Shamone
Hardaway, Sharman Hardaway, Sharrief Hardaway, Shawnell Hardaway, Shawntae Hardaway, Shimshon Hardaway, Shontay Hardaway, Shuaib Hardaway, Steveland Hardaway, Stonewall Hardaway, Sujal Hardaway, Tage
Hardaway, Tarin Hardaway, Tarrick Hardaway, Tarryl Hardaway, Tauras Hardaway, Tayvon Hardaway, Terrol Hardaway, Thearon Hardaway, Thinh Hardaway, Tirell Hardaway, Tobius Hardaway, Tobyn Hardaway,
Toderick Hardaway, Torben Hardaway, Torran Hardaway, Torray Hardaway, Traven Hardaway, Trek Hardaway, Trina Hardaway, Twayne Hardaway, Tyee Hardaway, Tyrick Hardaway, Tyvon Hardaway, Ubong Hardaway,
Ulysess Hardaway, Valerian Hardaway, Vennie Hardaway, Venus Hardaway, Vincen Hardaway, Vonn Hardaway, Wandell Hardaway, Warden Hardaway, Warnell Hardaway, Wille Hardaway, Wolfram Hardaway, Worley
Hardaway, Wren Hardaway, Yoseph Hardaway, Zacharian Hardaway, Zacheriah Hardaway, Anoop Hardaway, Antohny Hardaway, Bianco Hardaway, Billey Hardaway, Blanca Hardaway, Bradely Hardaway, Brison
Hardaway, Chanson Hardaway, Cheng Hardaway, Choice Hardaway, Cosmos Hardaway, Cymande Hardaway, Cyprian Hardaway, Dang Hardaway, Derrus Hardaway, Dorwin Hardaway, Everest Hardaway, Farzad Hardaway,
Gralin Hardaway, Idi Hardaway, Jakari Hardaway, Jezreel Hardaway, Jorell Hardaway, Kong Hardaway, Liza Hardaway, Mendy Hardaway, Onix Hardaway, Rajaee Hardaway, Random Hardaway, Rashean Hardaway,
Revis Hardaway, Saman Hardaway, Santini Hardaway, Shahin Hardaway, Talton Hardaway, Teco Hardaway, Temitope Hardaway, Vimal Hardaway, Wacey Hardaway, Aeric Hardaway, Alhaji Hardaway, Anas Hardaway,
Anselm Hardaway, Bamidele Hardaway, Behrang Hardaway, Binu Hardaway, Brenon Hardaway, Brittney Hardaway, Catina Hardaway, Chaunce Hardaway, Christerphor Hardaway, Claxton Hardaway, Colley Hardaway,
Courtnee Hardaway, Darrio Hardaway, Darweshi Hardaway, Dashiell Hardaway, Daymeon Hardaway, Deontay Hardaway, Devlyn Hardaway, Effrey Hardaway, Eldric Hardaway, Faith Hardaway, Garris Hardaway,
Holley Hardaway, Hombre Hardaway, Hussain Hardaway, Jamari Hardaway, Jaramy Hardaway, Jaydon Hardaway, Jeramee Hardaway, Jerediah Hardaway, Jermiane Hardaway, Jermont Hardaway, Jerud Hardaway,
Johncharles Hardaway, Jolan Hardaway, Kairaba Hardaway, Kawon Hardaway, Kemon Hardaway, Kennen Hardaway, Kenyun Hardaway, Khevin Hardaway, Kiowa Hardaway, Kwabene Hardaway, Ladrick Hardaway, Lamarc
Hardaway, Laramy Hardaway, Larance Hardaway, Lebrone Hardaway, Loc Hardaway, Lynda Hardaway, Marcius Hardaway, Mariel Hardaway, Marisela Hardaway, Meshach Hardaway, Milon Hardaway, Monterio Hardaway,
Najib Hardaway, Natha Hardaway, Nery Hardaway, Niklaus Hardaway, Pinchus Hardaway, Quaid Hardaway, Rael Hardaway, Rain Hardaway, Ralpheal Hardaway, Ramiah Hardaway, Redmond Hardaway, Regniald
Hardaway, Rennard Hardaway, Rhet Hardaway, Ritesh Hardaway, Rodolpho Hardaway, Rontrell Hardaway, Rossano Hardaway, Shale Hardaway, Shevin Hardaway, Shiraz Hardaway, Sirica Hardaway, Sridhar
Hardaway, Staton Hardaway, Suliman Hardaway, Tchalla Hardaway, Telvin Hardaway, Terren Hardaway, Traun Hardaway, Trong Hardaway, Trygve Hardaway, Unborn Hardaway, Vaden Hardaway, Valery Hardaway,
Vuong Hardaway, Waseem Hardaway, Zebulen Hardaway, Abed Hardaway, Abhay Hardaway, Adebayo Hardaway, Ahmid Hardaway, Akai Hardaway, Aldwin Hardaway, Altonia Hardaway, Alvertis Hardaway, Alvey
Hardaway, Amondo Hardaway, Amrom Hardaway, Andera Hardaway, Andrico Hardaway, Anterio Hardaway, Antolin Hardaway, Antorio Hardaway, Arkim Hardaway, Arran Hardaway, Aslan Hardaway, Ausencio Hardaway,
Aviel Hardaway, Avrum Hardaway, Ayron Hardaway, Azeem Hardaway, Azel Hardaway, Azell Hardaway, Basel Hardaway, Bengi Hardaway, Benjamyn Hardaway, Berish Hardaway, Bianca Hardaway, Bonny Hardaway, Boy
Hardaway, Brach Hardaway, Brishen Hardaway, Britney Hardaway, Burlin Hardaway, Buzz Hardaway, Carolina Hardaway, Casimer Hardaway, Cedeno Hardaway, Cemal Hardaway, Chadly Hardaway, Chen Hardaway,
Chirs Hardaway, Christion Hardaway, Christohpher Hardaway, Christpoher Hardaway, Clate Hardaway, Cleaven Hardaway, Clif Hardaway, Coburn Hardaway, Conroy Hardaway, Corbitt Hardaway, Crispus Hardaway,
Daiman Hardaway, Daimion Hardaway, Damarco Hardaway, Dammian Hardaway, Darr Hardaway, Darreyl Hardaway, Davar Hardaway, Dawain Hardaway, Dayron Hardaway, Dectrick Hardaway, Dederick Hardaway,
Demarkis Hardaway, Denman Hardaway, Derril Hardaway, Devanand Hardaway, Dharma Hardaway, Dionysus Hardaway, Domanick Hardaway, Dorey Hardaway, Duel Hardaway, Ebenezer Hardaway, Edon Hardaway, Effie
Hardaway, Ehron Hardaway, Eliceo Hardaway, Elija Hardaway, Elijha Hardaway, Elizandro Hardaway, Enriquez Hardaway, Eryn Hardaway, Essie Hardaway, Etoyi Hardaway, Fabrice Hardaway, Farouk Hardaway,
Garick Hardaway, Gean Hardaway, Geovani Hardaway, Germar Hardaway, Gillian Hardaway, Gilmore Hardaway, Giuseppi Hardaway, Governor Hardaway, Graden Hardaway, Gresham Hardaway, Haashim Hardaway, Hagop
Hardaway, Hamzah Hardaway, Harlie Hardaway, Hatem Hardaway, Hien Hardaway, Hose Hardaway, Ibe Hardaway, Ingo Hardaway, Irven Hardaway, Isam Hardaway, Ishi Hardaway, Istvan Hardaway, Jabriel Hardaway,
Jamont Hardaway, Janette Hardaway, Jaroslaw Hardaway, Jashawn Hardaway, Java Hardaway, Javid Hardaway, Jaysun Hardaway, Jemario Hardaway, Jemiah Hardaway, Jerek Hardaway, Jeren Hardaway, Jeriah
Hardaway, Jerode Hardaway, Jimmel Hardaway, Jitu Hardaway, Jmichael Hardaway, Johnas Hardaway, Johnwilliam Hardaway, Joran Hardaway, Jori Hardaway, Josehua Hardaway, Joshlin Hardaway, Joushua
Hardaway, Juane Hardaway, Juba Hardaway, Juvon Hardaway, Kamaal Hardaway, Kasib Hardaway, Kate Hardaway, Kayin Hardaway, Keffer Hardaway, Kentay Hardaway, Kentrail Hardaway, Kenyell Hardaway, Kerr
Hardaway, Khalik Hardaway, Khurram Hardaway, Kipchoge Hardaway, Koa Hardaway, Kowan Hardaway, Kreig Hardaway, Kristien Hardaway, Krystopher Hardaway, Kurry Hardaway, Kylee Hardaway, Lamaine Hardaway,
Lamell Hardaway, Lamour Hardaway, Lanoris Hardaway, Laquann Hardaway, Laquinn Hardaway, Lavone Hardaway, Leanord Hardaway, Leeandrew Hardaway, Lenzy Hardaway, Lequient Hardaway, Leshun Hardaway,
Lopaka Hardaway, Lopez Hardaway, Lyall Hardaway, Malcoln Hardaway, Mali Hardaway, March Hardaway, Marchant Hardaway, Matthen Hardaway, Medrick Hardaway, Mildred Hardaway, Mitchelle Hardaway, Moriah
Hardaway, Mykal Hardaway, Neema Hardaway, Nii Hardaway, Nikkia Hardaway, Okechukwu Hardaway, Olatunde Hardaway, Olu Hardaway, Oluseyi Hardaway, Orazio Hardaway, Orel Hardaway, Ortiz Hardaway, Orvel
Hardaway, Oseph Hardaway, Overton Hardaway, Peer Hardaway, Phyllis Hardaway, Pratik Hardaway, Quante Hardaway, Quindell Hardaway, Raeshawn Hardaway, Rahshon Hardaway, Ran Hardaway, Real Hardaway,
Rhodes Hardaway, Rhodney Hardaway, Rodeny Hardaway, Rondi Hardaway, Rozelle Hardaway, Rutilio Hardaway, Saif Hardaway, Sandford Hardaway, Seab Hardaway, Sevag Hardaway, Shahied Hardaway, Shamari
Hardaway, Shammah Hardaway, Sharmon Hardaway, Shervin Hardaway, Skippy Hardaway, Stefanie Hardaway, Stylianos Hardaway, Tafari Hardaway, Tag Hardaway, Tamon Hardaway, Tanisha Hardaway, Tearle
Hardaway, Tellys Hardaway, Terriance Hardaway, Timber Hardaway, Timthoy Hardaway, Tomiko Hardaway, Torr Hardaway, Torrez Hardaway, Toshiro Hardaway, Trammell Hardaway, Trance Hardaway, Tregg
Hardaway, Treston Hardaway, Tristian Hardaway, Tymaine Hardaway, Tyne Hardaway, Tyrane Hardaway, Vada Hardaway, Valerio Hardaway, Varion Hardaway, Vasean Hardaway, Vaugh Hardaway, Viviana Hardaway,
Welford Hardaway, Windel Hardaway, Yaacov Hardaway, Yaniv Hardaway, Yusuke Hardaway, Zebulah Hardaway, Zuberi Hardaway, Aarin Hardaway, Abbott Hardaway, Abdule Hardaway, Abdulrahman Hardaway, Abrahim
Hardaway, Abrian Hardaway, Adham Hardaway, Aditya Hardaway, Adre Hardaway, Akito Hardaway, Ala Hardaway, Alberico Hardaway, Alexsandro Hardaway, Aljandro Hardaway, Allin Hardaway, Alyssa Hardaway,
Amond Hardaway, Angelique Hardaway, Ankoma Hardaway, Antonne Hardaway, Antroine Hardaway, Antwanne Hardaway, Antwayne Hardaway, Arben Hardaway, Arby Hardaway, Arcangelo Hardaway, Arek Hardaway, Arel
Hardaway, Artavis Hardaway, Arvon Hardaway, Arye Hardaway, Ascension Hardaway, Ashanta Hardaway, Atilla Hardaway, Atrick Hardaway, Austreberto Hardaway, Avant Hardaway, Avian Hardaway, Avier
Hardaway, Ayatollah Hardaway, Baltasar Hardaway, Banning Hardaway, Barin Hardaway, Barlow Hardaway, Barnie Hardaway, Barrick Hardaway, Baruti Hardaway, Bascom Hardaway, Basheer Hardaway, Bear
Hardaway, Beatriz Hardaway, Bengt Hardaway, Bennette Hardaway, Berl Hardaway, Berley Hardaway, Berwin Hardaway, Biren Hardaway, Blase Hardaway, Bon Hardaway, Bowe Hardaway, Bowman Hardaway, Brance
Hardaway, Bravlio Hardaway, Brette Hardaway, Brewster Hardaway, Bria Hardaway, Brookes Hardaway, Buel Hardaway, Buffy Hardaway, Cairo Hardaway, Caldwell Hardaway, Cali Hardaway, Carlan Hardaway,
Carmello Hardaway, Carmichael Hardaway, Carney Hardaway, Carr Hardaway, Carrell Hardaway, Carzell Hardaway, Castro Hardaway, Caswell Hardaway, Catrell Hardaway, Cedrice Hardaway, Celeste Hardaway,
Celina Hardaway, Cesear Hardaway, Chade Hardaway, Chadwell Hardaway, Chalres Hardaway, Chamar Hardaway, Chanan Hardaway, Chanda Hardaway, Chanin Hardaway, Charo Hardaway, Chayim Hardaway, Chelton
Hardaway, Chevez Hardaway, Chih Hardaway, Chin Hardaway, Chino Hardaway, Christaphor Hardaway, Ciprian Hardaway, Conny Hardaway, Corbit Hardaway, Corinthians Hardaway, Cosby Hardaway, Curlee
Hardaway, Cyrano Hardaway, Daimian Hardaway, Daks Hardaway, Dalin Hardaway, Damir Hardaway, Damisi Hardaway, Damiso Hardaway, Dantae Hardaway, Danthony Hardaway, Dantoni Hardaway, Darcus Hardaway,
Darence Hardaway, Darrious Hardaway, Darylle Hardaway, Datril Hardaway, Davidlee Hardaway, Dawane Hardaway, Daylen Hardaway, Deandrew Hardaway, Deloy Hardaway, Delson Hardaway, Demaine Hardaway,
Dementrius Hardaway, Demetrise Hardaway, Demien Hardaway, Demingo Hardaway, Demitris Hardaway, Deni Hardaway, Dennys Hardaway, Denon Hardaway, Dent Hardaway, Deondrick Hardaway, Derelle Hardaway,
Derike Hardaway, Derrall Hardaway, Des Hardaway, Deshea Hardaway, Deston Hardaway, Deval Hardaway, Devery Hardaway, Deville Hardaway, Devrin Hardaway, Dewell Hardaway, Diondray Hardaway, Dishan
Hardaway, Dkwon Hardaway, Dmario Hardaway, Dolph Hardaway, Dominador Hardaway, Donae Hardaway, Donni Hardaway, Dontea Hardaway, Donye Hardaway, Dossie Hardaway, Dray Hardaway, Dwann Hardaway, Earic
Hardaway, Earley Hardaway, Eaton Hardaway, Edwar Hardaway, Eiad Hardaway, Eirc Hardaway, Eith Hardaway, Elba Hardaway, Elefterios Hardaway, Eleodoro Hardaway, Elisabeth Hardaway, Elrod Hardaway, Elzy
Hardaway, Emilo Hardaway, Emmanouel Hardaway, Enis Hardaway, Ephrain Hardaway, Erec Hardaway, Erique Hardaway, Eriverto Hardaway, Erven Hardaway, Esco Hardaway, Esquiel Hardaway, Essa Hardaway,
Ettore Hardaway, Eulis Hardaway, Evay Hardaway, Fabius Hardaway, Faiz Hardaway, Felando Hardaway, Fenwick Hardaway, Fernell Hardaway, Francesca Hardaway, Francisca Hardaway, Franke Hardaway, Gabiel
Hardaway, Garrit Hardaway, Garyn Hardaway, Geffery Hardaway, Generoso Hardaway, Genghis Hardaway, Georgia Hardaway, Geovanny Hardaway, Gerell Hardaway, Gerred Hardaway, Gerrett Hardaway, Ghassan
Hardaway, Gilliam Hardaway, Gilson Hardaway, Givon Hardaway, Goeffrey Hardaway, Gorgonio Hardaway, Grahame Hardaway, Grayland Hardaway, Grzegorz Hardaway, Guan Hardaway, Gumercindo Hardaway, Gwen
Hardaway, Hannah Hardaway, Harden Hardaway, Harsha Hardaway, Harvy Hardaway, Hassain Hardaway, Helio Hardaway, Hemant Hardaway, Herchel Hardaway, Herold Hardaway, Hershy Hardaway, Hillis Hardaway,
Hinton Hardaway, Hommy Hardaway, Hong Hardaway, Huberto Hardaway, Hughes Hardaway, Husain Hardaway, Hussien Hardaway, Ibis Hardaway, Ilias Hardaway, Irl Hardaway, Iron Hardaway, Isak Hardaway, Ishaq
Hardaway, Issaac Hardaway, Iva Hardaway, Jabon Hardaway, Jachin Hardaway, Jacobus Hardaway, Janeiro Hardaway, Japhet Hardaway, Jarion Hardaway, Jaris Hardaway, Jarmell Hardaway, Jarnell Hardaway,
Jarvon Hardaway, Jaryl Hardaway, Jasonn Hardaway, Jauan Hardaway, Javian Hardaway, Javoris Hardaway, Jawhar Hardaway, Jawwaad Hardaway, Jayesh Hardaway, Jeffree Hardaway, Jeffri Hardaway, Jerauld
Hardaway, Jereny Hardaway, Jereomy Hardaway, Jerom Hardaway, Jerrall Hardaway, Jerran Hardaway, Jerrico Hardaway, Jessiah Hardaway, Jesten Hardaway, Jevan Hardaway, Jimbob Hardaway, Jiro Hardaway,
Johnpatrick Hardaway, Jolon Hardaway, Jonothon Hardaway, Jonthomas Hardaway, Josephine Hardaway, Jovonne Hardaway, Juanantonio Hardaway, Julis Hardaway, Jumah Hardaway, Jwyanza Hardaway, Kail
Hardaway, Kaleem Hardaway, Kalil Hardaway, Kamari Hardaway, Kamien Hardaway, Kanon Hardaway, Kanta Hardaway, Karac Hardaway, Karega Hardaway, Karras Hardaway, Kealoha Hardaway, Kearston Hardaway,
Kedryn Hardaway, Keiven Hardaway, Keland Hardaway, Kelechi Hardaway, Kelson Hardaway, Kendle Hardaway, Kenjuan Hardaway, Kennell Hardaway, Kenson Hardaway, Kerick Hardaway, Ketric Hardaway, Kevine
Hardaway, Keylon Hardaway, Keynan Hardaway, Khaaliq Hardaway, Khali Hardaway, Khang Hardaway, Khoi Hardaway, Kinshasa Hardaway, Kono Hardaway, Konstandinos Hardaway, Kontar Hardaway, Kou Hardaway,
Kreston Hardaway, Kristapher Hardaway, Kristophor Hardaway, Kumasi Hardaway, Kuntakinte Hardaway, Kwami Hardaway, Kwon Hardaway, Laderick Hardaway, Lajon Hardaway, Lakesha Hardaway, Lakieth Hardaway,
Lakota Hardaway, Lamarkus Hardaway, Lameen Hardaway, Lamondo Hardaway, Lamonica Hardaway, Landers Hardaway, Landrick Hardaway, Lanis Hardaway, Lapriest Hardaway, Laquane Hardaway, Larvell Hardaway,
Larwence Hardaway, Latavius Hardaway, Latimer Hardaway, Laton Hardaway, Latrelle Hardaway, Latwon Hardaway, Laurencio Hardaway, Lavelton Hardaway, Laver Hardaway, Lavester Hardaway, Lavonn Hardaway,
Lawanda Hardaway, Lawarren Hardaway, Lazerick Hardaway, Lecedric Hardaway, Leconte Hardaway, Leib Hardaway, Lemel Hardaway, Lenis Hardaway, Lenox Hardaway, Lenward Hardaway, Leonid Hardaway, Leory
Hardaway, Lerin Hardaway, Leroyal Hardaway, Linnell Hardaway, Linville Hardaway, Lional Hardaway, Loel Hardaway, Lofton Hardaway, Londale Hardaway, Lonza Hardaway, Lovis Hardaway, Lucy Hardaway, Luie
Hardaway, Lyndall Hardaway, Macker Hardaway, Makarios Hardaway, Malcon Hardaway, Manly Hardaway, Marell Hardaway, Margo Hardaway, Markco Hardaway, Markeis Hardaway, Markian Hardaway, Marsden
Hardaway, Martina Hardaway, Martinus Hardaway, Mashon Hardaway, Masud Hardaway, Mathaniel Hardaway, Mazi Hardaway, Merel Hardaway, Merwyn Hardaway, Michaelanthony Hardaway, Michaeljames Hardaway,
Mico Hardaway, Mihran Hardaway, Mikki Hardaway, Milik Hardaway, Milos Hardaway, Miran Hardaway, Mirza Hardaway, Mishael Hardaway, Mondale Hardaway, Mondre Hardaway, Montsho Hardaway, Morse Hardaway,
Mostafa Hardaway, Mumin Hardaway, Mynor Hardaway, Nantambu Hardaway, Narayana Hardaway, Naseem Hardaway, Nathane Hardaway, Navada Hardaway, Nazar Hardaway, Nectarios Hardaway, Nell Hardaway, Nevelle
Hardaway, Nichalaus Hardaway, Nichlous Hardaway, Nicholson Hardaway, Nicklus Hardaway, Nickolai Hardaway, Nicolino Hardaway, Nihar Hardaway, Nishan Hardaway, Nizar Hardaway, Nocholas Hardaway, Noelle
Hardaway, Novel Hardaway, Nyerere Hardaway, Obbie Hardaway, Ocean Hardaway, Oded Hardaway, Oden Hardaway, Olaniyan Hardaway, Ondray Hardaway, Orentha Hardaway, Osby Hardaway, Othell Hardaway, Othniel
Hardaway, Ottie Hardaway, Panayiotis Hardaway, Parren Hardaway, Pedram Hardaway, Petey Hardaway, Petter Hardaway, Pharaoh Hardaway, Precious Hardaway, Pressley Hardaway, Procopio Hardaway, Pryor
Hardaway, Quaashie Hardaway, Quinte Hardaway, Quintez Hardaway, Quinzell Hardaway, Quirino Hardaway, Rabon Hardaway, Rachard Hardaway, Rade Hardaway, Raju Hardaway, Ramee Hardaway, Ranjan Hardaway,
Ransford Hardaway, Rashawd Hardaway, Raynald Hardaway, Regenal Hardaway, Reiko Hardaway, Reinhold Hardaway, Remijio Hardaway, Renaud Hardaway, Ritchard Hardaway, Roanld Hardaway, Roarke Hardaway,
Robley Hardaway, Rochester Hardaway, Rodolph Hardaway, Roi Hardaway, Romelle Hardaway, Romey Hardaway, Romolo Hardaway, Rondo Hardaway, Ronni Hardaway, Ronta Hardaway, Roselio Hardaway, Rosheen
Hardaway, Ruban Hardaway, Rustan Hardaway, Sajan Hardaway, Salik Hardaway, Samel Hardaway, Santonia Hardaway, Sargent Hardaway, Saud Hardaway, Saxton Hardaway, Schane Hardaway, Sebron Hardaway,
Shadley Hardaway, Shadron Hardaway, Shafi Hardaway, Shalako Hardaway, Shanell Hardaway, Shani Hardaway, Shantae Hardaway, Shari Hardaway, Shauntay Hardaway, Shauwn Hardaway, Shawkat Hardaway, Shell
Hardaway, Shephen Hardaway, Sherief Hardaway, Shin Hardaway, Shneur Hardaway, Shontell Hardaway, Shoun Hardaway, Shundell Hardaway, Sigmond Hardaway, Spanky Hardaway, Stpehen Hardaway, Suhail
Hardaway, Sulieman Hardaway, Sundown Hardaway, Taboris Hardaway, Taijuan Hardaway, Talion Hardaway, Tamario Hardaway, Tamel Hardaway, Tana Hardaway, Tari Hardaway, Tarnell Hardaway, Tashon Hardaway,
Tasos Hardaway, Tavar Hardaway, Tawon Hardaway, Tay Hardaway, Tederick Hardaway, Tedman Hardaway, Teejay Hardaway, Tephen Hardaway, Terranc Hardaway, Terric Hardaway, Tevita Hardaway, Thao Hardaway,
Theoplis Hardaway, Thuan Hardaway, Tiger Hardaway, Tigh Hardaway, Timmon Hardaway, Tin Hardaway, Tionne Hardaway, Tipton Hardaway, Todo Hardaway, Tomatra Hardaway, Tomeka Hardaway, Tomie Hardaway,
Toriono Hardaway, Torivio Hardaway, Toye Hardaway, Tramine Hardaway, Tre Hardaway, Treaver Hardaway, Tredell Hardaway, Treva Hardaway, Trinton Hardaway, TRUE Hardaway, Tydus Hardaway, Tyeson
Hardaway, Tyle Hardaway, Tyrae Hardaway, Tyrand Hardaway, Tyresse Hardaway, Tyri Hardaway, Tysean Hardaway, Tyus Hardaway, Uland Hardaway, Usman Hardaway, Vahid Hardaway, Vang Hardaway, Vassilios
Hardaway, Vernis Hardaway, Vidale Hardaway, Vikash Hardaway, Vishnu Hardaway, Volney Hardaway, Vondale Hardaway, Warees Hardaway, Webber Hardaway, Whitman Hardaway, Willilam Hardaway, Windy Hardaway,
Winn Hardaway, Wynell Hardaway, Xan Hardaway, Yeshaya Hardaway, Yuma Hardaway, Zackory Hardaway, Zahid Hardaway, Zarak Hardaway, Zayd Hardaway, Zebula Hardaway, Zef Hardaway, Zeljko Hardaway, Zephyr
Hardaway, Zeshan Hardaway, Aakash Hardaway, Aalon Hardaway, Aaraon Hardaway, Aaric Hardaway, Aarn Hardaway, Abduel Hardaway, Abid Hardaway, Abigail Hardaway, Abijah Hardaway, Achille Hardaway, Ada
Hardaway, Adali Hardaway, Adante Hardaway, Adarsh Hardaway, Adedayo Hardaway, Adeyinka Hardaway, Adib Hardaway, Adran Hardaway, Adrell Hardaway, Adrew Hardaway, Adriaan Hardaway, Adriene Hardaway,
Afton Hardaway, Agim Hardaway, Ahmond Hardaway, Aiman Hardaway, Akiba Hardaway, Alamin Hardaway, Aland Hardaway, Alanson Hardaway, Albertico Hardaway, Albertus Hardaway, Alcario Hardaway, Aldrin
Hardaway, Alejando Hardaway, Alexey Hardaway, Alfreda Hardaway, Alfreddie Hardaway, Allateef Hardaway, Almer Hardaway, Almondo Hardaway, Alpesh Hardaway, Alrahman Hardaway, Alson Hardaway, Altarik
Hardaway, Alven Hardaway, Amadi Hardaway, Amancio Hardaway, Amandeep Hardaway, Amauris Hardaway, Amdrew Hardaway, Amel Hardaway, Amelia Hardaway, Amhad Hardaway, Amrit Hardaway, Ande Hardaway, Anderw
Hardaway, Andon Hardaway, Andren Hardaway, Andretti Hardaway, Andrius Hardaway, Andrue Hardaway, Andrw Hardaway, Aneesh Hardaway, Aneudy Hardaway, Anglo Hardaway, Anissa Hardaway, Annie Hardaway,
Anquan Hardaway, Ansar Hardaway, Ansen Hardaway, Anslem Hardaway, Anthone Hardaway, Anthory Hardaway, Antoinio Hardaway, Antojuan Hardaway, Antonial Hardaway, Antowain Hardaway, Antowine Hardaway,
Antrione Hardaway, Antroy Hardaway, Anttwan Hardaway, Antwin Hardaway, Antwonn Hardaway, Antyon Hardaway, Anwon Hardaway, Aquan Hardaway, Aquarius Hardaway, Aracely Hardaway, Aragorn Hardaway, Arbie
Hardaway, Arhtur Hardaway, Aristede Hardaway, Aristotelis Hardaway, Arkee Hardaway, Arlandus Hardaway, Arlondo Hardaway, Armstead Hardaway, Arnab Hardaway, Aroldo Hardaway, Arric Hardaway, Arshad
Hardaway, Artice Hardaway, Arvelle Hardaway, Arvie Hardaway, Aryan Hardaway, Arzell Hardaway, Asbury Hardaway, Aseem Hardaway, Ash Hardaway, Ashan Hardaway, Ashkan Hardaway, Ashlin Hardaway, Ashur
Hardaway, Astor Hardaway, Asuncion Hardaway, Ata Hardaway, Atsushi Hardaway, Aubery Hardaway, Audi Hardaway, Augustino Hardaway, Aurther Hardaway, Aviv Hardaway, Ayan Hardaway, Azael Hardaway, Azar
Hardaway, Azhar Hardaway, Azizi Hardaway, Bai Hardaway, Bain Hardaway, Baldo Hardaway, Balentin Hardaway, Banyon Hardaway, Barbaro Hardaway, Barett Hardaway, Barnell Hardaway, Barren Hardaway,
Bartolomeo Hardaway, Basem Hardaway, Basilios Hardaway, Bassey Hardaway, Bayete Hardaway, Baylen Hardaway, Bayne Hardaway, Bayron Hardaway, Beaufort Hardaway, Bejan Hardaway, Belvin Hardaway,
Benajmin Hardaway, Bengamin Hardaway, Bently Hardaway, Berel Hardaway, Berman Hardaway, Bernadino Hardaway, Bernis Hardaway, Bernon Hardaway, Bertie Hardaway, Bertin Hardaway, Bibiano Hardaway, Biko
Hardaway, Binyamin Hardaway, Birche Hardaway, Blakley Hardaway, Bleu Hardaway, Blythe Hardaway, Bobak Hardaway, Bowden Hardaway, Boz Hardaway, Brahim Hardaway, Brahin Hardaway, Brandell Hardaway,
Brandley Hardaway, Brann Hardaway, Brannigan Hardaway, Brantly Hardaway, Brayan Hardaway, Braydon Hardaway, Breandan Hardaway, Breh Hardaway, Brence Hardaway, Brenner Hardaway, Brentwood Hardaway,
Bric Hardaway, Bridgette Hardaway, Brig Hardaway, Briggs Hardaway, Brionne Hardaway, Bronc Hardaway, Brunson Hardaway, Brycen Hardaway, Jennifer Hardaway, Amy Hardaway, Melissa Hardaway, Michelle
Hardaway, Kimberly Hardaway, Lisa Hardaway, Angela Hardaway, Heather Hardaway, Stephanie Hardaway, Nicole Hardaway, Jessica Hardaway, Elizabeth Hardaway, Rebecca Hardaway, Kelly Hardaway, Mary
Hardaway, Christina Hardaway, Amanda Hardaway, Julie Hardaway, Sarah Hardaway, Laura Hardaway, Shannon Hardaway, Christine Hardaway, Tammy Hardaway, Tracy Hardaway, Karen Hardaway, Dawn Hardaway,
Susan Hardaway, Andrea Hardaway, Tina Hardaway, Patricia Hardaway, Cynthia Hardaway, Lori Hardaway, Rachel Hardaway, April Hardaway, Maria Hardaway, Wendy Hardaway, Crystal Hardaway, Stacy Hardaway,
Erin Hardaway, Jamie Hardaway, Carrie Hardaway, Tiffany Hardaway, Tara Hardaway, Sandra Hardaway, Monica Hardaway, Danielle Hardaway, Stacey Hardaway, Pamela Hardaway, Tonya Hardaway, Sara Hardaway,
Michele Hardaway, Teresa Hardaway, Denise Hardaway, Jill Hardaway, Katherine Hardaway, Melanie Hardaway, Dana Hardaway, Holly Hardaway, Erica Hardaway, Brenda Hardaway, Deborah Hardaway, Tanya
Hardaway, Sharon Hardaway, Donna Hardaway, Amber Hardaway, Emily Hardaway, Linda Hardaway, Robin Hardaway, Kathleen Hardaway, Leslie Hardaway, Christy Hardaway, Kristen Hardaway, Catherine Hardaway,
Kristin Hardaway, Misty Hardaway, Barbara Hardaway, Heidi Hardaway, Nancy Hardaway, Cheryl Hardaway, Theresa Hardaway, Brandy Hardaway, Alicia Hardaway, Veronica Hardaway, Gina Hardaway, Jacqueline
Hardaway, Rhonda Hardaway, Anna Hardaway, Renee Hardaway, Megan Hardaway, Tamara Hardaway, Kathryn Hardaway, Melinda Hardaway, Debra Hardaway, Sherry Hardaway, Allison Hardaway, Valerie Hardaway,
Diana Hardaway, Paula Hardaway, Kristina Hardaway, Ann Hardaway, Margaret Hardaway, Cindy Hardaway, Victoria Hardaway, Jodi Hardaway, Natalie Hardaway, Brandi Hardaway, Kristi Hardaway, Suzanne
Hardaway, Samantha Hardaway, Beth Hardaway, Tracey Hardaway, Regina Hardaway, Vanessa Hardaway, Kristy Hardaway, Carolyn Hardaway, Yolanda Hardaway, Deanna Hardaway, Carla Hardaway, Sheila Hardaway,
Laurie Hardaway, Anne Hardaway, Shelly Hardaway, Diane Hardaway, Sabrina Hardaway, Janet Hardaway, Katrina Hardaway, Erika Hardaway, Courtney Hardaway, Colleen Hardaway, Carol Hardaway, Julia
Hardaway, Jenny Hardaway, Jaime Hardaway, Kathy Hardaway, Felicia Hardaway, Alison Hardaway, Lauren Hardaway, Kelli Hardaway, Leah Hardaway, Ashley Hardaway, Kim Hardaway, Traci Hardaway, Kristine
Hardaway, Tricia Hardaway, Joy Hardaway, Krista Hardaway, Kara Hardaway, Terri Hardaway, Sonya Hardaway, Aimee Hardaway, Natasha Hardaway, Cassandra Hardaway, Bridget Hardaway, Anita Hardaway, Kari
Hardaway, Nichole Hardaway, Christie Hardaway, Marie Hardaway, Virginia Hardaway, Connie Hardaway, Martha Hardaway, Carmen Hardaway, Stacie Hardaway, Lynn Hardaway, Monique Hardaway, Katie Hardaway,
Kristie Hardaway, Shelley Hardaway, Sherri Hardaway, Angel Hardaway, Bonnie Hardaway, Mandy Hardaway, Jody Hardaway, Shawna Hardaway, Kerry Hardaway, Annette Hardaway, Yvonne Hardaway, Toni Hardaway,
Meredith Hardaway, Molly Hardaway, Kendra Hardaway, Joanna Hardaway, Sonia Hardaway, Janice Hardaway, Robyn Hardaway, Brooke Hardaway, Kerri Hardaway, Sheri Hardaway, Becky Hardaway, Gloria Hardaway,
Mindy Hardaway, Tracie Hardaway, Angie Hardaway, Kellie Hardaway, Claudia Hardaway, Ruth Hardaway, Wanda Hardaway, Jeanette Hardaway, Cathy Hardaway, Adrienne Hardaway, Kelley Hardaway, Rachael
Hardaway, Beverly Hardaway, Candace Hardaway, Sylvia Hardaway, Penny Hardaway, Charlene Hardaway, Trisha Hardaway, Charlotte Hardaway, Belinda Hardaway, Candice Hardaway, Yvette Hardaway, Lindsay
Hardaway, Keri Hardaway, Melody Hardaway, Caroline Hardaway, Rosa Hardaway, Bethany Hardaway, Trina Hardaway, Joyce Hardaway, Joanne Hardaway, Tabitha Hardaway, Vicki Hardaway, Tamika Hardaway,
Gretchen Hardaway, Ginger Hardaway, Betty Hardaway, Dorothy Hardaway, Debbie Hardaway, Darlene Hardaway, Helen Hardaway, Latoya Hardaway, Ellen Hardaway, Leigh Hardaway, Rose Hardaway, Shawn
Hardaway, Karla Hardaway, Frances Hardaway, Ana Hardaway, Alice Hardaway, Jean Hardaway, Hope Hardaway, Tasha Hardaway, Latonya Hardaway, Latasha Hardaway, Nikki Hardaway, Maureen Hardaway, Bobbie
Hardaway, Rita Hardaway, Peggy Hardaway, Shirley Hardaway, Marsha Hardaway, Cara Hardaway, Christa Hardaway, Audrey Hardaway, Norma Hardaway, Shana Hardaway, Juanita Hardaway, Leticia Hardaway,
Kimberley Hardaway, Billie Hardaway, Evelyn Hardaway, Tonia Hardaway, Desiree Hardaway, Judith Hardaway, Joann Hardaway, Shanna Hardaway, Elaine Hardaway, Angelica Hardaway, Charity Hardaway, Staci
Hardaway, Tami Hardaway, Judy Hardaway, Rebekah Hardaway, Raquel Hardaway, Tammie Hardaway, Jane Hardaway, Meghan Hardaway, Jackie Hardaway, Sally Hardaway, Jana Hardaway, Priscilla Hardaway, Sandy
Hardaway, Rochelle Hardaway, Summer Hardaway, Jodie Hardaway, Marcia Hardaway, Alisha Hardaway, Keisha Hardaway, Stefanie Hardaway, Casey Hardaway, Loretta Hardaway, Eva Hardaway, Dena Hardaway,
Cristina Hardaway, Janelle Hardaway, Christi Hardaway, Naomi Hardaway, Rachelle Hardaway, Marisa Hardaway, Irene Hardaway, Kirsten Hardaway, Jeanne Hardaway, Jenifer Hardaway, Roxanne Hardaway,
Miranda Hardaway, Roberta Hardaway, Dina Hardaway, Shauna Hardaway, Sonja Hardaway, Marilyn Hardaway, Eileen Hardaway, Gwendolyn Hardaway, Vickie Hardaway, Katina Hardaway, Teri Hardaway, Olivia
Hardaway, Amie Hardaway, Lora Hardaway, Cheri Hardaway, Jennie Hardaway, Lindsey Hardaway, Lesley Hardaway, Lara Hardaway, Lydia Hardaway, Alisa Hardaway, Sheryl Hardaway, Autumn Hardaway, Jacquelyn
Hardaway, Lynette Hardaway, Esther Hardaway, Nina Hardaway, Antoinette Hardaway, Gail Hardaway, Deana Hardaway, Jaclyn Hardaway, Lorraine Hardaway, Alexandra Hardaway, Jami Hardaway, Ronda Hardaway,
Candy Hardaway, Bobbi Hardaway, Ericka Hardaway, Abigail Hardaway, Chandra Hardaway, Ebony Hardaway, Latisha Hardaway, Cherie Hardaway, Marla Hardaway, Lee Hardaway, Angelia Hardaway, Kenya Hardaway,
Joan Hardaway, Jeannie Hardaway, Shari Hardaway, Ruby Hardaway, Miriam Hardaway, Lakisha Hardaway, Tameka Hardaway, Marcy Hardaway, Angelina Hardaway, Faith Hardaway, Marisol Hardaway, Krystal
Hardaway, Dianna Hardaway, Adriana Hardaway, Audra Hardaway, Angelique Hardaway, Marissa Hardaway, Grace Hardaway, Alyssa Hardaway, Janine Hardaway, Alexis Hardaway, Elisa Hardaway, Jolene Hardaway,
Karin Hardaway, Shelby Hardaway, Wendi Hardaway, Tania Hardaway, Melisa Hardaway, Bernadette Hardaway, Lorena Hardaway, Shelia Hardaway, Rosemary Hardaway, Ramona Hardaway, Terry Hardaway, Nora
Hardaway, Marlene Hardaway, Camille Hardaway, Tanisha Hardaway, Carey Hardaway, Alma Hardaway, Sherrie Hardaway, Cecilia Hardaway, Celeste Hardaway, Kate Hardaway, Betsy Hardaway, Brandie Hardaway,
Annie Hardaway, Darla Hardaway, Lillian Hardaway, Maribel Hardaway, Glenda Hardaway, Chasity Hardaway, Patrice Hardaway, Amelia Hardaway, Lorie Hardaway, Tabatha Hardaway, Elena Hardaway, Lana
Hardaway, Jocelyn Hardaway, Jeannette Hardaway, Guadalupe Hardaway, Marcie Hardaway, Johanna Hardaway, Josephine Hardaway, Tisha Hardaway, Michael Hardaway, Leanne Hardaway, Vivian Hardaway, Whitney
Hardaway, Darcy Hardaway, Hilary Hardaway, Marci Hardaway, Doris Hardaway, Selena Hardaway, Constance Hardaway, Serena Hardaway, Daphne Hardaway, Hollie Hardaway, Bridgette Hardaway, Tammi Hardaway,
Jeanine Hardaway, Terra Hardaway, Margarita Hardaway, Vicky Hardaway, Allyson Hardaway, Marianne Hardaway, Elisabeth Hardaway, Paige Hardaway, Iris Hardaway, Lena Hardaway, Lakeisha Hardaway, Jillian
Hardaway, Arlene Hardaway, Latanya Hardaway, Emma Hardaway, Aisha Hardaway, Tia Hardaway, Alissa Hardaway, Sophia Hardaway, Hannah Hardaway, Lawanda Hardaway, Chrystal Hardaway, Lea Hardaway, Cari
Hardaway, Jessie Hardaway, Mia Hardaway, Chastity Hardaway, Lynda Hardaway, Cassie Hardaway, Marjorie Hardaway, Liza Hardaway, Edith Hardaway, Jenna Hardaway, Joni Hardaway, Misti Hardaway, Blanca
Hardaway, Dionne Hardaway, Lashonda Hardaway, Leann Hardaway, Yesenia Hardaway, Corinne Hardaway, Jasmine Hardaway, Malinda Hardaway, Gabriela Hardaway, Shonda Hardaway, Laurel Hardaway, Hillary
Hardaway, Kimberlee Hardaway, Karrie Hardaway, Irma Hardaway, Pauline Hardaway, Claire Hardaway, Dora Hardaway, Isabel Hardaway, Georgia Hardaway, Esmeralda Hardaway, Nadine Hardaway, Lakesha
Hardaway, Marcella Hardaway, Luz Hardaway, Ladonna Hardaway, Katharine Hardaway, Phyllis Hardaway, Sue Hardaway, Janie Hardaway, Kisha Hardaway, Dianne Hardaway, Myra Hardaway, Gabrielle Hardaway,
Ingrid Hardaway, Maryann Hardaway, Rene Hardaway, Lucy Hardaway, Bridgett Hardaway, Karyn Hardaway, Janel Hardaway, Mandi Hardaway, Paulette Hardaway, Lucinda Hardaway, Carissa Hardaway, Genevieve
Hardaway, Olga Hardaway, Dolores Hardaway, Catina Hardaway, Randi Hardaway, Tera Hardaway, Jeannine Hardaway, Nakia Hardaway, Janette Hardaway, June Hardaway, Alana Hardaway, Annmarie Hardaway,
Lashawn Hardaway, Noelle Hardaway, Chelsea Hardaway, Brittany Hardaway, Clarissa Hardaway, Shanda Hardaway, Cathleen Hardaway, Susana Hardaway, Susanne Hardaway, Chanda Hardaway, Edna Hardaway,
Catrina Hardaway, Kerrie Hardaway, Francine Hardaway, Mona Hardaway, Alyson Hardaway, Doreen Hardaway, Maggie Hardaway, Lynne Hardaway, Tori Hardaway, Clara Hardaway, Demetria Hardaway, Beatrice
Hardaway, Valarie Hardaway, Jeri Hardaway, Kasey Hardaway, Silvia Hardaway, Gena Hardaway, Janna Hardaway, Abby Hardaway, Cristy Hardaway, Deirdre Hardaway, Deidre Hardaway, Tawana Hardaway, Deena
Hardaway, Jan Hardaway, Maricela Hardaway, Tamra Hardaway, Daisy Hardaway, Sondra Hardaway, Caryn Hardaway, Nadia Hardaway, Lorrie Hardaway, Patty Hardaway, Anissa Hardaway, Farrah Hardaway, Bianca
Hardaway, Tiffani Hardaway, Carly Hardaway, Delia Hardaway, Susie Hardaway, Tessa Hardaway, Latrice Hardaway, Shellie Hardaway, Renae Hardaway, Corina Hardaway, Deanne Hardaway, Rena Hardaway, Louise
Hardaway, Larissa Hardaway, Latosha Hardaway, Lois Hardaway, Maritza Hardaway, Julianne Hardaway, Mildred Hardaway, Janell Hardaway, Karina Hardaway, Christal Hardaway, Antonia Hardaway, James
Hardaway, Ida Hardaway, Marina Hardaway, Lacey Hardaway, Kristal Hardaway, Ami Hardaway, Christopher Hardaway, Gladys Hardaway, Cori Hardaway, Gwen Hardaway, Eleanor Hardaway, Elisha Hardaway,
Justine Hardaway, Madeline Hardaway, Ursula Hardaway, Charmaine Hardaway, Rosemarie Hardaway, Suzette Hardaway, Elise Hardaway, Bertha Hardaway, Gayle Hardaway, Lynnette Hardaway, Sheree Hardaway,
Araceli Hardaway, Chantel Hardaway, Kesha Hardaway, Dayna Hardaway, Malissa Hardaway, Dara Hardaway, Latonia Hardaway, Jayme Hardaway, Robert Hardaway, Jerri Hardaway, Carolina Hardaway, Athena
Hardaway, Hilda Hardaway, Ivy Hardaway, Mara Hardaway, Tosha Hardaway, Marion Hardaway, Elsa Hardaway, Stella Hardaway, Devon Hardaway, Marlo Hardaway, Danette Hardaway, Mayra Hardaway, Kay Hardaway,
Bernice Hardaway, Adrianne Hardaway, Geneva Hardaway, Colette Hardaway, David Hardaway, Anitra Hardaway, Letitia Hardaway, Marian Hardaway, Windy Hardaway, Jason Hardaway, Katy Hardaway, Felecia
Hardaway, Elissa Hardaway, Susanna Hardaway, Lucia Hardaway, Celia Hardaway, Margie Hardaway, Kami Hardaway, Adrian Hardaway, Holli Hardaway, Delores Hardaway, Sasha Hardaway, Tamera Hardaway, Tonja
Hardaway, Taryn Hardaway, Stacia Hardaway, John Hardaway, Tomeka Hardaway, Anastasia Hardaway, Marta Hardaway, Patti Hardaway, Jeanna Hardaway, Georgina Hardaway, Eugenia Hardaway, Tawnya Hardaway,
Corey Hardaway, Leeann Hardaway, Juliet Hardaway, Chris Hardaway, Dominique Hardaway, Mitzi Hardaway, Margo Hardaway, Selina Hardaway, Marnie Hardaway, Casandra Hardaway, Deann Hardaway, Leanna
Hardaway, Vera Hardaway, Consuelo Hardaway, Buffy Hardaway, Lourdes Hardaway, Florence Hardaway, Geraldine Hardaway, Tyra Hardaway, Alecia Hardaway, Greta Hardaway, Carole Hardaway, Francesca
Hardaway, Meagan Hardaway, Jeanie Hardaway, Rosalinda Hardaway, Trudy Hardaway, Haley Hardaway, Sandi Hardaway, Corrie Hardaway, Maura Hardaway, Sharonda Hardaway, Graciela Hardaway, Michell
Hardaway, Terrie Hardaway, Eve Hardaway, Tiffanie Hardaway, Rosalind Hardaway, Marlena Hardaway, Shannan Hardaway, Thelma Hardaway, Carie Hardaway, Lashanda Hardaway, Shelli Hardaway, Karie Hardaway,
Danelle Hardaway, Tawanda Hardaway, Ryan Hardaway, Simone Hardaway, Maya Hardaway, Mollie Hardaway, Deidra Hardaway, Josie Hardaway, Noemi Hardaway, Briana Hardaway, Venus Hardaway, Lauri Hardaway,
Richelle Hardaway, Candi Hardaway, Ella Hardaway, Cora Hardaway, Latricia Hardaway, Marisela Hardaway, Beatriz Hardaway, Shanon Hardaway, Alejandra Hardaway, Aubrey Hardaway, Jeana Hardaway, Nanette
Hardaway, Brianne Hardaway, Leona Hardaway, Helena Hardaway, Kris Hardaway, Christian Hardaway, Annemarie Hardaway, Andria Hardaway, Mindi Hardaway, Rosalyn Hardaway, Shanta Hardaway, Mercedes
Hardaway, Polly Hardaway, Juana Hardaway, Rosie Hardaway, Martina Hardaway, Therese Hardaway, Cory Hardaway, Kayla Hardaway, Shantel Hardaway, Joey Hardaway, Shalonda Hardaway, Mellissa Hardaway, Ada
Hardaway, Brandee Hardaway, Sunshine Hardaway, Tamiko Hardaway, Danita Hardaway, Niki Hardaway, Rosanna Hardaway, Joelle Hardaway, Monika Hardaway, Trista Hardaway, Chiquita Hardaway, William
Hardaway, Brook Hardaway, Janis Hardaway, Rae Hardaway, Kizzy Hardaway, Cortney Hardaway, Cecelia Hardaway, Jaimie Hardaway, Christin Hardaway, Christen Hardaway, Tanika Hardaway, Darci Hardaway,
Claudine Hardaway, Sharla Hardaway, Toya Hardaway, Racheal Hardaway, Noel Hardaway, Jada Hardaway, Davina Hardaway, Della Hardaway, Kori Hardaway, Tomika Hardaway, Callie Hardaway, Jamila Hardaway,
Aileen Hardaway, Dusty Hardaway, Rocio Hardaway, Kia Hardaway, Benita Hardaway, Jena Hardaway, Aurora Hardaway, Penelope Hardaway, Sadie Hardaway, Patsy Hardaway, Valencia Hardaway, Lisette Hardaway,
Tamela Hardaway, Mari Hardaway, Tana Hardaway, Ivette Hardaway, Eliza Hardaway, Darcie Hardaway, Tawanna Hardaway, Bonita Hardaway, Debora Hardaway, Lacy Hardaway, Tarsha Hardaway, Stephenie
Hardaway, Johnna Hardaway, Kathrine Hardaway, Lakeshia Hardaway, Griselda Hardaway, Charla Hardaway, Lesa Hardaway, Sunny Hardaway, Celina Hardaway, Karri Hardaway, Corinna Hardaway, Dixie Hardaway,
Rosalie Hardaway, Rhiannon Hardaway, Eunice Hardaway, Mariah Hardaway, Angeline Hardaway, Danna Hardaway, Shayla Hardaway, Brenna Hardaway, Juliana Hardaway, Brianna Hardaway, Octavia Hardaway, Leila
Hardaway, Liliana Hardaway, Shameka Hardaway, Lissette Hardaway, Cary Hardaway, Chrissy Hardaway, Corrine Hardaway, Destiny Hardaway, India Hardaway, Adriane Hardaway, Kandi Hardaway, Coleen
Hardaway, Angelita Hardaway, Denice Hardaway, Janeen Hardaway, Suzanna Hardaway, Dee Hardaway, Magdalena Hardaway, Lola Hardaway, Brian Hardaway, Nikole Hardaway, Sofia Hardaway, Nicolle Hardaway,
Lucretia Hardaway, Mellisa Hardaway, Morgan Hardaway, Alexa Hardaway, Faye Hardaway, Daniel Hardaway, Antionette Hardaway, Tennille Hardaway, Ayanna Hardaway, Maxine Hardaway, Christel Hardaway,
Alysia Hardaway, Alexandria Hardaway, Lenora Hardaway, Aida Hardaway, Kyla Hardaway, Joseph Hardaway, Bree Hardaway, Kira Hardaway, Renita Hardaway, Sharron Hardaway, Jenni Hardaway, Ethel Hardaway,
Fatima Hardaway, Hazel Hardaway, Nellie Hardaway, Juli Hardaway, Delilah Hardaway, Pearl Hardaway, Caren Hardaway, Melonie Hardaway, Robbie Hardaway, Lindy Hardaway, Tangela Hardaway, Trena Hardaway,
Daniela Hardaway, Michaela Hardaway, Carri Hardaway, Sommer Hardaway, Alanna Hardaway, Tressa Hardaway, Georgette Hardaway, Nikita Hardaway, Chanel Hardaway, Aretha Hardaway, Charisse Hardaway,
Laquita Hardaway, Natalia Hardaway, Josette Hardaway, Roxanna Hardaway, Kirstin Hardaway, Shani Hardaway, Nichol Hardaway, Regan Hardaway, Salina Hardaway, Anjanette Hardaway, Susannah Hardaway,
Jammie Hardaway, Kenyatta Hardaway, Jasmin Hardaway, Anthony Hardaway, Ginny Hardaway, Lillie Hardaway, Missy Hardaway, Starla Hardaway, Lucille Hardaway, Lorna Hardaway, Tiana Hardaway, Melodie
Hardaway, Fawn Hardaway, Candida Hardaway, Carin Hardaway, Elvira Hardaway, Shonna Hardaway, Carmela Hardaway, Spring Hardaway, Shara Hardaway, Yolonda Hardaway, Josefina Hardaway, Evette Hardaway,
Carisa Hardaway, Sherita Hardaway, Roseann Hardaway, Saundra Hardaway, Myrna Hardaway, Bambi Hardaway, Layla Hardaway, Lilia Hardaway, Mechelle Hardaway, Cathryn Hardaway, Jayne Hardaway, Lolita
Hardaway, Rosario Hardaway, Eric Hardaway, Valeria Hardaway, Jade Hardaway, Nicki Hardaway, Maranda Hardaway, Felisha Hardaway, Raina Hardaway, Tracee Hardaway, Cindi Hardaway, Freda Hardaway,
Gillian Hardaway, Damaris Hardaway, Ava Hardaway, Brigitte Hardaway, Nicola Hardaway, Esperanza Hardaway, Leilani Hardaway, Sharlene Hardaway, Marni Hardaway, Claudette Hardaway, Keely Hardaway,
Hayley Hardaway, Chantelle Hardaway, Joleen Hardaway, Elsie Hardaway, Rebeca Hardaway, Mendy Hardaway, Adria Hardaway, Harmony Hardaway, Francisca Hardaway, Jolie Hardaway, Vikki Hardaway, Nikia
Hardaway, Tamatha Hardaway, Keli Hardaway, Keshia Hardaway, Charissa Hardaway, Imelda Hardaway, Francis Hardaway, Richard Hardaway, Shantell Hardaway, Shawanda Hardaway, Caitlin Hardaway, Frankie
Hardaway, Shea Hardaway, Cherry Hardaway, Sheena Hardaway, Brigette Hardaway, Keesha Hardaway, Cristi Hardaway, Justina Hardaway, Rashida Hardaway, Emilie Hardaway, Ashlee Hardaway, Shane Hardaway,
Cherise Hardaway, Qiana Hardaway, Latarsha Hardaway, Crissy Hardaway, Celena Hardaway, Kylie Hardaway, Rolanda Hardaway, Crista Hardaway, Roslyn Hardaway, Angella Hardaway, Shamika Hardaway, Wilma
Hardaway, Violet Hardaway, Kyra Hardaway, Kevin Hardaway, Marybeth Hardaway, Alesha Hardaway, Matthew Hardaway, Shanika Hardaway, Shawnda Hardaway, Cristal Hardaway, Julissa Hardaway, Lila Hardaway,
Alycia Hardaway, Camilla Hardaway, Johnnie Hardaway, Dori Hardaway, Nichelle Hardaway, Venessa Hardaway, Anika Hardaway, Lily Hardaway, Tamica Hardaway, Rhoda Hardaway, Alethea Hardaway, Carina
Hardaway, Roxann Hardaway, Latrina Hardaway, Charles Hardaway, Bessie Hardaway, Julianna Hardaway, Marquita Hardaway, Gayla Hardaway, Kandy Hardaway, Estella Hardaway, Marguerite Hardaway, Chana
Hardaway, Robbin Hardaway, Cristin Hardaway, Raven Hardaway, Tanesha Hardaway, Alina Hardaway, Juliette Hardaway, Milagros Hardaway, Inez Hardaway, Vonda Hardaway, Jenelle Hardaway, Elaina Hardaway,
Dedra Hardaway, Verna Hardaway, Starr Hardaway, Bobby Hardaway, Lizette Hardaway, Dawna Hardaway, Danyelle Hardaway, Petra Hardaway, Stephaine Hardaway, Angelic Hardaway, Brittney Hardaway, Corrina
Hardaway, Britt Hardaway, Stephany Hardaway, Lorri Hardaway, Shayna Hardaway, Flora Hardaway, Jesse Hardaway, Velma Hardaway, Racquel Hardaway, Stefani Hardaway, Phoebe Hardaway, Yadira Hardaway,
Liana Hardaway, Milissa Hardaway, Pam Hardaway, Zoe Hardaway, Libby Hardaway, Shasta Hardaway, Luisa Hardaway, Barbra Hardaway, Janae Hardaway, Shandra Hardaway, Thomas Hardaway, Kelsey Hardaway,
Shante Hardaway, Bettina Hardaway, Lashunda Hardaway, Melina Hardaway, Mattie Hardaway, Meaghan Hardaway, Timothy Hardaway, Rosetta Hardaway, Mistie Hardaway, Steven Hardaway, Patrica Hardaway, Shay
Hardaway, Takisha Hardaway, Geri Hardaway, Teressa Hardaway, Aaron Hardaway, Clare Hardaway, Noreen Hardaway, Adina Hardaway, Rosanne Hardaway, Luciana Hardaway, Nakisha Hardaway, Trinity Hardaway,
Cicely Hardaway, Minerva Hardaway, Devin Hardaway, Gabriella Hardaway, Lela Hardaway, Kendall Hardaway, Treva Hardaway, Hallie Hardaway, Kristan Hardaway, Lia Hardaway, Viola Hardaway, Portia
Hardaway, Jonna Hardaway, Kellee Hardaway, Danica Hardaway, Kristyn Hardaway, Lina Hardaway, Jewel Hardaway, Tamala Hardaway, Jenniffer Hardaway, Melony Hardaway, Kacey Hardaway, Quiana Hardaway,
Shawnna Hardaway, Rana Hardaway, Dannielle Hardaway, Andra Hardaway, Lenore Hardaway, Carlene Hardaway, Malia Hardaway, Lanette Hardaway, Jacquline Hardaway, Willie Hardaway, Precious Hardaway,
Collette Hardaway, Mariana Hardaway, Ayana Hardaway, Daniella Hardaway, Thea Hardaway, Viviana Hardaway, Kindra Hardaway, Gracie Hardaway, Louisa Hardaway, Adele Hardaway, Annamarie Hardaway, Kiesha
Hardaway, Sherie Hardaway, Althea Hardaway, Catalina Hardaway, January Hardaway, Ernestine Hardaway, Corie Hardaway, Lesli Hardaway, Shanita Hardaway, Tresa Hardaway, Maryanne Hardaway, Tamar
Hardaway, Rayna Hardaway, Siobhan Hardaway, Demetra Hardaway, Kimberlie Hardaway, Rhea Hardaway, Elva Hardaway, Elvia Hardaway, Karissa Hardaway, Maryellen Hardaway, Valorie Hardaway, Adrianna
Hardaway, Nicolette Hardaway, Daniele Hardaway, Trish Hardaway, Twila Hardaway, Abbie Hardaway, Towanda Hardaway, Lorinda Hardaway, Marcela Hardaway, Kyle Hardaway, Carmelita Hardaway, Helene
Hardaway, Mark Hardaway, Jeffrey Hardaway, Lidia Hardaway, Kenyetta Hardaway, Lupe Hardaway, Cami Hardaway, Cher Hardaway, Minnie Hardaway, Toby Hardaway, Karena Hardaway, Katheryn Hardaway, Renea
Hardaway, Kathie Hardaway, Laverne Hardaway, Shona Hardaway, Linette Hardaway, Kecia Hardaway, Elana Hardaway, Eboni Hardaway, Loren Hardaway, Tabetha Hardaway, Dione Hardaway, Glenna Hardaway, Marti
Hardaway, Gia Hardaway, Tena Hardaway, Cameron Hardaway, Roxana Hardaway, Alberta Hardaway, Dolly Hardaway, Pamala Hardaway, Talia Hardaway, Tobi Hardaway, Latesha Hardaway, Larhonda Hardaway, Sybil
Hardaway, Kandice Hardaway, Yasmin Hardaway, Christiana Hardaway, Sydney Hardaway, Chantal Hardaway, Estela Hardaway, Jaimee Hardaway, Salena Hardaway, Tamekia Hardaway, Dalia Hardaway, Twana
Hardaway, Jacklyn Hardaway, Scarlett Hardaway, Tanja Hardaway, Kenisha Hardaway, Daria Hardaway, Agnes Hardaway, Mandie Hardaway, Lani Hardaway, May Hardaway, Sharee Hardaway, Tenisha Hardaway, Twyla
Hardaway, Chantell Hardaway, Shiela Hardaway, Isela Hardaway, Karey Hardaway, Alaina Hardaway, Loriann Hardaway, Star Hardaway, Samara Hardaway, Janene Hardaway, Gerri Hardaway, Lasonya Hardaway,
Lavonne Hardaway, Rikki Hardaway, Meridith Hardaway, Lorene Hardaway, Scott Hardaway, Stephani Hardaway, Kimberli Hardaway, Zandra Hardaway, Sallie Hardaway, Cecily Hardaway, Rachele Hardaway,
Melynda Hardaway, Reagan Hardaway, Carmella Hardaway, Danyell Hardaway, Ivonne Hardaway, Jamey Hardaway, Shawana Hardaway, Aja Hardaway, Casie Hardaway, Alena Hardaway, Barbie Hardaway, Clarice
Hardaway, Kassandra Hardaway, Tiffiny Hardaway, Madelyn Hardaway, Cherish Hardaway, Lavonda Hardaway, Malisa Hardaway, Stormy Hardaway, Amalia Hardaway, Machelle Hardaway, Abbey Hardaway, Evangelina
Hardaway, Serina Hardaway, Fannie Hardaway, Joi Hardaway, Tarra Hardaway, Dale Hardaway, Shaunna Hardaway, Jeremy Hardaway, Sarina Hardaway, Harriet Hardaway, Holley Hardaway, Reyna Hardaway, Ayesha
Hardaway, Natosha Hardaway, Princess Hardaway, Shavon Hardaway, Ashleigh Hardaway, Laticia Hardaway, Cristine Hardaway, Asia Hardaway, Antonette Hardaway, Dinah Hardaway, Jordan Hardaway, Cherilyn
Hardaway, Mackenzie Hardaway, Giselle Hardaway, Edwina Hardaway, Mae Hardaway, Kathi Hardaway, Shalanda Hardaway, Donita Hardaway, Melissia Hardaway, Cristie Hardaway, Mirna Hardaway, Cheyenne
Hardaway, Leslee Hardaway, Jeni Hardaway, Jacinda Hardaway, Consuela Hardaway, Evangeline Hardaway, Joanie Hardaway, Nickie Hardaway, Jolynn Hardaway, Malika Hardaway, Nathalie Hardaway, Mamie
Hardaway, Paul Hardaway, Davida Hardaway, Erinn Hardaway, Laronda Hardaway, Latina Hardaway, Liberty Hardaway, Nicol Hardaway, Danyel Hardaway, Tommie Hardaway, Joshua Hardaway, Shaun Hardaway, Jodee
Hardaway, Teena Hardaway, Shanell Hardaway, Calandra Hardaway, Iesha Hardaway, Devona Hardaway, Aracely Hardaway, Brandon Hardaway, Dona Hardaway, Micah Hardaway, Adela Hardaway, Lashon Hardaway,
Reina Hardaway, Sierra Hardaway, Addie Hardaway, Carlotta Hardaway, Nia Hardaway, Ariana Hardaway, Chante Hardaway, Tamisha Hardaway, Ali Hardaway, Deedee Hardaway, Sarita Hardaway, Jeniffer
Hardaway, Kiley Hardaway, Ronnie Hardaway, Cherri Hardaway, Nickole Hardaway, Hattie Hardaway, Syreeta Hardaway, Chaya Hardaway, Dani Hardaway, Catharine Hardaway, Tamie Hardaway, Perla Hardaway,
Billiejo Hardaway, Deloris Hardaway, Lashawnda Hardaway, Meggan Hardaway, Temeka Hardaway, Dorothea Hardaway, Marianna Hardaway, Lou Hardaway, Tesha Hardaway, Karmen Hardaway, Luann Hardaway, Rona
Hardaway, Elyse Hardaway, Latashia Hardaway, Dacia Hardaway, Nita Hardaway, Tashia Hardaway, Kenneth Hardaway, Mireya Hardaway, Carleen Hardaway, Alfreda Hardaway, Candie Hardaway, Delicia Hardaway,
Lakiesha Hardaway, Renata Hardaway, Juliann Hardaway, Patience Hardaway, Alexia Hardaway, Lakeesha Hardaway, Petrina Hardaway, Liane Hardaway, Ester Hardaway, Kimberely Hardaway, Millie Hardaway,
Roni Hardaway, Reba Hardaway, Tory Hardaway, Maryjo Hardaway, Aundrea Hardaway, Kasandra Hardaway, Melba Hardaway, Tiffaney Hardaway, Maisha Hardaway, Kenna Hardaway, Detra Hardaway, Jerry Hardaway,
Meghann Hardaway, Sanjuanita Hardaway, Emilia Hardaway, Denita Hardaway, Shaunda Hardaway, Shavonne Hardaway, Jose Hardaway, Roseanne Hardaway, Kiersten Hardaway, Sophie Hardaway, Maren Hardaway,
Giovanna Hardaway, Bethann Hardaway, Ilene Hardaway, Dorian Hardaway, Lanita Hardaway, Ariel Hardaway, Piper Hardaway, Shawnee Hardaway, Cammie Hardaway, Joellen Hardaway, Millicent Hardaway, Rachell
Hardaway, Billy Hardaway, Felica Hardaway, Jacinta Hardaway, Krissy Hardaway, Letisha Hardaway, Pennie Hardaway, Delisa Hardaway, Evonne Hardaway, Keysha Hardaway, Migdalia Hardaway, Opal Hardaway,
Taunya Hardaway, Farah Hardaway, Ina Hardaway, Samatha Hardaway, Linnea Hardaway, Lyn Hardaway, Merry Hardaway, Patrina Hardaway, Terese Hardaway, Leandra Hardaway, Cornelia Hardaway, Manda Hardaway,
Lissa Hardaway, Coretta Hardaway, Gidget Hardaway, Demetrice Hardaway, Mabel Hardaway, Necole Hardaway, Torrie Hardaway, Kandace Hardaway, Loraine Hardaway, Tarah Hardaway, Cody Hardaway, Cinnamon
Hardaway, Shonta Hardaway, Rosalia Hardaway, Delana Hardaway, Nannette Hardaway, Amee Hardaway, Ofelia Hardaway, Shakira Hardaway, Aletha Hardaway, Kacy Hardaway, Rosita Hardaway, Talisha Hardaway,
Moira Hardaway, Sebrina Hardaway, Katrice Hardaway, Eden Hardaway, Jaqueline Hardaway, Kirstie Hardaway, Gregory Hardaway, Letha Hardaway, Apryl Hardaway, Kristel Hardaway, Fabiola Hardaway,
Steffanie Hardaway, Elicia Hardaway, Nedra Hardaway, Maia Hardaway, Sharita Hardaway, Tammara Hardaway, Charise Hardaway, Cheree Hardaway, Andrew Hardaway, Honey Hardaway, Velvet Hardaway, Anglea
Hardaway, Brett Hardaway, Henrietta Hardaway, Jenine Hardaway, Meg Hardaway, Khalilah Hardaway, Sheronda Hardaway, Dyan Hardaway, Marty Hardaway, Suzan Hardaway, Liz Hardaway, Brande Hardaway, Flor
Hardaway, Tameika Hardaway, Manuela Hardaway, Stephen Hardaway, Kamilah Hardaway, Karolyn Hardaway, Demetrius Hardaway, Lacie Hardaway, Lakeysha Hardaway, Jannette Hardaway, Vonetta Hardaway, Gale
Hardaway, Shiloh Hardaway, Shira Hardaway, Gisela Hardaway, Jacquelin Hardaway, Natisha Hardaway, Billi Hardaway, Charleen Hardaway, Brigid Hardaway, Dorinda Hardaway, Jacki Hardaway, Keena Hardaway,
Lamonica Hardaway, Shena Hardaway, Michel Hardaway, Nanci Hardaway, Donya Hardaway, Leisa Hardaway, Rosalba Hardaway, Anabel Hardaway, Annalisa Hardaway, Amity Hardaway, Mimi Hardaway, Erma Hardaway,
Lyndsey Hardaway, Marlana Hardaway, Skye Hardaway, Chloe Hardaway, Inga Hardaway, Jonathan Hardaway, Melaine Hardaway, Lorelei Hardaway, Marivel Hardaway, Nelly Hardaway, Amiee Hardaway, Cathrine
Hardaway, Jonelle Hardaway, Sacha Hardaway, Jennette Hardaway, Lilly Hardaway, Shawnta Hardaway, Tisa Hardaway, Judi Hardaway, Cherrie Hardaway, Annetta Hardaway, Charlette Hardaway, Christiane
Hardaway, Jesica Hardaway, Pilar Hardaway, Elida Hardaway, Twanna Hardaway, Madonna Hardaway, Tausha Hardaway, Carletta Hardaway, Leesa Hardaway, Matilda Hardaway, Nakita Hardaway, Natashia Hardaway,
Kaci Hardaway, Mica Hardaway, Dionna Hardaway, Margot Hardaway, Maribeth Hardaway, Darby Hardaway, Shondra Hardaway, Socorro Hardaway, Misha Hardaway, Taylor Hardaway, Becki Hardaway, Kiana Hardaway,
Valentina Hardaway, Ashlie Hardaway, Desirae Hardaway, Lizabeth Hardaway, Alesia Hardaway, Roxane Hardaway, Donielle Hardaway, Kelle Hardaway, Larisa Hardaway, Paola Hardaway, Aleta Hardaway, Dorene
Hardaway, Kitty Hardaway, Shenita Hardaway, Tatiana Hardaway, Felisa Hardaway, Yashica Hardaway, Jeanetta Hardaway, Tony Hardaway, Latrisha Hardaway, Nyree Hardaway, Ronald Hardaway, Romona Hardaway,
Shawanna Hardaway, Edward Hardaway, Justin Hardaway, Tuesday Hardaway, Avis Hardaway, Edie Hardaway, Aurelia Hardaway, Nova Hardaway, Ruthie Hardaway, Estelle Hardaway, Mikki Hardaway, Jamee
Hardaway, Micaela Hardaway, Phaedra Hardaway, Sean Hardaway, Fiona Hardaway, Shilo Hardaway, Trudi Hardaway, Blythe Hardaway, Julee Hardaway, Alia Hardaway, Lachelle Hardaway, Roseanna Hardaway,
Donald Hardaway, Nydia Hardaway, Effie Hardaway, Jenell Hardaway, Carolee Hardaway, Karma Hardaway, Violeta Hardaway, Ilana Hardaway, Leonor Hardaway, Talitha Hardaway, Deedra Hardaway, Jimmie
Hardaway, Buffie Hardaway, Iva Hardaway, Joycelyn Hardaway, Randa Hardaway, Tawny Hardaway, Bonny Hardaway, Dallas Hardaway, Felice Hardaway, Lawana Hardaway, Corine Hardaway, Kizzie Hardaway, Ladawn
Hardaway, Torri Hardaway, Marva Hardaway, Sabina Hardaway, Taneka Hardaway, Deandra Hardaway, Kary Hardaway, Juliane Hardaway, Asha Hardaway, Mika Hardaway, Twanda Hardaway, Dawne Hardaway, Ellie
Hardaway, Lawanna Hardaway, Rowena Hardaway, Caron Hardaway, Chad Hardaway, Ronna Hardaway, Tatum Hardaway, Caprice Hardaway, Cinthia Hardaway, Debby Hardaway, Deeann Hardaway, Shalon Hardaway,
Ernestina Hardaway, Andre Hardaway, Annamaria Hardaway, Latoshia Hardaway, Shemeka Hardaway, Stacee Hardaway, Tyesha Hardaway, Annabelle Hardaway, Germaine Hardaway, Risa Hardaway, Latoria Hardaway,
Patrick Hardaway, Wendie Hardaway, Laureen Hardaway, Cristen Hardaway, Kati Hardaway, Niccole Hardaway, Sabra Hardaway, Shoshana Hardaway, Tequila Hardaway, Melva Hardaway, Micki Hardaway, Darcey
Hardaway, Coral Hardaway, Fonda Hardaway, Lashaunda Hardaway, Pepper Hardaway, Trinette Hardaway, Eloisa Hardaway, Goldie Hardaway, Tijuana Hardaway, Venita Hardaway, Danae Hardaway, Tessie Hardaway,
Rebecka Hardaway, Taisha Hardaway, Heide Hardaway, Joslyn Hardaway, Rebbecca Hardaway, Magda Hardaway, Roxie Hardaway, Daniell Hardaway, Kellye Hardaway, Timika Hardaway, Dannette Hardaway, Lakita
Hardaway, Shawnte Hardaway, Dusti Hardaway, Shenika Hardaway, Amberly Hardaway, Koren Hardaway, Carolynn Hardaway, Thomasina Hardaway, Bobbiejo Hardaway, Candance Hardaway, Jacque Hardaway, Nikisha
Hardaway, Torie Hardaway, Cyndi Hardaway, Diann Hardaway, Marylou Hardaway, Nelda Hardaway, Prudence Hardaway, Sharyn Hardaway, Carman Hardaway, Coreen Hardaway, Amina Hardaway, Neva Hardaway,
Scarlet Hardaway, Ammie Hardaway, Haydee Hardaway, Jaymie Hardaway, Rina Hardaway, Lyndsay Hardaway, Shantelle Hardaway, Alyce Hardaway, Danika Hardaway, Leeanne Hardaway, Tamesha Hardaway, Iliana
Hardaway, Kylee Hardaway, Lisha Hardaway, Mercy Hardaway, Nona Hardaway, Teresita Hardaway, Xochitl Hardaway, Carmel Hardaway, Jessi Hardaway, Kourtney Hardaway, Elia Hardaway, Lasandra Hardaway,
Alisia Hardaway, Candis Hardaway, Nicholle Hardaway, Nissa Hardaway, Micheal Hardaway, Kali Hardaway, Tajuana Hardaway, Aleshia Hardaway, Daina Hardaway, Fay Hardaway, Rivka Hardaway, Jennefer
Hardaway, Kena Hardaway, Cassidy Hardaway, Toshia Hardaway, Larry Hardaway, Travis Hardaway, Chaka Hardaway, Anya Hardaway, Leia Hardaway, Nola Hardaway, Rosalva Hardaway, Arica Hardaway, Hanna
Hardaway, Kenda Hardaway, Tiesha Hardaway, Jewell Hardaway, Roshanda Hardaway, Arianne Hardaway, Eryn Hardaway, Denna Hardaway, George Hardaway, Jerilyn Hardaway, Lakia Hardaway, Melani Hardaway,
Raeann Hardaway, Season Hardaway, Sherilyn Hardaway, Lizbeth Hardaway, Myesha Hardaway, Suzy Hardaway, Emilee Hardaway, Kristian Hardaway, Leighann Hardaway, Akilah Hardaway, Britney Hardaway, Juan
Hardaway, Bobbijo Hardaway, Dalila Hardaway, Ileana Hardaway, Contessa Hardaway, Darleen Hardaway, Deshawn Hardaway, Audrea Hardaway, Kawana Hardaway, Lona Hardaway, Margret Hardaway, Melany
Hardaway, Michal Hardaway, Odessa Hardaway, Reva Hardaway, Lula Hardaway, Mariann Hardaway, Kanika Hardaway, Paris Hardaway, Shaneka Hardaway, Sherrell Hardaway, Suzann Hardaway, Chelsey Hardaway,
Laquisha Hardaway, Willow Hardaway, Kaye Hardaway, Muriel Hardaway, Britta Hardaway, Dania Hardaway, Jesenia Hardaway, Nilda Hardaway, Dottie Hardaway, Nereida Hardaway, Kellyann Hardaway, Alishia
Hardaway, Denisha Hardaway, Marietta Hardaway, Merideth Hardaway, Enid Hardaway, Isis Hardaway, Rosio Hardaway, America Hardaway, Belen Hardaway, Randy Hardaway, Karol Hardaway, Kasie Hardaway,
Julieann Hardaway, Sari Hardaway, Bronwyn Hardaway, Concepcion Hardaway, Criselda Hardaway, Natacha Hardaway, Ariane Hardaway, Breanna Hardaway, Shayne Hardaway, Alysha Hardaway, Gary Hardaway,
Rasheeda Hardaway, Sherice Hardaway, Tonisha Hardaway, Katrena Hardaway, Shemika Hardaway, Sirena Hardaway, Breanne Hardaway, Simona Hardaway, Lakenya Hardaway, Winona Hardaway, Adam Hardaway,
Caridad Hardaway, Deonna Hardaway, Kimber Hardaway, Lashaun Hardaway, Maile Hardaway, Tomekia Hardaway, Bernadine Hardaway, Miesha Hardaway, Adelina Hardaway, Tiara Hardaway, Selene Hardaway, Shae
Hardaway, Tomi Hardaway, Laila Hardaway, Keith Hardaway, Lilian Hardaway, Sherika Hardaway, Mariela Hardaway, Moriah Hardaway, Page Hardaway, Raegan Hardaway, Timeka Hardaway, Tonda Hardaway, Danya
Hardaway, Kacie Hardaway, Sabrena Hardaway, Shala Hardaway, Yalonda Hardaway, Jermaine Hardaway, Ranee Hardaway, Sherron Hardaway, Tameca Hardaway, Winter Hardaway, Yulanda Hardaway, Gabriel
Hardaway, Johna Hardaway, Keya Hardaway, Tobie Hardaway, Johnetta Hardaway, Rochell Hardaway, Celestine Hardaway, Christinia Hardaway, Kymberly Hardaway, Vernita Hardaway, Jina Hardaway, Lashawna
Hardaway, Malena Hardaway, Gilda Hardaway, Somer Hardaway, Aliza Hardaway, Christene Hardaway, Marilee Hardaway, Ricki Hardaway, Tiffiney Hardaway, Winifred Hardaway, Antonio Hardaway, Cassondra
Hardaway, Suzie Hardaway, Etta Hardaway, Toi Hardaway, Ashli Hardaway, Charis Hardaway, Isabelle Hardaway, Jayna Hardaway, Nena Hardaway, Nisha Hardaway, Jenise Hardaway, Lizzie Hardaway, Maegan
Hardaway, Contina Hardaway, Joe Hardaway, Sunday Hardaway, Tanna Hardaway, Eleni Hardaway, Mirella Hardaway, Concetta Hardaway, Jeneen Hardaway, Nerissa Hardaway, Bethanie Hardaway, Latoyia Hardaway,
Maryjane Hardaway, Treena Hardaway, Benjamin Hardaway, Keturah Hardaway, Paulina Hardaway, Rasheda Hardaway, Shaunta Hardaway, Vickey Hardaway, Ashanti Hardaway, Georgiana Hardaway, Noelia Hardaway,
Georgianna Hardaway, Troy Hardaway, Yulonda Hardaway, Carley Hardaway, Stephine Hardaway, Oralia Hardaway, Pricilla Hardaway, Ranae Hardaway, Lory Hardaway, Africa Hardaway, Lekisha Hardaway, Rosaura
Hardaway, Shelita Hardaway, Louann Hardaway, Shanti Hardaway, Takesha Hardaway, Tonie Hardaway, Santa Hardaway, Zulema Hardaway, Dahlia Hardaway, Jacquelynn Hardaway, Karon Hardaway, Teisha Hardaway,
Carlos Hardaway, Jerrie Hardaway, Shyla Hardaway, Jacqulyn Hardaway, Janee Hardaway, Lupita Hardaway, Ora Hardaway, Pia Hardaway, Dodie Hardaway, Mariam Hardaway, Meegan Hardaway, Shanelle Hardaway,
Chanelle Hardaway, Janetta Hardaway, Chanell Hardaway, Hiedi Hardaway, Lesia Hardaway, Nikkia Hardaway, Ivory Hardaway, Karry Hardaway, Marcey Hardaway, Theodora Hardaway, Lonnie Hardaway, Kai
Hardaway, Mina Hardaway, Seana Hardaway, Kathlene Hardaway, Melvina Hardaway, Temika Hardaway, Kala Hardaway, Leta Hardaway, Shaunte Hardaway, Sheron Hardaway, Tameeka Hardaway, Eloise Hardaway,
Tatanisha Hardaway, Catrice Hardaway, Malaika Hardaway, Raymond Hardaway, Sharhonda Hardaway, Tandra Hardaway, Tess Hardaway, Linsey Hardaway, Lottie Hardaway, Miki Hardaway, Ronni Hardaway, Tiffini
Hardaway, Jenee Hardaway, Alta Hardaway, Elba Hardaway, Jamillah Hardaway, Jannie Hardaway, Loree Hardaway, Roselyn Hardaway, Winnie Hardaway, Alvina Hardaway, Dorcas Hardaway, Laci Hardaway, Merri
Hardaway, Mickie Hardaway, Penni Hardaway, Allegra Hardaway, Dreama Hardaway, Nidia Hardaway, Pattie Hardaway, Gianna Hardaway, Kimberlyn Hardaway, Roshonda Hardaway, Joel Hardaway, Nettie Hardaway,
Blair Hardaway, Terina Hardaway, Ebonie Hardaway, Kera Hardaway, Ruthann Hardaway, Randee Hardaway, Bryan Hardaway, Jolyn Hardaway, Stephannie Hardaway, Delta Hardaway, Janella Hardaway, Laquanda
Hardaway, Lonna Hardaway, Marcelle Hardaway, Tavia Hardaway, Terrell Hardaway, Chasidy Hardaway, Danell Hardaway, Khadijah Hardaway, Neely Hardaway, Nicholas Hardaway, Nicky Hardaway, Jennine
Hardaway, Tomica Hardaway, Xiomara Hardaway, Deneen Hardaway, Magaly Hardaway, Zoraida Hardaway, Marquetta Hardaway, Tianna Hardaway, Annabel Hardaway, Brie Hardaway, Caterina Hardaway, Dennise
Hardaway, Madeleine Hardaway, Marika Hardaway, Angelena Hardaway, Doretha Hardaway, Quintina Hardaway, Rashonda Hardaway, Carlie Hardaway, Krysta Hardaway, Lianne Hardaway, Martine Hardaway, Nacole
Hardaway, Regena Hardaway, Jeanene Hardaway, Kassie Hardaway, Larae Hardaway, Leana Hardaway, Ronica Hardaway, Tenesha Hardaway, Deeanna Hardaway, Echo Hardaway, Melita Hardaway, Johnny Hardaway,
Lorianne Hardaway, Makeba Hardaway, Nakesha Hardaway, Zenaida Hardaway, Ophelia Hardaway, Gennifer Hardaway, Rasheedah Hardaway, Jenette Hardaway, Lashana Hardaway, Lashandra Hardaway, Blanche
Hardaway, Kina Hardaway, Dia Hardaway, Emmy Hardaway, Janina Hardaway, Krishna Hardaway, Shawnette Hardaway, Wenona Hardaway, Hailey Hardaway, Marita Hardaway, Charita Hardaway, Diedra Hardaway,
Annika Hardaway, Gay Hardaway, Karima Hardaway, Keila Hardaway, Lajuana Hardaway, Marya Hardaway, Omayra Hardaway, Carl Hardaway, Gricelda Hardaway, Lynsey Hardaway, Jama Hardaway, Lynelle Hardaway,
Telisha Hardaway, Todd Hardaway, Camie Hardaway, Luanne Hardaway, Mira Hardaway, Raylene Hardaway, Rosana Hardaway, Nakeisha Hardaway, Brea Hardaway, Leisha Hardaway, Maira Hardaway, Shera Hardaway,
Tangie Hardaway, Tari Hardaway, Treasa Hardaway, Angle Hardaway, Jaimi Hardaway, Joie Hardaway, Ryann Hardaway, Sueann Hardaway, Tiffney Hardaway, Alida Hardaway, Jackeline Hardaway, Tonika Hardaway,
Lakeitha Hardaway, Lezlie Hardaway, Stephania Hardaway, Zulma Hardaway, Candra Hardaway, Charlie Hardaway, Aleisha Hardaway, Andreana Hardaway, Berta Hardaway, Deon Hardaway, Gigi Hardaway, Meri
Hardaway, Cherice Hardaway, Mya Hardaway, Rani Hardaway, Sunni Hardaway, Felicity Hardaway, Marcus Hardaway, Myisha Hardaway, Eureka Hardaway, Gertrude Hardaway, Lavette Hardaway, Karisa Hardaway,
Lacresha Hardaway, Olympia Hardaway, Bryn Hardaway, Frank Hardaway, Jovita Hardaway, Leola Hardaway, Augusta Hardaway, Karlene Hardaway, Shaina Hardaway, Tenille Hardaway, Fanny Hardaway, Katerina
Hardaway, Lainie Hardaway, Natalee Hardaway, Tandy Hardaway, Thalia Hardaway, Elda Hardaway, Jacalyn Hardaway, Beckie Hardaway, Chenoa Hardaway, Leeanna Hardaway, Sharie Hardaway, Tyisha Hardaway,
Zenobia Hardaway, Diedre Hardaway, Charline Hardaway, Janey Hardaway, Shanan Hardaway, Keva Hardaway, Samuel Hardaway, Sandie Hardaway, Zabrina Hardaway, Earlene Hardaway, Jacy Hardaway, Janessa
Hardaway, Sarai Hardaway, Sharice Hardaway, Tambra Hardaway, Cammy Hardaway, Lashun Hardaway, Leonora Hardaway, Dennis Hardaway, Lelia Hardaway, Cherita Hardaway, Courtenay Hardaway, Keira Hardaway,
Tommy Hardaway, Genia Hardaway, Jordana Hardaway, Kaylene Hardaway, Loni Hardaway, Mickey Hardaway, Sheba Hardaway, Sherell Hardaway, Anisa Hardaway, Camelia Hardaway, Domenica Hardaway, Ines
Hardaway, Nathan Hardaway, Sherrill Hardaway, Delma Hardaway, Essie Hardaway, Lavon Hardaway, Marilynn Hardaway, Myrtle Hardaway, Takiyah Hardaway, Alba Hardaway, Delena Hardaway, Leyla Hardaway,
Lynell Hardaway, Tayna Hardaway, Jazmin Hardaway, Vannessa Hardaway, Awilda Hardaway, Carma Hardaway, Derrick Hardaway, Kasi Hardaway, Malka Hardaway, Carli Hardaway, Cordelia Hardaway, Jesus
Hardaway, Kristeen Hardaway, Chrissie Hardaway, Guillermina Hardaway, Joana Hardaway, Lauralee Hardaway, Serenity Hardaway, Alethia Hardaway, Dulce Hardaway, Lavinia Hardaway, Nilsa Hardaway, Tinisha
Hardaway, Veda Hardaway, Demetris Hardaway, Jamilah Hardaway, Janean Hardaway, Soledad Hardaway, Talisa Hardaway, Yael Hardaway, Adeline Hardaway, Amparo Hardaway, Shannah Hardaway, Anel Hardaway,
Corissa Hardaway, Myriam Hardaway, September Hardaway, Shawntel Hardaway, Camisha Hardaway, Cathie Hardaway, Reanna Hardaway, Shavonda Hardaway, Azure Hardaway, Darnell Hardaway, Delphine Hardaway,
Keren Hardaway, Lashondra Hardaway, Yajaira Hardaway, Yessenia Hardaway, Adelaida Hardaway, Aura Hardaway, Dorthy Hardaway, Karly Hardaway, Latonja Hardaway, Mendi Hardaway, Wende Hardaway, Adriene
Hardaway, Arin Hardaway, Cinda Hardaway, Jeanmarie Hardaway, Khristina Hardaway, Rory Hardaway, Shanel Hardaway, Zena Hardaway, Astrid Hardaway, Darice Hardaway, Melia Hardaway, Nell Hardaway, Sage
Hardaway, Tedra Hardaway, Tyler Hardaway, Inger Hardaway, Lekesha Hardaway, Celine Hardaway, Marchelle Hardaway, Shaundra Hardaway, Bethanne Hardaway, Kathryne Hardaway, Yanira Hardaway, Jacquetta
Hardaway, Jessika Hardaway, Niesha Hardaway, Shantae Hardaway, Tira Hardaway, Elke Hardaway, Freida Hardaway, Gretta Hardaway, Kerstin Hardaway, San Hardaway, Suzannah Hardaway, Tamira Hardaway, Zina
Hardaway, Charmain Hardaway, Cherisse Hardaway, Kylene Hardaway, Lacretia Hardaway, Mavis Hardaway, Telisa Hardaway, Alysa Hardaway, Carlita Hardaway, Christianne Hardaway, Franchesca Hardaway,
Joetta Hardaway, Lashelle Hardaway, Rosamaria Hardaway, Seema Hardaway, Youlanda Hardaway, Clarisa Hardaway, Donella Hardaway, Markita Hardaway, Mesha Hardaway, Poppy Hardaway, Arleen Hardaway,
Cecile Hardaway, Krisha Hardaway, Mckenzie Hardaway, Sharmaine Hardaway, Willa Hardaway, Anessa Hardaway, Donetta Hardaway, Donnetta Hardaway, Leora Hardaway, Christena Hardaway, Dorie Hardaway,
Eddie Hardaway, Herlinda Hardaway, Lise Hardaway, Rosalee Hardaway, Temple Hardaway, Tonette Hardaway, Cheryle Hardaway, Cleo Hardaway, Elma Hardaway, Erlinda Hardaway, Peter Hardaway, Sparkle
Hardaway, Evie Hardaway, Lavina Hardaway, Marna Hardaway, Meranda Hardaway, Tynisha Hardaway, Alyse Hardaway, Anisha Hardaway, Bevin Hardaway, Glory Hardaway, Jonnie Hardaway, Marinda Hardaway,
Natascha Hardaway, Shanetta Hardaway, Angelle Hardaway, Dion Hardaway, Quinn Hardaway, Shanette Hardaway, Shaniqua Hardaway, Shantay Hardaway, Tamarra Hardaway, Diona Hardaway, Hortencia Hardaway,
Latara Hardaway, Lisamarie Hardaway, Sidney Hardaway, Alessandra Hardaway, Idalia Hardaway, Malina Hardaway, Sindy Hardaway, Antonella Hardaway, Kasha Hardaway, Laquinta Hardaway, Tysha Hardaway,
Adena Hardaway, Aide Hardaway, Bernita Hardaway, Bettie Hardaway, Delinda Hardaway, Lashundra Hardaway, Sanjuana Hardaway, Susann Hardaway, Vilma Hardaway, Deandrea Hardaway, Devorah Hardaway,
Marisha Hardaway, Reena Hardaway, Takeisha Hardaway, Tora Hardaway, Carlyn Hardaway, Carrieann Hardaway, Chanta Hardaway, China Hardaway, Rhianna Hardaway, Sharleen Hardaway, Jessenia Hardaway,
Kemberly Hardaway, Rosalina Hardaway, Rosella Hardaway, Latrese Hardaway, Mecca Hardaway, Nneka Hardaway, Alva Hardaway, Douglas Hardaway, Mandee Hardaway, Sherese Hardaway, Taya Hardaway, Tenika
Hardaway, Donelle Hardaway, Frieda Hardaway, Joye Hardaway, Kayce Hardaway, Sammie Hardaway, Shereen Hardaway, Tomiko Hardaway, Tommi Hardaway, Marleen Hardaway, Shonte Hardaway, Avery Hardaway,
Delfina Hardaway, Dollie Hardaway, Kaycee Hardaway, Lakesia Hardaway, Penney Hardaway, Tinika Hardaway, Tywanda Hardaway, Betina Hardaway, Debrah Hardaway, Latia Hardaway, Lynetta Hardaway, Michaelle
Hardaway, Shunta Hardaway, Soraya Hardaway, Anastacia Hardaway, Charisma Hardaway, Jonie Hardaway, Shunda Hardaway, Terah Hardaway, Trinidad Hardaway, Valeri Hardaway, Charlena Hardaway, Elizebeth
Hardaway, Joella Hardaway, Johnette Hardaway, Nikol Hardaway, Rosalynn Hardaway, Aime Hardaway, Aviva Hardaway, Joselyn Hardaway, Lashanna Hardaway, Tarrah Hardaway, Candyce Hardaway, Estrella
Hardaway, Mable Hardaway, Renay Hardaway, Shontel Hardaway, Wendee Hardaway, Angelene Hardaway, Karli Hardaway, Maricruz Hardaway, Minda Hardaway, Robbi Hardaway, Towana Hardaway, Charolette
Hardaway, Kandis Hardaway, Keeley Hardaway, Kimbley Hardaway, Kristene Hardaway, Lashell Hardaway, Merissa Hardaway, Taneshia Hardaway, Tristan Hardaway, Vida Hardaway, Zelda Hardaway, Angele
Hardaway, Beronica Hardaway, Damita Hardaway, Debbra Hardaway, Lanetta Hardaway, Priya Hardaway, Tamia Hardaway, Arlinda Hardaway, Armida Hardaway, Chere Hardaway, Errin Hardaway, Kamala Hardaway,
Lashan Hardaway, Luana Hardaway, Yalanda Hardaway, Antonietta Hardaway, Colby Hardaway, Jona Hardaway, Keitha Hardaway, Dava Hardaway, Herminia Hardaway, Allie Hardaway, Dawnmarie Hardaway, Eula
Hardaway, Monet Hardaway, Shirlene Hardaway, Tarina Hardaway, Yasmine Hardaway, Cassaundra Hardaway, Chelsie Hardaway, Fern Hardaway, Rosina Hardaway, Dyana Hardaway, Lavern Hardaway, Mario Hardaway,
Ollie Hardaway, Rodney Hardaway, Bridgit Hardaway, Julieta Hardaway, Korie Hardaway, Sumer Hardaway, Arianna Hardaway, Carmon Hardaway, Chassidy Hardaway, Ivana Hardaway, Jerusha Hardaway, Marne
Hardaway, Melisha Hardaway, Regenia Hardaway, Sarena Hardaway, Shanae Hardaway, Tona Hardaway, Cherese Hardaway, Deitra Hardaway, Dustin Hardaway, Kory Hardaway, Suzana Hardaway, Synthia Hardaway,
Zakia Hardaway, Lisbeth Hardaway, Matilde Hardaway, Terresa Hardaway, Tiffeny Hardaway, Toinette Hardaway, Charmin Hardaway, Julienne Hardaway, Marcelina Hardaway, Kama Hardaway, Katisha Hardaway,
Tiffiany Hardaway, Tobey Hardaway, Adelita Hardaway, Atiya Hardaway, Bradley Hardaway, Cathi Hardaway, Dominica Hardaway, Jeanelle Hardaway, Jeffery Hardaway, Rachal Hardaway, Sammi Hardaway,
Anamaria Hardaway, Arcelia Hardaway, Isabella Hardaway, Janay Hardaway, Janett Hardaway, Jonell Hardaway, Lakeya Hardaway, Leigha Hardaway, Alexander Hardaway, Denae Hardaway, Dustie Hardaway,
Felicita Hardaway, Freedom Hardaway, Harriett Hardaway, Katia Hardaway, Londa Hardaway, Marlyn Hardaway, Tereasa Hardaway, Analisa Hardaway, Annissa Hardaway, August Hardaway, Jenae Hardaway, Jenice
Hardaway, Kamie Hardaway, Shanee Hardaway, Sherise Hardaway, Tamieka Hardaway, Karianne Hardaway, Naima Hardaway, Natarsha Hardaway, Vita Hardaway, Anjali Hardaway, Ena Hardaway, Jaci Hardaway,
Jolanda Hardaway, Kimi Hardaway, Celestina Hardaway, Devonna Hardaway, Latrece Hardaway, Leatrice Hardaway, Lucila Hardaway, Mignon Hardaway, Stefany Hardaway, Tamyra Hardaway, Tracye Hardaway,
Beulah Hardaway, Danetta Hardaway, Dede Hardaway, Dwana Hardaway, Julisa Hardaway, Kanisha Hardaway, Lanae Hardaway, Ronette Hardaway, Sandee Hardaway, Veronique Hardaway, Deirdra Hardaway, Ember
Hardaway, Kerin Hardaway, Lekeisha Hardaway, Letisia Hardaway, Pandora Hardaway, Zaneta Hardaway, Earline Hardaway, Evon Hardaway, Lisandra Hardaway, Lura Hardaway, Raelene Hardaway, Shontell
Hardaway, Ila Hardaway, Kamisha Hardaway, Tinamarie Hardaway, Tish Hardaway, Jay Hardaway, Kayleen Hardaway, Maryam Hardaway, Phillip Hardaway, Santina Hardaway, Taneisha Hardaway, Zakiya Hardaway,
Brandice Hardaway, Jennifier Hardaway, Mariaelena Hardaway, Mirian Hardaway, Shalene Hardaway, Tammera Hardaway, Tesa Hardaway, Zaida Hardaway, Anjelica Hardaway, Arika Hardaway, Babette Hardaway,
Cameo Hardaway, Dwan Hardaway, Kiera Hardaway, Michella Hardaway, Nekia Hardaway, Rashelle Hardaway, Robynn Hardaway, Sheilah Hardaway, Velia Hardaway, Betsey Hardaway, Chiara Hardaway, Kortney
Hardaway, Tressie Hardaway, Tywana Hardaway, Valery Hardaway, Arnetta Hardaway, Carlee Hardaway, Craig Hardaway, Debbi Hardaway, Genesis Hardaway, Genny Hardaway, Jori Hardaway, Kandie Hardaway,
Lorin Hardaway, Melodee Hardaway, Cherlyn Hardaway, Karleen Hardaway, Windi Hardaway, Beverley Hardaway, Loreen Hardaway, Malynda Hardaway, Nada Hardaway, Nichola Hardaway, Sakinah Hardaway, Steffany
Hardaway, Tunisia Hardaway, Vanita Hardaway, Gemma Hardaway, Jennell Hardaway, Lakecia Hardaway, Shameeka Hardaway, Towanna Hardaway, Tywanna Hardaway, Vanesa Hardaway, Dea Hardaway, Desire Hardaway,
Donnie Hardaway, Kiki Hardaway, Makesha Hardaway, Rebeccah Hardaway, Anglia Hardaway, Branda Hardaway, Kawanna Hardaway, Tawnia Hardaway, Debi Hardaway, Elysia Hardaway, Lorretta Hardaway, Renetta
Hardaway, Tamu Hardaway, Anneliese Hardaway, Devan Hardaway, Rochel Hardaway, Vernetta Hardaway, Anetra Hardaway, Brynn Hardaway, Claribel Hardaway, Denielle Hardaway, Eugena Hardaway, Fara Hardaway,
Genie Hardaway, Joya Hardaway, Keisa Hardaway, Loria Hardaway, Tahira Hardaway, Tawna Hardaway, Angila Hardaway, Chara Hardaway, Cybil Hardaway, Jimmy Hardaway, Kendal Hardaway, Serita Hardaway, Tova
Hardaway, Vernice Hardaway, Cruz Hardaway, Jameelah Hardaway, Karan Hardaway, Kyna Hardaway, Larita Hardaway, Maricella Hardaway, Mariel Hardaway, Elyssa Hardaway, Kathyrn Hardaway, Raechel Hardaway,
Jolee Hardaway, Earnestine Hardaway, Livia Hardaway, Nechama Hardaway, Shawndra Hardaway, Antonina Hardaway, Chassity Hardaway, Codi Hardaway, Georgetta Hardaway, Gwyn Hardaway, Jacqualine Hardaway,
Latania Hardaway, Tawni Hardaway, Tiwana Hardaway, Toyia Hardaway, Willette Hardaway, Anette Hardaway, Brandye Hardaway, Danny Hardaway, Enedina Hardaway, Korina Hardaway, Latisa Hardaway, Vasiliki
Hardaway, Victor Hardaway, Daneen Hardaway, Elnora Hardaway, Shareen Hardaway, Amey Hardaway, Carroll Hardaway, Channon Hardaway, Charletta Hardaway, Deserie Hardaway, Jovan Hardaway, Latrica
Hardaway, Lorina Hardaway, Love Hardaway, Carolann Hardaway, Fran Hardaway, Ilona Hardaway, Kandra Hardaway, Keela Hardaway, Lasonia Hardaway, Leatha Hardaway, Lizzette Hardaway, Macy Hardaway, Magan
Hardaway, Rebekka Hardaway, Vanetta Hardaway, Azucena Hardaway, Brigit Hardaway, Jamaica Hardaway, Jameka Hardaway, Sheridan Hardaway, Aliya Hardaway, Chantay Hardaway, Giuseppina Hardaway, Lyndi
Hardaway, Ragan Hardaway, Retha Hardaway, Bari Hardaway, Daryl Hardaway, Elonda Hardaway, Grisel Hardaway, Joline Hardaway, Korey Hardaway, Milinda Hardaway, Rubi Hardaway, Sabine Hardaway, Silvana
Hardaway, Turkessa Hardaway, Myla Hardaway, Sharese Hardaway, Cheron Hardaway, Joell Hardaway, Lachandra Hardaway, Lateefah Hardaway, Sanya Hardaway, Wesley Hardaway, Amira Hardaway, Imani Hardaway,
Joette Hardaway, Kimmie Hardaway, Lynna Hardaway, Sheneka Hardaway, Sunnie Hardaway, Andi Hardaway, Angelyn Hardaway, Maurice Hardaway, Melodi Hardaway, Nelida Hardaway, Sherida Hardaway, Shronda
Hardaway, Takia Hardaway, Toshiba Hardaway, Trishia Hardaway, Alonda Hardaway, Araseli Hardaway, Laketa Hardaway, Lesha Hardaway, Magen Hardaway, Neisha Hardaway, Selma Hardaway, Shannen Hardaway,
Toria Hardaway, Aline Hardaway, Aubree Hardaway, Delaina Hardaway, Joannie Hardaway, Mai Hardaway, Niya Hardaway, Dimitra Hardaway, Karl Hardaway, Kimberle Hardaway, Quanda Hardaway, Alica Hardaway,
Chaundra Hardaway, Cherly Hardaway, Donnell Hardaway, Ellyn Hardaway, Lanell Hardaway, Lawrence Hardaway, Tonita Hardaway, Vania Hardaway, Genea Hardaway, Kathaleen Hardaway, Laquetta Hardaway,
Lorien Hardaway, Mallory Hardaway, Margery Hardaway, Porsha Hardaway, Agatha Hardaway, Alayna Hardaway, Barbi Hardaway, Fatimah Hardaway, Jaclynn Hardaway, Kalisha Hardaway, Liesl Hardaway, Alita
Hardaway, Janita Hardaway, Ola Hardaway, Shawne Hardaway, Talena Hardaway, Angla Hardaway, Autum Hardaway, Bertina Hardaway, Samira Hardaway, Shalena Hardaway, Consuella Hardaway, Corin Hardaway,
Delora Hardaway, Lanna Hardaway, Pat Hardaway, Roshunda Hardaway, Jonette Hardaway, Latishia Hardaway, Reginia Hardaway, Ronnette Hardaway, Sherree Hardaway, Tamicka Hardaway, Alejandrina Hardaway,
Carlisa Hardaway, Jene Hardaway, Loralee Hardaway, Lorenza Hardaway, Luis Hardaway, Yasmeen Hardaway, Zakiyyah Hardaway, Ashly Hardaway, Cassy Hardaway, Chari Hardaway, Indira Hardaway, Ivelisse
Hardaway, Jaymi Hardaway, Karalee Hardaway, Lauretta Hardaway, Raelynn Hardaway, Tifany Hardaway, Torrey Hardaway, Amyjo Hardaway, Annalee Hardaway, Hester Hardaway, Josefa Hardaway, Latresa
Hardaway, Lita Hardaway, Marysol Hardaway, Mindie Hardaway, Shan Hardaway, Susy Hardaway, Teneka Hardaway, Arnita Hardaway, Charice Hardaway, Cleopatra Hardaway, Cotina Hardaway, Dawnette Hardaway,
Julene Hardaway, Lashay Hardaway, Latrell Hardaway, Lucrecia Hardaway, Ricky Hardaway, Shamekia Hardaway, Camellia Hardaway, Cassi Hardaway, Curtis Hardaway, Hana Hardaway, Isha Hardaway, Jacquelyne
Hardaway, Jenica Hardaway, Norah Hardaway, Oneida Hardaway, Subrina Hardaway, Tyrone Hardaway, Angi Hardaway, Anjeanette Hardaway, Bella Hardaway, Donica Hardaway, Kareen Hardaway, Laina Hardaway,
Luci Hardaway, Mischelle Hardaway, Tanita Hardaway, Heaven Hardaway, Madelin Hardaway, Michaele Hardaway, Rashell Hardaway, Tamarah Hardaway, Viki Hardaway, Candee Hardaway, Danyale Hardaway, Karis
Hardaway, Mishelle Hardaway, Shanequa Hardaway, Shellee Hardaway, Despina Hardaway, Dodi Hardaway, Katherin Hardaway, Mechele Hardaway, Pebbles Hardaway, Roshelle Hardaway, Carry Hardaway, Denine
Hardaway, Dory Hardaway, Jolinda Hardaway, Julieanne Hardaway, Karlyn Hardaway, Kathern Hardaway, Kerensa Hardaway, Kerrin Hardaway, Lafonda Hardaway, Launa Hardaway, Marin Hardaway, Ona Hardaway,
Sheli Hardaway, Trixie Hardaway, Yahaira Hardaway, Casi Hardaway, Ciara Hardaway, Darline Hardaway, Drema Hardaway, Francie Hardaway, Gerald Hardaway, Khara Hardaway, Lashone Hardaway, Meloney
Hardaway, Melyssa Hardaway, Merrie Hardaway, Neysa Hardaway, Queen Hardaway, Rashawn Hardaway, Robbyn Hardaway, Sharilyn Hardaway, Sharyl Hardaway, Tamitha Hardaway, Tresha Hardaway, Amaris Hardaway,
Bernetta Hardaway, Channel Hardaway, Corrin Hardaway, Deadra Hardaway, Filomena Hardaway, Jaye Hardaway, Kerriann Hardaway, Kysha Hardaway, Lanie Hardaway, Lashauna Hardaway, Lashona Hardaway,
Laurinda Hardaway, Monifa Hardaway, Sharri Hardaway, Shontae Hardaway, Terrica Hardaway, Thresa Hardaway, Tondra Hardaway, Geralyn Hardaway, Ginamarie Hardaway, Jesseca Hardaway, Lasonja Hardaway,
Marica Hardaway, Marlen Hardaway, Virgie Hardaway, Genna Hardaway, Kriste Hardaway, Kyndra Hardaway, Tequilla Hardaway, Vernell Hardaway, Zoila Hardaway, Aiesha Hardaway, Andera Hardaway, Blake
Hardaway, Carlena Hardaway, Deangela Hardaway, Lavonia Hardaway, Lesly Hardaway, Savannah Hardaway, Danisha Hardaway, Dierdre Hardaway, Hermelinda Hardaway, Kam Hardaway, Milena Hardaway, Omega
Hardaway, Rosaria Hardaway, Russell Hardaway, Shaquita Hardaway, Tashika Hardaway, Arlena Hardaway, Delisha Hardaway, Demetric Hardaway, Lakendra Hardaway, Marilu Hardaway, Neha Hardaway, Terria
Hardaway, Carita Hardaway, Kallie Hardaway, Makisha Hardaway, Martie Hardaway, Mitzie Hardaway, Stephane Hardaway, Donette Hardaway, Joli Hardaway, Kathlyn Hardaway, Keona Hardaway, Lakishia
Hardaway, Laurice Hardaway, Melena Hardaway, Nan Hardaway, Rafaela Hardaway, Taura Hardaway, Wynette Hardaway, Arielle Hardaway, Carlette Hardaway, Cyrstal Hardaway, Dewanda Hardaway, Kimmy Hardaway,
Pauletta Hardaway, Shannel Hardaway, Shelbi Hardaway, Camillia Hardaway, Charlyn Hardaway, Christianna Hardaway, Gerry Hardaway, Jamy Hardaway, Karoline Hardaway, Katrinia Hardaway, Lorissa Hardaway,
Magdalene Hardaway, Mariza Hardaway, Markisha Hardaway, Mindee Hardaway, Terica Hardaway, Terisa Hardaway, Tyronda Hardaway, Aracelis Hardaway, Betzaida Hardaway, Charon Hardaway, Chastidy Hardaway,
Henry Hardaway, Jamika Hardaway, Janiece Hardaway, Kariann Hardaway, Kawanda Hardaway, Saprina Hardaway, Chera Hardaway, Genoveva Hardaway, Jacey Hardaway, Jannifer Hardaway, Jaylene Hardaway, Meisha
Hardaway, Daphney Hardaway, Delois Hardaway, Denelle Hardaway, Diamond Hardaway, Felita Hardaway, Hilarie Hardaway, Jamica Hardaway, Joelene Hardaway, Kisa Hardaway, Lilliana Hardaway, Sulema
Hardaway, Wendolyn Hardaway, Calista Hardaway, Darrell Hardaway, Elinor Hardaway, Faviola Hardaway, Jenean Hardaway, Lalena Hardaway, Mischa Hardaway, Tamikia Hardaway, Alanda Hardaway, Ashia
Hardaway, Bryna Hardaway, Serene Hardaway, Shauntel Hardaway, Sheria Hardaway, Talya Hardaway, Thersa Hardaway, Walter Hardaway, Yevette Hardaway, Yolande Hardaway, Ashlea Hardaway, Gaylene Hardaway,
Khadija Hardaway, Latitia Hardaway, Micheline Hardaway, Monalisa Hardaway, Sakina Hardaway, Stacye Hardaway, Taffy Hardaway, Tifani Hardaway, Wilhelmina Hardaway, Amara Hardaway, Gypsy Hardaway,
Joely Hardaway, Kameron Hardaway, Keasha Hardaway, Louella Hardaway, Riki Hardaway, Torey Hardaway, Adrien Hardaway, Arwen Hardaway, Crysta Hardaway, Delaine Hardaway, Denyse Hardaway, Georgie
Hardaway, Janea Hardaway, Medina Hardaway, Meka Hardaway, Nika Hardaway, Rashanda Hardaway, Corena Hardaway, Denee Hardaway, Kimya Hardaway, Lanisha Hardaway, Luella Hardaway, Marylin Hardaway,
Merritt Hardaway, Mysti Hardaway, Shawntay Hardaway, Shirelle Hardaway, Siri Hardaway, Cayce Hardaway, Chava Hardaway, Jovanna Hardaway, Lorine Hardaway, Maryelizabeth Hardaway, Sonji Hardaway,
Titania Hardaway, Brent Hardaway, Jamye Hardaway, Kela Hardaway, Korin Hardaway, Monette Hardaway, Rebbeca Hardaway, Rebeka Hardaway, Stormie Hardaway, Angelo Hardaway, Camela Hardaway, Darlena
Hardaway, Dietra Hardaway, Gwendolen Hardaway, Kenia Hardaway, Kesia Hardaway, Kolleen Hardaway, Leighanne Hardaway, Lindi Hardaway, Marcee Hardaway, Miya Hardaway, Sami Hardaway, Tameko Hardaway,
Tamora Hardaway, Tanaya Hardaway, Tiki Hardaway, Toniann Hardaway, Jeaneen Hardaway, Lizeth Hardaway, Manuel Hardaway, Marlina Hardaway, Monisha Hardaway, Nichele Hardaway, Rechelle Hardaway, Richele
Hardaway, Shenna Hardaway, Sky Hardaway, Tahnee Hardaway, Brita Hardaway, Erikka Hardaway, Jani Hardaway, Kattie Hardaway, Kimbra Hardaway, Quincy Hardaway, Shaquana Hardaway, Sherelle Hardaway,
Sherryl Hardaway, Shireen Hardaway, Theressa Hardaway, Yecenia Hardaway, Albert Hardaway, Ansley Hardaway, Clair Hardaway, Danille Hardaway, Evelia Hardaway, Jacob Hardaway, Jon Hardaway, Katrin
Hardaway, Kiva Hardaway, Mirta Hardaway, Pearlie Hardaway, Roger Hardaway, Shawntell Hardaway, Terrance Hardaway, Vincent Hardaway, Aimie Hardaway, Alise Hardaway, Caryl Hardaway, Devora Hardaway,
Dian Hardaway, Genelle Hardaway, Johannah Hardaway, Lanora Hardaway, Memory Hardaway, Mila Hardaway, Shandy Hardaway, Shella Hardaway, Shila Hardaway, Tahisha Hardaway, Channa Hardaway, Christeen
Hardaway, Janise Hardaway, Kristee Hardaway, Melaney Hardaway, Tanis Hardaway, Teasha Hardaway, Arthur Hardaway, Derek Hardaway, Ginette Hardaway, Kamesha Hardaway, Keriann Hardaway, Lateisha
Hardaway, Merrilee Hardaway, Nikesha Hardaway, Sherrita Hardaway, Shontay Hardaway, Bess Hardaway, Candelaria Hardaway, Charo Hardaway, Dominga Hardaway, Kateri Hardaway, Kelsie Hardaway, Kenyata
Hardaway, Lane Hardaway, Lateshia Hardaway, Lili Hardaway, Linnette Hardaway, Odette Hardaway, Sharika Hardaway, Sherra Hardaway, Tama Hardaway, Cicily Hardaway, Clarinda Hardaway, Corri Hardaway,
Donnita Hardaway, Duana Hardaway, Gisele Hardaway, Laquitta Hardaway, Mahogany Hardaway, Seanna Hardaway, Sena Hardaway, Shelbie Hardaway, Steffani Hardaway, Tauna Hardaway, Zara Hardaway, Aminah
Hardaway, Amye Hardaway, Damon Hardaway, Doria Hardaway, Frederica Hardaway, Hollis Hardaway, Katey Hardaway, Nila Hardaway, Pamella Hardaway, Ray Hardaway, Taina Hardaway, Zenia Hardaway, Ananda
Hardaway, Becca Hardaway, Dessie Hardaway, Donnette Hardaway, Marney Hardaway, Nekisha Hardaway, Sharell Hardaway, Stefania Hardaway, Tandi Hardaway, Allana Hardaway, Andree Hardaway, Arlette
Hardaway, Denisa Hardaway, Ieshia Hardaway, Latrecia Hardaway, Meryl Hardaway, Shakia Hardaway, Shaye Hardaway, Clover Hardaway, Cresta Hardaway, Francina Hardaway, Fredericka Hardaway, Kimbery
Hardaway, Shameika Hardaway, Sherene Hardaway, Armanda Hardaway, Cally Hardaway, Cydney Hardaway, Dasha Hardaway, Gwynne Hardaway, Josephina Hardaway, Lavonna Hardaway, Melessa Hardaway, Nycole
Hardaway, Nyoka Hardaway, Pansy Hardaway, Shakita Hardaway, Sonjia Hardaway, Wanita Hardaway, Xenia Hardaway, Any Hardaway, Bobi Hardaway, Bonni Hardaway, Chevelle Hardaway, Chrystie Hardaway, Genine
Hardaway, Iona Hardaway, Kezia Hardaway, Latresha Hardaway, Lucie Hardaway, Malikah Hardaway, Micha Hardaway, Reginald Hardaway, Ronita Hardaway, Sharina Hardaway, Sharise Hardaway, Stormi Hardaway,
Tammey Hardaway, Angelika Hardaway, Anja Hardaway, Carianne Hardaway, Dalene Hardaway, Florinda Hardaway, Irasema Hardaway, Karren Hardaway, Kedra Hardaway, Koreen Hardaway, Leena Hardaway, Libra
Hardaway, Moncia Hardaway, Rosaline Hardaway, Roshawn Hardaway, Rusty Hardaway, Sonda Hardaway, Talina Hardaway, Trudie Hardaway, Audry Hardaway, Celinda Hardaway, Deona Hardaway, Jennipher Hardaway,
Jolena Hardaway, Kimerly Hardaway, Kym Hardaway, Laquana Hardaway, Madalyn Hardaway, Renate Hardaway, Una Hardaway, Barrie Hardaway, Chanin Hardaway, Deven Hardaway, Hadley Hardaway, Hether Hardaway,
Ninfa Hardaway, Sha Hardaway, Sloane Hardaway, Tamla Hardaway, Amada Hardaway, Belynda Hardaway, Celesta Hardaway, Chelsa Hardaway, Corene Hardaway, Dawana Hardaway, Elly Hardaway, Laure Hardaway,
Pascha Hardaway, Shenell Hardaway, Tashana Hardaway, Trecia Hardaway, Angelea Hardaway, Dominque Hardaway, Frederick Hardaway, Ira Hardaway, Karine Hardaway, Lachanda Hardaway, Lyla Hardaway, Melanee
Hardaway, Nikkole Hardaway, Ronetta Hardaway, Shanise Hardaway, Shiree Hardaway, Vincenza Hardaway, Yolando Hardaway, Adelle Hardaway, Ara Hardaway, Bria Hardaway, Cheryll Hardaway, Delila Hardaway,
Delina Hardaway, Jonetta Hardaway, Kamela Hardaway, Lacinda Hardaway, Lashanta Hardaway, Laurene Hardaway, Lenita Hardaway, Martin Hardaway, Neda Hardaway, Sharen Hardaway, Tanishia Hardaway, Bronwen
Hardaway, Jamilla Hardaway, Jennelle Hardaway, Lavetta Hardaway, Makeda Hardaway, Omaira Hardaway, Randie Hardaway, Ricci Hardaway, Romy Hardaway, Sherlyn Hardaway, Sonal Hardaway, Tierra Hardaway,
Tonyia Hardaway, Caressa Hardaway, Carmina Hardaway, Dann Hardaway, Dawnelle Hardaway, Isa Hardaway, Lameka Hardaway, Lavada Hardaway, Lavita Hardaway, Lessie Hardaway, Miracle Hardaway, Mylinda
Hardaway, Nancie Hardaway, Raeanne Hardaway, Shanique Hardaway, Sharolyn Hardaway, Alex Hardaway, Cheyanne Hardaway, Christan Hardaway, Halima Hardaway, Karlee Hardaway, Katherina Hardaway, Lisabeth
Hardaway, Marylynn Hardaway, Naimah Hardaway, Ria Hardaway, Tierney Hardaway, Velda Hardaway, Abbe Hardaway, Alane Hardaway, Belia Hardaway, Falisha Hardaway, Kaylee Hardaway, Latausha Hardaway,
Lynnea Hardaway, Myranda Hardaway, Nyla Hardaway, Rima Hardaway, Seneca Hardaway, Aprille Hardaway, Carlina Hardaway, Christle Hardaway, Corliss Hardaway, Crystle Hardaway, Desha Hardaway, Kea
Hardaway, Kristena Hardaway, Kristiana Hardaway, Lari Hardaway, Lorita Hardaway, Marlaina Hardaway, Marquitta Hardaway, Ondrea Hardaway, Tashara Hardaway, Devonne Hardaway, Domonique Hardaway, Jannet
Hardaway, Kila Hardaway, Krysten Hardaway, Leslye Hardaway, Lynnae Hardaway, Shela Hardaway, Sherica Hardaway, Timberly Hardaway, Arminda Hardaway, Essence Hardaway, Gala Hardaway, Hellen Hardaway,
Jannine Hardaway, Kamille Hardaway, Lalita Hardaway, Layne Hardaway, Letecia Hardaway, Panagiota Hardaway, Philana Hardaway, Trisa Hardaway, Akisha Hardaway, Audria Hardaway, Berenice Hardaway, Bette
Hardaway, Denese Hardaway, Donia Hardaway, Johana Hardaway, Kamika Hardaway, Katelyn Hardaway, Laine Hardaway, Lajuan Hardaway, Lasondra Hardaway, Marketta Hardaway, Patrizia Hardaway, Sascha
Hardaway, Shamara Hardaway, Shamica Hardaway, Tirzah Hardaway, Trinia Hardaway, Tyla Hardaway, Venice Hardaway, Carline Hardaway, Deyanira Hardaway, Isabell Hardaway, Jackqueline Hardaway, Kendria
Hardaway, Marlee Hardaway, Sharisse Hardaway, Shawntae Hardaway, Sholanda Hardaway, Thomasine Hardaway, Tikisha Hardaway, Tyna Hardaway, Aria Hardaway, Colene Hardaway, Daphine Hardaway, Dorthea
Hardaway, Genise Hardaway, Keia Hardaway, Leasa Hardaway, Nohemi Hardaway, Rosann Hardaway, Ruthanne Hardaway, Shundra Hardaway, Taria Hardaway, Valinda Hardaway, Adrain Hardaway, Brunilda Hardaway,
Caralee Hardaway, Charlee Hardaway, Katasha Hardaway, Krystina Hardaway, Krystyna Hardaway, Lovie Hardaway, Marquette Hardaway, Marybel Hardaway, Melvin Hardaway, Nathaniel Hardaway, Rickie Hardaway,
Romana Hardaway, Tene Hardaway, Tinesha Hardaway, Tyann Hardaway, Wednesday Hardaway, Abra Hardaway, Adia Hardaway, Casondra Hardaway, Charmane Hardaway, Chevonne Hardaway, Gwenn Hardaway, Laketha
Hardaway, Louis Hardaway, Marvin Hardaway, Micole Hardaway, Novella Hardaway, Oriana Hardaway, Raena Hardaway, Sharone Hardaway, Suellen Hardaway, Thais Hardaway, Trenda Hardaway, Zuleika Hardaway,
Aundria Hardaway, Ayisha Hardaway, Catrena Hardaway, Corinthia Hardaway, Dorine Hardaway, Dulcie Hardaway, Dyanna Hardaway, Kimyatta Hardaway, Lekeshia Hardaway, Lenette Hardaway, Nesha Hardaway,
Reta Hardaway, Rian Hardaway, Rori Hardaway, Samia Hardaway, Shelonda Hardaway, Sima Hardaway, Adriann Hardaway, Beata Hardaway, Dewanna Hardaway, Elisia Hardaway, Lida Hardaway, Myriah Hardaway,
Porsche Hardaway, Regine Hardaway, Roma Hardaway, Shandi Hardaway, Sharae Hardaway, Brady Hardaway, Denitra Hardaway, Erik Hardaway, Jeane Hardaway, Jinny Hardaway, Karah Hardaway, Mechell Hardaway,
Mikel Hardaway, Nikkie Hardaway, Nisa Hardaway, Nubia Hardaway, Peaches Hardaway, Philip Hardaway, Ramie Hardaway, Romelia Hardaway, Roshell Hardaway, Tai Hardaway, Tracia Hardaway, Wonda Hardaway,
Alix Hardaway, Aprile Hardaway, Ilda Hardaway, Kacee Hardaway, Kathrin Hardaway, Lettie Hardaway, Michaelene Hardaway, Rania Hardaway, Shilpa Hardaway, Teddi Hardaway, Tegan Hardaway, Terrilyn
Hardaway, Albertina Hardaway, Aryn Hardaway, Augustina Hardaway, Belle Hardaway, Britton Hardaway, Irena Hardaway, Jacquie Hardaway, Katrine Hardaway, Lorenda Hardaway, Magali Hardaway, Mikaela
Hardaway, Remona Hardaway, Rossana Hardaway, Shree Hardaway, Adrea Hardaway, Ashaki Hardaway, Beatris Hardaway, Elisabet Hardaway, Gene Hardaway, Greer Hardaway, Hillery Hardaway, Jillene Hardaway,
Keyona Hardaway, Latifa Hardaway, Marshell Hardaway, Miguel Hardaway, Resa Hardaway, Roy Hardaway, Tarnisha Hardaway, Tekisha Hardaway, Teshia Hardaway, Tomara Hardaway, Vashti Hardaway, Brienne
Hardaway, Calvin Hardaway, Corrinne Hardaway, Dorina Hardaway, Errica Hardaway, Eulalia Hardaway, Jacelyn Hardaway, Jimi Hardaway, Kavita Hardaway, Keyana Hardaway, Kiara Hardaway, Lahoma Hardaway,
Merilee Hardaway, Milisa Hardaway, Pollyanna Hardaway, Rabecca Hardaway, Shalisa Hardaway, Tyree Hardaway, Baby Hardaway, Cali Hardaway, Cherron Hardaway, Conswella Hardaway, Dayle Hardaway, Deniece
Hardaway, Elesha Hardaway, Eugene Hardaway, Fredricka Hardaway, Letty Hardaway, Marlisa Hardaway, Meadow Hardaway, Nana Hardaway, Ricardo Hardaway, Sunita Hardaway, Tahirah Hardaway, Aleasha
Hardaway, Alene Hardaway, Jerica Hardaway, Jonni Hardaway, Jorie Hardaway, Keenya Hardaway, Phillis Hardaway, Scottie Hardaway, Tamy Hardaway, Adelaide Hardaway, Akia Hardaway, Alona Hardaway,
Brandilyn Hardaway, Davita Hardaway, Deondra Hardaway, Guinevere Hardaway, Jaquetta Hardaway, Joyelle Hardaway, Kaitlin Hardaway, Kammy Hardaway, Lashonna Hardaway, Manisha Hardaway, Nyesha Hardaway,
Phebe Hardaway, Shadonna Hardaway, Tamila Hardaway, Treasure Hardaway, Wayne Hardaway, Charmayne Hardaway, Corry Hardaway, Doni Hardaway, Jannell Hardaway, Kaia Hardaway, Karna Hardaway, Kinya
Hardaway, Lasha Hardaway, Lecia Hardaway, Letricia Hardaway, Paulita Hardaway, Sharnell Hardaway, Sheniqua Hardaway, Toy Hardaway, Unique Hardaway, Wynona Hardaway, Audrie Hardaway, Cambria Hardaway,
Crissie Hardaway, Deseree Hardaway, Deva Hardaway, Jenney Hardaway, Jessamyn Hardaway, Leda Hardaway, Lorry Hardaway, Marjory Hardaway, Marnee Hardaway, Monigue Hardaway, Nakeya Hardaway, Passion
Hardaway, Ryanne Hardaway, Sharona Hardaway, Shelina Hardaway, Tabbatha Hardaway, Tameshia Hardaway, Adelia Hardaway, Anica Hardaway, Calli Hardaway, Carolyne Hardaway, Chiffon Hardaway, Elizabet
Hardaway, Ermelinda Hardaway, Freddie Hardaway, Jamia Hardaway, Korrie Hardaway, Latrenda Hardaway, Makia Hardaway, Marcel Hardaway, Meta Hardaway, Noni Hardaway, Sigrid Hardaway, Zita Hardaway,
Angelee Hardaway, Balinda Hardaway, Brandyn Hardaway, Christia Hardaway, Eneida Hardaway, Irish Hardaway, Jackelyn Hardaway, Jacqui Hardaway, Kammie Hardaway, Kendell Hardaway, Kiya Hardaway,
Mellanie Hardaway, Sherina Hardaway, Syretta Hardaway, Teia Hardaway, Teneshia Hardaway, Tomasa Hardaway, Tonna Hardaway, Virgina Hardaway, Zelma Hardaway, Adella Hardaway, Alea Hardaway, Annita
Hardaway, Denetra Hardaway, Georgeann Hardaway, Gwyneth Hardaway, Jenea Hardaway, Kalli Hardaway, Khristine Hardaway, Krissie Hardaway, Lanelle Hardaway, Letrice Hardaway, Meshell Hardaway, Michon
Hardaway, Nekita Hardaway, Raelyn Hardaway, Shaleen Hardaway, Sharanda Hardaway, Sol Hardaway, Suzi Hardaway, Ulanda Hardaway, Ame Hardaway, Aprill Hardaway, Brooks Hardaway, Carlye Hardaway,
Francisco Hardaway, Gaynell Hardaway, Gwenda Hardaway, Heidy Hardaway, Helga Hardaway, Janese Hardaway, Kathrina Hardaway, Katonya Hardaway, Kodi Hardaway, Leshawn Hardaway, London Hardaway, Lyndee
Hardaway, Marc Hardaway, Miyoshi Hardaway, Onika Hardaway, Peyton Hardaway, Shandell Hardaway, Terena Hardaway, Acacia Hardaway, Aleida Hardaway, Antoine Hardaway, Arletha Hardaway, Carlin Hardaway,
Danni Hardaway, Davette Hardaway, December Hardaway, Jenene Hardaway, Karalyn Hardaway, Kimbely Hardaway, Lakisa Hardaway, Lynae Hardaway, Margaux Hardaway, Michela Hardaway, Renne Hardaway, Tanzania
Hardaway, Tenia Hardaway, Toyna Hardaway, Veronika Hardaway, Amal Hardaway, Camila Hardaway, Chi Hardaway, Dominic Hardaway, Macie Hardaway, Maida Hardaway, Malisha Hardaway, Milagro Hardaway,
Mitchell Hardaway, Shondell Hardaway, Taira Hardaway, Argelia Hardaway, Casaundra Hardaway, Courtnie Hardaway, Dalena Hardaway, Donyell Hardaway, Ericia Hardaway, Kelsi Hardaway, Madelene Hardaway,
Mikelle Hardaway, Moria Hardaway, Salome Hardaway, Starlet Hardaway, Stepanie Hardaway, Stephnie Hardaway, Teesha Hardaway, Allisa Hardaway, Aneesah Hardaway, Breann Hardaway, Chevon Hardaway,
Deatrice Hardaway, Delaney Hardaway, Jodene Hardaway, Julian Hardaway, Lorilee Hardaway, Margarette Hardaway, Marshall Hardaway, Meleah Hardaway, Nadene Hardaway, Nefertiti Hardaway, Rebekkah
Hardaway, Shakina Hardaway, Sibyl Hardaway, Teria Hardaway, Briget Hardaway, Cortina Hardaway, Gennie Hardaway, Georgeanna Hardaway, Jael Hardaway, Katharina Hardaway, Kennetha Hardaway, Kerrianne
Hardaway, Meggin Hardaway, Rochele Hardaway, Santos Hardaway, Senta Hardaway, Shanie Hardaway, Sona Hardaway, Yana Hardaway, Alicea Hardaway, Capri Hardaway, Cassey Hardaway, Daun Hardaway, Denette
Hardaway, Harold Hardaway, Jacie Hardaway, Jennica Hardaway, Krissa Hardaway, Lawonda Hardaway, Lenna Hardaway, Maija Hardaway, Ninette Hardaway, Tomeika Hardaway, Trenna Hardaway, Amberlee Hardaway,
Avril Hardaway, Charnell Hardaway, Chenita Hardaway, Crystel Hardaway, Eugenie Hardaway, Ginnie Hardaway, Jerome Hardaway, Karee Hardaway, Latunya Hardaway, Lianna Hardaway, Lisaann Hardaway,
Maribell Hardaway, Marylyn Hardaway, Molli Hardaway, Raye Hardaway, Risha Hardaway, Sherrice Hardaway, Taralyn Hardaway, Tika Hardaway, Tyanna Hardaway, Wilda Hardaway, Brandalyn Hardaway, Camesha
Hardaway, Ericca Hardaway, Ernest Hardaway, Glynis Hardaway, Larena Hardaway, Latecia Hardaway, Lindsy Hardaway, Melea Hardaway, Monic Hardaway, Suanne Hardaway, Tammra Hardaway, Tyeisha Hardaway,
Allen Hardaway, Ambra Hardaway, Cindee Hardaway, Deja Hardaway, Denika Hardaway, Easter Hardaway, Emerald Hardaway, Jann Hardaway, Jeannetta Hardaway, Keyla Hardaway, Lataya Hardaway, Natividad
Hardaway, Nikola Hardaway, Quanita Hardaway, Shalyn Hardaway, Temekia Hardaway, Tishia Hardaway, Allena Hardaway, Andrena Hardaway, Aurea Hardaway, Buffi Hardaway, Darren Hardaway, Dovie Hardaway,
Evelina Hardaway, Fatina Hardaway, Ivey Hardaway, Jennafer Hardaway, Kimberlea Hardaway, Kishia Hardaway, Lacrecia Hardaway, Laney Hardaway, Latora Hardaway, Robinette Hardaway, Shalita Hardaway,
Shanin Hardaway, Terasa Hardaway, Tiphanie Hardaway, Tiwanna Hardaway, Ashlyn Hardaway, Breana Hardaway, Emely Hardaway, Evan Hardaway, Evangelia Hardaway, Halle Hardaway, Lynnetta Hardaway, Mandisa
Hardaway, Mardi Hardaway, Neomi Hardaway, Peggie Hardaway, Senaida Hardaway, Shantee Hardaway, Alondra Hardaway, Bettyjo Hardaway, Dawnell Hardaway, Jaunita Hardaway, Jaymee Hardaway, Jeneane
Hardaway, Kamara Hardaway, Korinne Hardaway, Lacosta Hardaway, Latrena Hardaway, Letticia Hardaway, Luanna Hardaway, Meighan Hardaway, Merinda Hardaway, Raechelle Hardaway, Randall Hardaway,
Shermaine Hardaway, Tacy Hardaway, Telicia Hardaway, Trica Hardaway, Unknown Hardaway, Venetia Hardaway, Virgen Hardaway, Chanita Hardaway, Christe Hardaway, Collene Hardaway, Deni Hardaway, Eliana
Hardaway, Evelin Hardaway, Jessa Hardaway, Katarina Hardaway, Kersten Hardaway, Krysti Hardaway, Lanise Hardaway, Lannette Hardaway, Latacha Hardaway, Lavenia Hardaway, Magnolia Hardaway, Mayte
Hardaway, Missie Hardaway, Renda Hardaway, Sapna Hardaway, Shawan Hardaway, Starlene Hardaway, Stasia Hardaway, Tamaria Hardaway, Tiffanee Hardaway, Yazmin Hardaway, Ammy Hardaway, Blossom Hardaway,
Cherokee Hardaway, Chrissa Hardaway, Dean Hardaway, Donell Hardaway, Jannelle Hardaway, Jenessa Hardaway, Jenne Hardaway, Kimiko Hardaway, Lenise Hardaway, Lotoya Hardaway, Mariko Hardaway, Myeshia
Hardaway, Nadja Hardaway, Shalana Hardaway, Tangee Hardaway, Waynette Hardaway, Alan Hardaway, Bennie Hardaway, Berlinda Hardaway, Brinda Hardaway, Brooklyn Hardaway, Bruce Hardaway, Daisha Hardaway,
Devina Hardaway, Donisha Hardaway, Judie Hardaway, Khristy Hardaway, Margarett Hardaway, Markesha Hardaway, Melannie Hardaway, Nickol Hardaway, Pamelia Hardaway, Roselle Hardaway, Shirleen Hardaway,
Terrence Hardaway, Trenise Hardaway, Alda Hardaway, Ariella Hardaway, Aubrie Hardaway, Aziza Hardaway, Carrol Hardaway, Charnita Hardaway, Deshanna Hardaway, Donyale Hardaway, Dorrie Hardaway, Elayne
Hardaway, Feather Hardaway, Gregoria Hardaway, Kanesha Hardaway, Kirsti Hardaway, Maryalice Hardaway, Maude Hardaway, Millissa Hardaway, Monya Hardaway, Shareese Hardaway, Shereka Hardaway, Tenaya
Hardaway, Vonnie Hardaway, Bunny Hardaway, Carmin Hardaway, Cissy Hardaway, Damika Hardaway, Dayla Hardaway, Dewana Hardaway, Jorge Hardaway, Kaila Hardaway, Kaisha Hardaway, Kaylynn Hardaway, Kemba
Hardaway, Lael Hardaway, Lysa Hardaway, Roberto Hardaway, Tanasha Hardaway, Tynisa Hardaway, Vina Hardaway, Angelette Hardaway, Angenette Hardaway, Aquila Hardaway, Carrianne Hardaway, Charlynn
Hardaway, Denean Hardaway, Dondi Hardaway, Enriqueta Hardaway, Georganna Hardaway, Glinda Hardaway, Jenel Hardaway, Larinda Hardaway, Latika Hardaway, Lequita Hardaway, Licia Hardaway, Marilou
Hardaway, Sylvie Hardaway, Aaliyah Hardaway, Cozette Hardaway, Damian Hardaway, Ginna Hardaway, Janalee Hardaway, Jenniefer Hardaway, Kierstin Hardaway, Lexie Hardaway, Maja Hardaway, Michelene
Hardaway, Norine Hardaway, Sana Hardaway, Sarrah Hardaway, Sharna Hardaway, Sherre Hardaway, Shoshanna Hardaway, Tangi Hardaway, Tillie Hardaway, Aishia Hardaway, Allene Hardaway, Antoniette
Hardaway, Brittani Hardaway, Candise Hardaway, Carlota Hardaway, Debroah Hardaway, Demeka Hardaway, Dondra Hardaway, Franca Hardaway, Gaye Hardaway, Jeanell Hardaway, Lark Hardaway, Laurin Hardaway,
Lourie Hardaway, Machell Hardaway, Pasha Hardaway, Shelena Hardaway, Tangelia Hardaway, Tani Hardaway, Twilla Hardaway, Tynesha Hardaway, Vena Hardaway, Vernessa Hardaway, Aleesha Hardaway, Cassia
Hardaway, Catheryn Hardaway, Chavon Hardaway, Codie Hardaway, Conchita Hardaway, Felipa Hardaway, Jeanice Hardaway, Kelleen Hardaway, Latona Hardaway, Loreal Hardaway, Lutricia Hardaway, Natanya
Hardaway, Perry Hardaway, Samaria Hardaway, Sharene Hardaway, Shavone Hardaway, Stavroula Hardaway, Tatia Hardaway, Terrina Hardaway, Veroncia Hardaway, Alaine Hardaway, Carmalita Hardaway, Chirstina
Hardaway, Dawanna Hardaway, Delonda Hardaway, Ebonee Hardaway, Jamela Hardaway, Kerie Hardaway, Lanika Hardaway, Latressa Hardaway, Priscella Hardaway, Sharifa Hardaway, Shenequa Hardaway, Shon
Hardaway, Taresa Hardaway, Viva Hardaway, Annabell Hardaway, Arelis Hardaway, Carra Hardaway, Char Hardaway, Chekesha Hardaway, Cherylann Hardaway, Crystall Hardaway, Diandra Hardaway, Dwayne
Hardaway, Elishia Hardaway, Emi Hardaway, Jennife Hardaway, Kassi Hardaway, Keyonna Hardaway, Kresta Hardaway, Kriston Hardaway, Laree Hardaway, Lashea Hardaway, Lauryn Hardaway, Lennie Hardaway,
Nailah Hardaway, Neena Hardaway, Nelia Hardaway, Reiko Hardaway, Sejal Hardaway, Shanea Hardaway, Sharman Hardaway, Talesha Hardaway, Teresia Hardaway, Tunya Hardaway, Annett Hardaway, Catherina
Hardaway, Charlita Hardaway, Dandrea Hardaway, Darcel Hardaway, Eleanore Hardaway, Falicia Hardaway, Glynda Hardaway, Karra Hardaway, Kerra Hardaway, Ladona Hardaway, Loretha Hardaway, Lus Hardaway,
Marketa Hardaway, Megen Hardaway, Shalini Hardaway, Sheela Hardaway, Tally Hardaway, Thuy Hardaway, Tristen Hardaway, Yashika Hardaway, Aleah Hardaway, Arline Hardaway, Catarina Hardaway, Cris
Hardaway, Dawnita Hardaway, Kelliann Hardaway, Kristle Hardaway, Lanesha Hardaway, Malanie Hardaway, Nekesha Hardaway, Rhona Hardaway, Roslynn Hardaway, Sariah Hardaway, Takeshia Hardaway, Theda
Hardaway, Altagracia Hardaway, Anneke Hardaway, Aspen Hardaway, Azalea Hardaway, Jamelle Hardaway, Kalyn Hardaway, Lavonya Hardaway, Malea Hardaway, Marykate Hardaway, Meliss Hardaway, Noelani
Hardaway, Patina Hardaway, Riann Hardaway, Rolonda Hardaway, Shaney Hardaway, Shanice Hardaway, Wynne Hardaway, Yocheved Hardaway, Aysha Hardaway, Cindie Hardaway, Daphanie Hardaway, Darcee Hardaway,
Demitra Hardaway, Francene Hardaway, Ian Hardaway, Janaya Hardaway, Jera Hardaway, Jeremiah Hardaway, Karlie Hardaway, Kiwana Hardaway, Krystin Hardaway, Larina Hardaway, Marylee Hardaway, Maurine
Hardaway, Mistee Hardaway, Rainey Hardaway, Raynette Hardaway, Robyne Hardaway, Ronelle Hardaway, Sharma Hardaway, Shelene Hardaway, Stevie Hardaway, Wynter Hardaway, Ylonda Hardaway, Zondra
Hardaway, Aesha Hardaway, Alpha Hardaway, Angala Hardaway, Che Hardaway, Courtnay Hardaway, Deatra Hardaway, Genell Hardaway, Jerrica Hardaway, Jozette Hardaway, Kerianne Hardaway, Krissi Hardaway,
Millisa Hardaway, Minna Hardaway, Naeemah Hardaway, Shantrell Hardaway, Theresia Hardaway, Tomeko Hardaway, Vanda Hardaway, Annelise Hardaway, Bettye Hardaway, Brina Hardaway, Carmelina Hardaway,
Charlotta Hardaway, Charmine Hardaway, Darbi Hardaway, Devra Hardaway, Donnamarie Hardaway, Enza Hardaway, Jack Hardaway, Lamika Hardaway, Lashaundra Hardaway, Latavia Hardaway, Mauri Hardaway,
Nevada Hardaway, Patrisha Hardaway, Rashel Hardaway, Rayann Hardaway, Shanay Hardaway, Shareka Hardaway, Sharlyn Hardaway, Sharra Hardaway, Suzzanne Hardaway, Tascha Hardaway, Tomeca Hardaway, Annisa
Hardaway, Anthea Hardaway, Bethel Hardaway, Charlesetta Hardaway, Davetta Hardaway, Deonne Hardaway, Emmie Hardaway, Fanta Hardaway, Felicitas Hardaway, Flavia Hardaway, Glenn Hardaway, Gwendolynn
Hardaway, Hollye Hardaway, Jalene Hardaway, Jamel Hardaway, Jenita Hardaway, Jonda Hardaway, Kameka Hardaway, Lagina Hardaway, Lateasha Hardaway, Louanne Hardaway, Ludivina Hardaway, Ramonita
Hardaway, Reshonda Hardaway, Romina Hardaway, Samone Hardaway, Shauntae Hardaway, Shelle Hardaway, Tasheka Hardaway, Tashima Hardaway, Tekesha Hardaway, Trini Hardaway, Tyese Hardaway, Alesa
Hardaway, Brigett Hardaway, Camella Hardaway, Charee Hardaway, Coby Hardaway, Constantina Hardaway, Contrina Hardaway, Danise Hardaway, Delanie Hardaway, Denia Hardaway, Detrice Hardaway, Donyelle
Hardaway, Estee Hardaway, Freya Hardaway, Gabriele Hardaway, Heidie Hardaway, Idella Hardaway, Iraida Hardaway, Jonica Hardaway, Kaley Hardaway, Kirsty Hardaway, Laurette Hardaway, Nataki Hardaway,
Patrisia Hardaway, Ronnetta Hardaway, Signe Hardaway, Tarita Hardaway, Trenia Hardaway, Annessa Hardaway, Celene Hardaway, Cerissa Hardaway, Chinita Hardaway, Danah Hardaway, Dannelle Hardaway,
Dawnielle Hardaway, Doretta Hardaway, Dwanna Hardaway, Jinger Hardaway, Karilyn Hardaway, Loida Hardaway, Miko Hardaway, Mirtha Hardaway, Myia Hardaway, Perri Hardaway, Ragina Hardaway, Rea Hardaway,
Saralyn Hardaway, Saskia Hardaway, Sharry Hardaway, Tiare Hardaway, Tonica Hardaway, Waleska Hardaway, Xochilt Hardaway, Zipporah Hardaway, Delynn Hardaway, Detria Hardaway, Earl Hardaway, Enedelia
Hardaway, Jeanett Hardaway, Jonita Hardaway, Kenita Hardaway, Laural Hardaway, Leonard Hardaway, Ligia Hardaway, Linnie Hardaway, Lowanda Hardaway, Ltanya Hardaway, Monquie Hardaway, Paloma Hardaway,
Rafael Hardaway, Steve Hardaway, Torina Hardaway, Yolunda Hardaway, Cariann Hardaway, Carmita Hardaway, Cerise Hardaway, Dalana Hardaway, Dawnya Hardaway, Deshannon Hardaway, Deshonda Hardaway,
Desirea Hardaway, Eartha Hardaway, Emelia Hardaway, Francoise Hardaway, Gerrie Hardaway, Gitty Hardaway, Jared Hardaway, Kianna Hardaway, Lamanda Hardaway, Laraine Hardaway, Latonda Hardaway,
Latrease Hardaway, Leaann Hardaway, Liv Hardaway, Maha Hardaway, Meliza Hardaway, Shawnetta Hardaway, Shontelle Hardaway, Sloan Hardaway, Taja Hardaway, Tammye Hardaway, Terika Hardaway, Vinita
Hardaway, Charese Hardaway, Charisa Hardaway, Danella Hardaway, Halley Hardaway, Jara Hardaway, Jawana Hardaway, Jobina Hardaway, Juniper Hardaway, Leondra Hardaway, Letonya Hardaway, Marianela
Hardaway, Mariella Hardaway, Marisel Hardaway, Marlinda Hardaway, Maurita Hardaway, Merrill Hardaway, Nalani Hardaway, Natika Hardaway, Ralph Hardaway, Shajuana Hardaway, Sharmon Hardaway, Tanda
Hardaway, Clarence Hardaway, Enjoli Hardaway, Hali Hardaway, Isaura Hardaway, Joellyn Hardaway, Kaylyn Hardaway, Ketra Hardaway, Latayna Hardaway, Melisse Hardaway, Natoshia Hardaway, Niko Hardaway,
Shalina Hardaway, Sharrie Hardaway, Silva Hardaway, Taren Hardaway, Tawnie Hardaway, Venetta Hardaway, Chenelle Hardaway, Chonda Hardaway, Drucilla Hardaway, Genene Hardaway, Jaquita Hardaway, Kally
Hardaway, Laury Hardaway, Lucina Hardaway, Lynde Hardaway, Marley Hardaway, Scherrie Hardaway, Shaila Hardaway, Shalimar Hardaway, Shannyn Hardaway, Shantina Hardaway, Shirl Hardaway, Takita
Hardaway, Tineka Hardaway, Valisa Hardaway, Adrina Hardaway, Ambre Hardaway, Arlisa Hardaway, Cecila Hardaway, Demetrica Hardaway, Dyann Hardaway, Erina Hardaway, Haven Hardaway, Hector Hardaway,
Janika Hardaway, Johnnetta Hardaway, Kellyanne Hardaway, Lacrisha Hardaway, Lamar Hardaway, Litisha Hardaway, Malkia Hardaway, Marcellina Hardaway, Otilia Hardaway, Pascale Hardaway, Ramonda
Hardaway, Safiya Hardaway, Sebrena Hardaway, Shifra Hardaway, Sina Hardaway, Tashina Hardaway, Teffany Hardaway, Trevor Hardaway, Veleka Hardaway, Zanetta Hardaway, Arie Hardaway, Azalia Hardaway,
Cedric Hardaway, Chasta Hardaway, Cherelle Hardaway, Cyndee Hardaway, Darnetta Hardaway, Dinora Hardaway, Dorena Hardaway, Howard Hardaway, Joyel Hardaway, July Hardaway, Lance Hardaway, Lore
Hardaway, Michelina Hardaway, Olive Hardaway, Reem Hardaway, Rica Hardaway, Shaneen Hardaway, Sharlotte Hardaway, Shavonna Hardaway, Shawnie Hardaway, Sherria Hardaway, Sian Hardaway, Stephanee
Hardaway, Susette Hardaway, Terilyn Hardaway, Tiajuana Hardaway, Tristin Hardaway, Vada Hardaway, Vesta Hardaway, Yara Hardaway, Alyssia Hardaway, Arian Hardaway, Chianti Hardaway, Chriselda
Hardaway, Darian Hardaway, Dawnetta Hardaway, Desarae Hardaway, Dinorah Hardaway, Djuana Hardaway, Elka Hardaway, Golda Hardaway, Honor Hardaway, Jamille Hardaway, Jesusita Hardaway, Junita Hardaway,
Kenyada Hardaway, Konnie Hardaway, Kwanza Hardaway, Lissett Hardaway, Marlon Hardaway, Marlys Hardaway, Marshelle Hardaway, Meagen Hardaway, Myka Hardaway, Mylissa Hardaway, Patrece Hardaway, Raguel
Hardaway, Sharia Hardaway, Shawnell Hardaway, Teanna Hardaway, Adrena Hardaway, Brad Hardaway, Charlisa Hardaway, Dannie Hardaway, Fernanda Hardaway, Jeania Hardaway, Kareema Hardaway, Karyl
Hardaway, Kellianne Hardaway, Lilah Hardaway, Lorelle Hardaway, Lynnell Hardaway, Quana Hardaway, Rondi Hardaway, Rosiland Hardaway, Ruben Hardaway, Sharmin Hardaway, Taniesha Hardaway, Tashonda
Hardaway, Valecia Hardaway, Alnisa Hardaway, Cherity Hardaway, Coletta Hardaway, Damary Hardaway, Darcia Hardaway, Dashawn Hardaway, Garnet Hardaway, Hadassah Hardaway, Javon Hardaway, Kareem
Hardaway, Keara Hardaway, Kerryann Hardaway, Loleta Hardaway, Marijo Hardaway, Maritsa Hardaway, Marvella Hardaway, Miriah Hardaway, Necia Hardaway, Priscila Hardaway, Ramon Hardaway, Shalawn
Hardaway, Sidra Hardaway, Sundee Hardaway, Tasia Hardaway, Tiwanda Hardaway, Xavier Hardaway, Alayne Hardaway, Anesha Hardaway, Anndrea Hardaway, Astra Hardaway, Breezy Hardaway, Carletha Hardaway,
Chantele Hardaway, Damara Hardaway, Denell Hardaway, Dessa Hardaway, Marcell Hardaway, Maretta Hardaway, Marline Hardaway, Melania Hardaway, Modesta Hardaway, Montoya Hardaway, Rashidah Hardaway,
Rusti Hardaway, Shannell Hardaway, Shaune Hardaway, Sheresa Hardaway, Stasha Hardaway, Talaya Hardaway, Taletha Hardaway, Tashawna Hardaway, Terrah Hardaway, Yessica Hardaway, Yolander Hardaway, Ani
Hardaway, Cyndy Hardaway, Dannell Hardaway, Edythe Hardaway, Elodia Hardaway, Felissa Hardaway, Jamison Hardaway, Janalyn Hardaway, Jodell Hardaway, Josetta Hardaway, Karmin Hardaway, Kenesha
Hardaway, Keyna Hardaway, Kimara Hardaway, Kiran Hardaway, Ladonya Hardaway, Latorya Hardaway, Marizol Hardaway, Nadya Hardaway, Pamula Hardaway, Pleshette Hardaway, Tamaka Hardaway, Tifanie
Hardaway, Tunisha Hardaway, Verona Hardaway, Veronda Hardaway, Allicia Hardaway, Allisha Hardaway, Allissa Hardaway, Allyn Hardaway, Andriana Hardaway, Anneka Hardaway, Bridgitte Hardaway, Brigida
Hardaway, Chaunte Hardaway, Christol Hardaway, Elita Hardaway, Jacquelene Hardaway, Jenefer Hardaway, Jessyca Hardaway, Kammi Hardaway, Kasondra Hardaway, Katricia Hardaway, Katrinka Hardaway,
Lakasha Hardaway, Maryrose Hardaway, Mirinda Hardaway, Natoya Hardaway, Omar Hardaway, Santana Hardaway, Shakima Hardaway, Shatina Hardaway, Shawnita Hardaway, Sherine Hardaway, Starlett Hardaway,
Tanga Hardaway, Teal Hardaway, Teara Hardaway, Tiffanny Hardaway, Vivien Hardaway, Altovise Hardaway, Conni Hardaway, Dawanda Hardaway, Ellena Hardaway, Janica Hardaway, Jeralyn Hardaway, Kameelah
Hardaway, Kasia Hardaway, Katharyn Hardaway, Lamesha Hardaway, Laurieann Hardaway, Leon Hardaway, Margherita Hardaway, Marguerita Hardaway, Mariama Hardaway, Mariea Hardaway, Maryfrances Hardaway,
Mayda Hardaway, Meena Hardaway, Mikal Hardaway, Nicholette Hardaway, Nitza Hardaway, Norman Hardaway, Raychelle Hardaway, Riva Hardaway, Ronit Hardaway, Rosenda Hardaway, Royce Hardaway, Saira
Hardaway, Samona Hardaway, Toye Hardaway, Zinnia Hardaway, Alfred Hardaway, Amita Hardaway, Aqueelah Hardaway, Blenda Hardaway, Charron Hardaway, Chrisie Hardaway, Ema Hardaway, Endia Hardaway,
Georgine Hardaway, Gretel Hardaway, Johnita Hardaway, Joscelyn Hardaway, Kiona Hardaway, Lamont Hardaway, Lateesha Hardaway, Liisa Hardaway, Lizet Hardaway, Marny Hardaway, Martisha Hardaway,
Marvette Hardaway, Natonya Hardaway, Oona Hardaway, Philomena Hardaway, Sahar Hardaway, Shantia Hardaway, Starlette Hardaway, Talana Hardaway, Tereza Hardaway, Torsha Hardaway, Tyeshia Hardaway,
Aislinn Hardaway, Alexandrea Hardaway, Artisha Hardaway, Assunta Hardaway, Belva Hardaway, Britany Hardaway, Bryanna Hardaway, Chauntel Hardaway, Cristel Hardaway, Damali Hardaway, Daphane Hardaway,
Dorianne Hardaway, Imogene Hardaway, Janai Hardaway, Laveda Hardaway, Lucero Hardaway, Lyssa Hardaway, Marielle Hardaway, Natalya Hardaway, Pammy Hardaway, Patches Hardaway, Rhiana Hardaway, Sakeena
Hardaway, Tanea Hardaway, Tiffinie Hardaway, Xaviera Hardaway, Arla Hardaway, Collen Hardaway, Deane Hardaway, Don Hardaway, Gari Hardaway, Jazmine Hardaway, Joani Hardaway, Jude Hardaway, Lacee
Hardaway, Lakeasha Hardaway, Leandrea Hardaway, Marvina Hardaway, Minette Hardaway, Nico Hardaway, Noell Hardaway, Ouida Hardaway, Rainbow Hardaway, Shania Hardaway, Shemekia Hardaway, Velinda
Hardaway, Yetta Hardaway, Adell Hardaway, Alyshia Hardaway, Arlana Hardaway, Athina Hardaway, Calley Hardaway, Carson Hardaway, Cassandre Hardaway, Channing Hardaway, Cherith Hardaway, Darnella
Hardaway, Deepa Hardaway, Eboney Hardaway, Florentina Hardaway, Fredia Hardaway, Jerilynn Hardaway, Kamila Hardaway, Keeli Hardaway, Korena Hardaway, Korri Hardaway, Lakeia Hardaway, Lulu Hardaway,
Madelaine Hardaway, Marja Hardaway, Maryhelen Hardaway, Mei Hardaway, Quintella Hardaway, Rainy Hardaway, Rashunda Hardaway, Rheanna Hardaway, Saran Hardaway, Shanese Hardaway, Shelisa Hardaway,
Tamecia Hardaway, Tanica Hardaway, Tawnee Hardaway, Van Hardaway, Vivienne Hardaway, Yoland Hardaway, Yvonna Hardaway, Abbi Hardaway, Afrika Hardaway, Anise Hardaway, Aron Hardaway, Carrieanne
Hardaway, Chauna Hardaway, Daena Hardaway, Elin Hardaway, Jerrilyn Hardaway, Landa Hardaway, Latoyna Hardaway, Leane Hardaway, Lilliam Hardaway, Marquisha Hardaway, Marry Hardaway, Matasha Hardaway,
Matina Hardaway, Megin Hardaway, Mitchelle Hardaway, Nakeshia Hardaway, Pearline Hardaway, Rayne Hardaway, Reeshemah Hardaway, Romaine Hardaway, Shameca Hardaway, Sterling Hardaway, Suzzette
Hardaway, Tangy Hardaway, Tonetta Hardaway, Trenice Hardaway, Zahra Hardaway, Abena Hardaway, Aliesha Hardaway, Andreia Hardaway, Beryl Hardaway, Cathey Hardaway, Crystalyn Hardaway, Davonna
Hardaway, Delphia Hardaway, Denisse Hardaway, Elina Hardaway, Elisheva Hardaway, Evangela Hardaway, Felina Hardaway, Gisella Hardaway, Ilka Hardaway, Janeth Hardaway, Katara Hardaway, Laquesha
Hardaway, Lysandra Hardaway, Malana Hardaway, Morgen Hardaway, Natali Hardaway, Niambi Hardaway, Niketa Hardaway, Pamla Hardaway, Rebeckah Hardaway, Revonda Hardaway, Sarabeth Hardaway, Sharmila
Hardaway, Shylo Hardaway, Takiya Hardaway, Trevia Hardaway, Aleatha Hardaway, Anabelle Hardaway, Blima Hardaway, Bliss Hardaway, Christyn Hardaway, Chrysta Hardaway, Cristela Hardaway, Deliah
Hardaway, Ellisa Hardaway, Genice Hardaway, Isadora Hardaway, Jawanna Hardaway, Kamaria Hardaway, Ladena Hardaway, Larenda Hardaway, Laurissa Hardaway, Leela Hardaway, Lovina Hardaway, Lyra Hardaway,
Maryanna Hardaway, Marybell Hardaway, Meika Hardaway, Meshelle Hardaway, Moana Hardaway, Nicoletta Hardaway, Radiah Hardaway, Rupal Hardaway, Shannin Hardaway, Shyra Hardaway, Stanley Hardaway,
Sundae Hardaway, Taunia Hardaway, Tomorrow Hardaway, Abbigail Hardaway, Amoreena Hardaway, Anamarie Hardaway, Chani Hardaway, Dayanara Hardaway, Delane Hardaway, Demika Hardaway, Desi Hardaway, Desma
Hardaway, Eugina Hardaway, Florine Hardaway, Johanne Hardaway, Karman Hardaway, Katheryne Hardaway, Kema Hardaway, Kenitra Hardaway, Lareina Hardaway, Laryssa Hardaway, Lasaundra Hardaway, Lorriane
Hardaway, Lurdes Hardaway, Meredyth Hardaway, Mireille Hardaway, Naomie Hardaway, Rewa Hardaway, Rosette Hardaway, Rukiya Hardaway, Shanica Hardaway, Sherill Hardaway, Shonette Hardaway, Adonica
Hardaway, Aquilla Hardaway, Billijo Hardaway, Bretta Hardaway, Charmian Hardaway, Chondra Hardaway, Coty Hardaway, Danine Hardaway, Davena Hardaway, Davia Hardaway, Falana Hardaway, Fredrica
Hardaway, Harry Hardaway, Jae Hardaway, Jawanda Hardaway, Keana Hardaway, Krisann Hardaway, Laqueta Hardaway, Lashannon Hardaway, Lavelle Hardaway, Layna Hardaway, Linde Hardaway, Lorisa Hardaway,
Lyndie Hardaway, Macey Hardaway, Maricel Hardaway, Nadina Hardaway, Nadiyah Hardaway, Nereyda Hardaway, Sequoia Hardaway, Stephene Hardaway, Tahesha Hardaway, Takeya Hardaway, Tali Hardaway, Tiera
Hardaway, Tomasina Hardaway, Verena Hardaway, Yvonda Hardaway, Aracelia Hardaway, Arletta Hardaway, Augustine Hardaway, Barry Hardaway, Brittny Hardaway, Christella Hardaway, Courtnee Hardaway,
Dakota Hardaway, Denesha Hardaway, Donnella Hardaway, Fotini Hardaway, Gita Hardaway, Holland Hardaway, Jacquiline Hardaway, Jamilyn Hardaway, Kafi Hardaway, Kalena Hardaway, Katya Hardaway, Kit
Hardaway, Laniece Hardaway, Lauree Hardaway, Leslea Hardaway, Maite Hardaway, Mitzy Hardaway, Myria Hardaway, Njeri Hardaway, Oscar Hardaway, Rain Hardaway, Rema Hardaway, Ronee Hardaway, Shevon
Hardaway, Shirlee Hardaway, Sissy Hardaway, Sonali Hardaway, Sundra Hardaway, Taralee Hardaway, Trace Hardaway, Tyana Hardaway, Agustina Hardaway, Akua Hardaway, Allyssa Hardaway, Andrina Hardaway,
Angell Hardaway, Antwanette Hardaway, Carmelia Hardaway, Chelsi Hardaway, Cheronda Hardaway, Clorinda Hardaway, Crissa Hardaway, Cyrena Hardaway, Dante Hardaway, Destini Hardaway, Emiko Hardaway, Era
Hardaway, Gentry Hardaway, Gini Hardaway, Hanan Hardaway, Haylee Hardaway, Hollee Hardaway, Jannett Hardaway, Javier Hardaway, Jenay Hardaway, Jennier Hardaway, Kandee Hardaway, Kerilyn Hardaway,
Kissy Hardaway, Konstantina Hardaway, Ladana Hardaway, Logan Hardaway, Lovette Hardaway, Manal Hardaway, Marykay Hardaway, Phoenix Hardaway, Rosalin Hardaway, Shatara Hardaway, Shawnya Hardaway,
Sheralyn Hardaway, Sheretta Hardaway, Suann Hardaway, Tava Hardaway, Bobette Hardaway, Carrissa Hardaway, Coralee Hardaway, Daneille Hardaway, Edwin Hardaway, Emelda Hardaway, Flecia Hardaway, Jame
Hardaway, Janeane Hardaway, Jelena Hardaway, Katja Hardaway, Kelia Hardaway, Lanee Hardaway, Laverna Hardaway, Lorra Hardaway, Marlin Hardaway, Mieke Hardaway, Miroslava Hardaway, Nekeisha Hardaway,
Nikeya Hardaway, Noami Hardaway, Saida Hardaway, Sienna Hardaway, Sundi Hardaway, Tylene Hardaway, Yakima Hardaway, Alicha Hardaway, Amani Hardaway, Armandina Hardaway, Bethani Hardaway, Carine
Hardaway, Carmencita Hardaway, Chantee Hardaway, Chavonne Hardaway, Chinyere Hardaway, Clarisse Hardaway, Evett Hardaway, Felisia Hardaway, Flossie Hardaway, Indra Hardaway, Inge Hardaway, Keiko
Hardaway, Kristia Hardaway, Lady Hardaway, Latrelle Hardaway, Lean Hardaway, Lewanda Hardaway, Marit Hardaway, Melodye Hardaway, Moneka Hardaway, Naisha Hardaway, Najah Hardaway, Ronisha Hardaway,
Shamona Hardaway, Shantal Hardaway, Sharay Hardaway, Shawndell Hardaway, Shekita Hardaway, Shelah Hardaway, Shenise Hardaway, Soraida Hardaway, Sujey Hardaway, Suni Hardaway, Tiona Hardaway, Tuwana
Hardaway, Wynetta Hardaway, Angelie Hardaway, Annalise Hardaway, Carlen Hardaway, Cherryl Hardaway, Donnelle Hardaway, Florencia Hardaway, Gema Hardaway, Genita Hardaway, Happy Hardaway, Jessy
Hardaway, Karimah Hardaway, Lanitra Hardaway, Louanna Hardaway, Maisie Hardaway, Marco Hardaway, Margaretta Hardaway, Marguita Hardaway, Marice Hardaway, Marygrace Hardaway, Retta Hardaway, Roderick
Hardaway, Ronny Hardaway, Shalom Hardaway, Sherena Hardaway, Tamura Hardaway, Threasa Hardaway, Ulonda Hardaway, Zachary Hardaway, Aletta Hardaway, Amaryllis Hardaway, Amethyst Hardaway, Antoinetta
Hardaway, Aricka Hardaway, Bobbyjo Hardaway, Camile Hardaway, Charyl Hardaway, Elysa Hardaway, Jameela Hardaway, Jeannett Hardaway, Jyl Hardaway, Kinda Hardaway, Kristianne Hardaway, Latifah
Hardaway, Lavera Hardaway, Madaline Hardaway, Marea Hardaway, Melaina Hardaway, Natesha Hardaway, Oliva Hardaway, Renell Hardaway, Salli Hardaway, Satina Hardaway, Shanyn Hardaway, Sharelle Hardaway,
Sharena Hardaway, Shenetta Hardaway, Sherlonda Hardaway, Sun Hardaway, Teah Hardaway, Teana Hardaway, Tramaine Hardaway, Wenda Hardaway, Wyndi Hardaway, Yoko Hardaway, Adrean Hardaway, Aishah
Hardaway, Aundra Hardaway, Betsaida Hardaway, Chrisann Hardaway, Clarita Hardaway, Correna Hardaway, Cynitha Hardaway, Cythia Hardaway, Damien Hardaway, Dedria Hardaway, Divina Hardaway, Dody
Hardaway, Elanda Hardaway, Elizbeth Hardaway, Estrellita Hardaway, Eydie Hardaway, Gerilyn Hardaway, Hedy Hardaway, Jacklynn Hardaway, Kimberleigh Hardaway, Kristinia Hardaway, Lakeeta Hardaway,
Larie Hardaway, Latrisa Hardaway, Lesleigh Hardaway, Marella Hardaway, Martinique Hardaway, Melessia Hardaway, Nicolasa Hardaway, Quantina Hardaway, Remy Hardaway, Robina Hardaway, Roya Hardaway,
Saba Hardaway, Sacheen Hardaway, Shalee Hardaway, Sharlet Hardaway, Shereese Hardaway, Sherronda Hardaway, Shewanda Hardaway, Skyla Hardaway, Tallie Hardaway, Tanganyika Hardaway, Taressa Hardaway,
Tarshia Hardaway, Timi Hardaway, Zola Hardaway, Ainsley Hardaway, Alivia Hardaway, Ardith Hardaway, Artina Hardaway, Ashely Hardaway, Athanasia Hardaway, Chantale Hardaway, Chimene Hardaway, Corinn
Hardaway, Daffney Hardaway, Danene Hardaway, Daphnie Hardaway, Deeanne Hardaway, Erynn Hardaway, Glennis Hardaway, Kalee Hardaway, Kandyce Hardaway, Kathren Hardaway, Kelcey Hardaway, Laticha
Hardaway, Lenee Hardaway, Luvenia Hardaway, Machele Hardaway, Maris Hardaway, Marisella Hardaway, Nakeia Hardaway, Nickey Hardaway, Nikkita Hardaway, Noella Hardaway, Peri Hardaway, Raushanah
Hardaway, Renisha Hardaway, Shahidah Hardaway, Shakisha Hardaway, Shevonne Hardaway, Tashema Hardaway, Tondalaya Hardaway, Trinita Hardaway, Akiko Hardaway, Alfredia Hardaway, Bathsheba Hardaway,
Billye Hardaway, Carter Hardaway, Clementina Hardaway, Danyele Hardaway, Darnisha Hardaway, Darryl Hardaway, Eda Hardaway, Emy Hardaway, Faustina Hardaway, Hayden Hardaway, Hortensia Hardaway, Ivon
Hardaway, Javonna Hardaway, Keita Hardaway, Lakeisa Hardaway, Lan Hardaway, Leota Hardaway, Letasha Hardaway, Maggi Hardaway, Monita Hardaway, Nadirah Hardaway, Nikka Hardaway, Noemy Hardaway, Roxy
Hardaway, Sabrinia Hardaway, Sanda Hardaway, Sharan Hardaway, Sharnette Hardaway, Swati Hardaway, Tiny Hardaway, Tonjia Hardaway, Trella Hardaway, Venecia Hardaway, Zana Hardaway, Zorana Hardaway,
Adrienna Hardaway, Ama Hardaway, Angelisa Hardaway, Beckey Hardaway, Bobbye Hardaway, Cammi Hardaway, Chequita Hardaway, Clinton Hardaway, Dama Hardaway, Danee Hardaway, Dari Hardaway, Deandre
Hardaway, Deetta Hardaway, Devonda Hardaway, Edelmira Hardaway, Ieasha Hardaway, Joylynn Hardaway, Krisinda Hardaway, Kristol Hardaway, Kwana Hardaway, Lacresia Hardaway, Ladawna Hardaway, Lavanda
Hardaway, Lavena Hardaway, Leaha Hardaway, Massiel Hardaway, Michiko Hardaway, Naketa Hardaway, Persephone Hardaway, Raynell Hardaway, Ronell Hardaway, Shalynn Hardaway, Shimeka Hardaway, Shiquita
Hardaway, Tanaka Hardaway, Tomasita Hardaway, Trixy Hardaway, Tyson Hardaway, Andy Hardaway, Arden Hardaway, Carlissa Hardaway, Chery Hardaway, Duane Hardaway, Ereka Hardaway, Gracia Hardaway, Halona
Hardaway, Jamal Hardaway, Kechia Hardaway, Kelda Hardaway, Latessa Hardaway, Lynett Hardaway, Mikala Hardaway, Naida Hardaway, Nancee Hardaway, Renna Hardaway, Resha Hardaway, Rifka Hardaway, Rolinda
Hardaway, Shaneika Hardaway, Syliva Hardaway, Tamikka Hardaway, Tamyka Hardaway, Tarika Hardaway, Tawania Hardaway, Trinda Hardaway, Trinetta Hardaway, Valrie Hardaway, Anabela Hardaway, Andretta
Hardaway, Capricia Hardaway, Chistina Hardaway, Coree Hardaway, Darrah Hardaway, Diahann Hardaway, Faigy Hardaway, Georgiann Hardaway, Hillarie Hardaway, Ieisha Hardaway, Joeann Hardaway, Jovana
Hardaway, Kesa Hardaway, Ladeana Hardaway, Laresa Hardaway, Lisset Hardaway, Lynnann Hardaway, Lysette Hardaway, Melenie Hardaway, Natina Hardaway, Nicolina Hardaway, Queena Hardaway, Tameaka
Hardaway, Tansy Hardaway, Terressa Hardaway, Topeka Hardaway, Alicen Hardaway, Aneka Hardaway, Arlyn Hardaway, Bethzaida Hardaway, Caralyn Hardaway, Cartina Hardaway, Cathlene Hardaway, Chaquita
Hardaway, Clifford Hardaway, Danitra Hardaway, Dayana Hardaway, Delania Hardaway, Dennie Hardaway, Edina Hardaway, Electra Hardaway, Elouise Hardaway, Ginelle Hardaway, Heath Hardaway, Kassia
Hardaway, Katrenia Hardaway, Keidra Hardaway, Laconda Hardaway, Loura Hardaway, Marchell Hardaway, Mayela Hardaway, Merle Hardaway, Merlinda Hardaway, Mychelle Hardaway, Nannie Hardaway, Nikko
Hardaway, Nori Hardaway, Prescilla Hardaway, Rika Hardaway, Ronika Hardaway, Sahara Hardaway, Sandrea Hardaway, Senora Hardaway, Shawntee Hardaway, Sheryll Hardaway, Shina Hardaway, Terre Hardaway,
Tinna Hardaway, Tonnette Hardaway, Torre Hardaway, Tya Hardaway, Vianey Hardaway, Zora Hardaway, Adonna Hardaway, Alberto Hardaway, Almee Hardaway, Ashanta Hardaway, Bridgid Hardaway, Cindia
Hardaway, Cricket Hardaway, Darah Hardaway, Dineen Hardaway, Elan Hardaway, Iman Hardaway, Jacquelynne Hardaway, Jennilyn Hardaway, Jessicca Hardaway, Julio Hardaway, Kenosha Hardaway, Kimyata
Hardaway, Lanessa Hardaway, Leesha Hardaway, Letoya Hardaway, Nkenge Hardaway, Priti Hardaway, Raizel Hardaway, Rosaisela Hardaway, Sameerah Hardaway, Sera Hardaway, Shallon Hardaway, Sharmane
Hardaway, Shauntay Hardaway, Sherhonda Hardaway, Shonnie Hardaway, Sueanne Hardaway, Taneesha Hardaway, Telina Hardaway, Tenecia Hardaway, Tyanne Hardaway, Alexsandra Hardaway, Angeles Hardaway,
Charlean Hardaway, Danel Hardaway, Davi Hardaway, Daysha Hardaway, Demaris Hardaway, Dorenda Hardaway, Dorita Hardaway, Dorlisa Hardaway, Genee Hardaway, Jamella Hardaway, Kathyjo Hardaway, Kaya
Hardaway, Latondra Hardaway, Leshia Hardaway, Mahala Hardaway, Marija Hardaway, Maudie Hardaway, Megann Hardaway, Phuong Hardaway, Reesa Hardaway, Ronya Hardaway, Selinda Hardaway, Shama Hardaway,
Shirell Hardaway, Shonita Hardaway, Taiwana Hardaway, Takenya Hardaway, Talonda Hardaway, Tassie Hardaway, Telena Hardaway, Theodore Hardaway, Violetta Hardaway, Willetta Hardaway, Adora Hardaway,
Alvin Hardaway, Chauncey Hardaway, Chrisy Hardaway, Clancy Hardaway, Claressa Hardaway, Corrinna Hardaway, Darra Hardaway, Deshawna Hardaway, Donise Hardaway, Elona Hardaway, Gara Hardaway, Georgene
Hardaway, Gila Hardaway, Jania Hardaway, Jocelynn Hardaway, Joei Hardaway, Kassy Hardaway, Liset Hardaway, Maleka Hardaway, Mashell Hardaway, Melida Hardaway, Michalene Hardaway, Moya Hardaway,
Nyisha Hardaway, Rayanne Hardaway, Renia Hardaway, Salima Hardaway, Samanthia Hardaway, Sammy Hardaway, Santosha Hardaway, Shaneeka Hardaway, Shawntelle Hardaway, Shwanda Hardaway, Sita Hardaway,
Tarin Hardaway, Tawn Hardaway, Tosca Hardaway, Valisha Hardaway, Vangie Hardaway, Vernon Hardaway, Zella Hardaway, Amarilis Hardaway, Amelie Hardaway, Aubri Hardaway, Bobbette Hardaway, Byron
Hardaway, Chance Hardaway, Deshon Hardaway, Eraina Hardaway, Jaycee Hardaway, Joyell Hardaway, Keirsten Hardaway, Kendi Hardaway, Kendrea Hardaway, Keyanna Hardaway, Khristie Hardaway, Kimberlyann
Hardaway, Kimyetta Hardaway, Kinberly Hardaway, Lakeita Hardaway, Lakina Hardaway, Leida Hardaway, Lenae Hardaway, Mariette Hardaway, Marymargaret Hardaway, Meera Hardaway, Micky Hardaway, Naja
Hardaway, Oneka Hardaway, Pedro Hardaway, Rennie Hardaway, Sharlee Hardaway, Sharonne Hardaway, Shonya Hardaway, Solange Hardaway, Tashawn Hardaway, Amorette Hardaway, Andrienne Hardaway, Ari
Hardaway, Carolin Hardaway, Chalonda Hardaway, Cinthya Hardaway, Drusilla Hardaway, Erendira Hardaway, Franki Hardaway, Fredrick Hardaway, Isaac Hardaway, Jacklin Hardaway, Jamese Hardaway, Jeani
Hardaway, Josi Hardaway, Jullie Hardaway, Kadie Hardaway, Kennetta Hardaway, Keonna Hardaway, Krisanne Hardaway, Lakrisha Hardaway, Lakysha Hardaway, Lasheka Hardaway, Levette Hardaway, Lexi
Hardaway, Lezlee Hardaway, Lovely Hardaway, Marah Hardaway, Markeeta Hardaway, Marlow Hardaway, Meloni Hardaway, Natausha Hardaway, Nate Hardaway, Neoma Hardaway, Nicci Hardaway, Niema Hardaway,
Nykia Hardaway, Oma Hardaway, Quandra Hardaway, Rasha Hardaway, Shenelle Hardaway, Sheryle Hardaway, Sylena Hardaway, Tasheba Hardaway, Tennile Hardaway, Terriann Hardaway, Tesia Hardaway, Tuere
Hardaway, Tynika Hardaway, Valena Hardaway, Beverlee Hardaway, Carriann Hardaway, Chaney Hardaway, Chantae Hardaway, Charie Hardaway, Cherell Hardaway, Chrystina Hardaway, Cookie Hardaway, Delphina
Hardaway, Denys Hardaway, Digna Hardaway, Doriann Hardaway, Elecia Hardaway, Elmira Hardaway, Irina Hardaway, Jacky Hardaway, Jimmi Hardaway, Joslin Hardaway, Katty Hardaway, Kimberlin Hardaway,
Laquan Hardaway, Latonga Hardaway, Lauran Hardaway, Lei Hardaway, Maralee Hardaway, Marilena Hardaway, Markeisha Hardaway, Mena Hardaway, Mike Hardaway, Nadeen Hardaway, Ranelle Hardaway, Renika
Hardaway, Ritu Hardaway, Roshawnda Hardaway, Sandria Hardaway, Shann Hardaway, Shellene Hardaway, Shemica Hardaway, Shenandoah Hardaway, Sherrye Hardaway, Sholonda Hardaway, Teka Hardaway, Temica
Hardaway, Thomasena Hardaway, Vallerie Hardaway, Velva Hardaway, Annjanette Hardaway, Arnitra Hardaway, Asa Hardaway, Barri Hardaway, Deborha Hardaway, Deshaun Hardaway, Disa Hardaway, Doree
Hardaway, Enrique Hardaway, Fredrika Hardaway, Gisel Hardaway, Jemima Hardaway, Kellene Hardaway, Kinsey Hardaway, Lanice Hardaway, Leonore Hardaway, Liesel Hardaway, Loredana Hardaway, Lynita
Hardaway, Maeve Hardaway, Michale Hardaway, Michalle Hardaway, Nadira Hardaway, Orlando Hardaway, Radonna Hardaway, Rivkah Hardaway, Sanita Hardaway, Shaketa Hardaway, Shatika Hardaway, Stepheny
Hardaway, Tashanda Hardaway, Tashanna Hardaway, Thera Hardaway, Vernette Hardaway, Virna Hardaway, Wanetta Hardaway, Andee Hardaway, Aparna Hardaway, Armando Hardaway, Brynne Hardaway, Carolynne
Hardaway, Danyal Hardaway, Drena Hardaway, Faydra Hardaway, Felishia Hardaway, Gwendalyn Hardaway, Jennyfer Hardaway, Jesika Hardaway, Johnell Hardaway, Kalen Hardaway, Keelie Hardaway, Kirby
Hardaway, Kristopher Hardaway, Lynnett Hardaway, Mikka Hardaway, Moraima Hardaway, Nasha Hardaway, Nechelle Hardaway, Nelson Hardaway, Rashawnda Hardaway, Rianna Hardaway, Rogina Hardaway, Rudy
Hardaway, Schwanda Hardaway, Serafina Hardaway, Seth Hardaway, Shantrice Hardaway, Shenee Hardaway, Sherrel Hardaway, Shirin Hardaway, Soila Hardaway, Sujata Hardaway, Tannia Hardaway, Tiya Hardaway,
Tonnie Hardaway, Tuwanda Hardaway, Young Hardaway, Anjela Hardaway, Aquanetta Hardaway, Ardis Hardaway, Brigitta Hardaway, Carmilla Hardaway, Catherin Hardaway, Charna Hardaway, Chaunda Hardaway,
Chela Hardaway, Cherylyn Hardaway, Clifton Hardaway, Danice Hardaway, Daylene Hardaway, Delecia Hardaway, Dell Hardaway, Eleonora Hardaway, Gesenia Hardaway, Giana Hardaway, Iisha Hardaway, Jacqualyn
Hardaway, Jule Hardaway, Kanita Hardaway, Karrin Hardaway, Keishia Hardaway, Kerith Hardaway, Kima Hardaway, Letica Hardaway, Letta Hardaway, Lucianna Hardaway, Mala Hardaway, Maresa Hardaway,
Michaeline Hardaway, Mitra Hardaway, Norene Hardaway, Raul Hardaway, Robi Hardaway, Sativa Hardaway, Shakeema Hardaway, Shalisha Hardaway, Sherece Hardaway, Sherlene Hardaway, Talicia Hardaway, Tinia
Hardaway, Toneka Hardaway, Tuwanna Hardaway, Tymeka Hardaway, Vesna Hardaway, Yvetta Hardaway, Akeisha Hardaway, Almeta Hardaway, Alvita Hardaway, Ameerah Hardaway, Amena Hardaway, Anntoinette
Hardaway, Asusena Hardaway, Benetta Hardaway, Bernard Hardaway, Bibiana Hardaway, Bracha Hardaway, Branden Hardaway, Carrisa Hardaway, Channell Hardaway, Christee Hardaway, Daisey Hardaway, Dandra
Hardaway, Dene Hardaway, Destinee Hardaway, Emmalee Hardaway, Evamarie Hardaway, Felicha Hardaway, Hally Hardaway, Harmonie Hardaway, Ivone Hardaway, Janira Hardaway, Jennfier Hardaway, Josalyn
Hardaway, Karley Hardaway, Kellyjo Hardaway, Kinshasa Hardaway, Laurena Hardaway, Lewis Hardaway, Louvenia Hardaway, Lynley Hardaway, Maleah Hardaway, Marceline Hardaway, Maree Hardaway, Maxie
Hardaway, Melora Hardaway, Minta Hardaway, Nella Hardaway, Nikea Hardaway, Orlanda Hardaway, Radhika Hardaway, Renelle Hardaway, Robynne Hardaway, Rosland Hardaway, Saudia Hardaway, Shannette
Hardaway, Sharetta Hardaway, Sharnita Hardaway, Shequita Hardaway, Tanessa Hardaway, Tannis Hardaway, Thu Hardaway, Toiya Hardaway, Umeka Hardaway, Yadhira Hardaway, Birdie Hardaway, Bradi Hardaway,
Carola Hardaway, Chelle Hardaway, Chermaine Hardaway, Christelle Hardaway, Cortnie Hardaway, Dalinda Hardaway, Daphene Hardaway, Demetri Hardaway, Ennifer Hardaway, Esta Hardaway, Francheska
Hardaway, Griselle Hardaway, Ife Hardaway, Jahaira Hardaway, Jammi Hardaway, Javonne Hardaway, Jenie Hardaway, Jocelin Hardaway, Karole Hardaway, Kelee Hardaway, Khrista Hardaway, Leya Hardaway,
Luwanda Hardaway, Makayla Hardaway, Marly Hardaway, Nzinga Hardaway, Paquita Hardaway, Ramanda Hardaway, Rayleen Hardaway, Rinda Hardaway, Shauntell Hardaway, Tamecka Hardaway, Tanji Hardaway, Tela
Hardaway, Teneisha Hardaway, Valicia Hardaway, Willona Hardaway, Alejandro Hardaway, Ameka Hardaway, Ashante Hardaway, Brisa Hardaway, Charli Hardaway, Charmagne Hardaway, Charrise Hardaway, Crystol
Hardaway, Darya Hardaway, Destiney Hardaway, Elidia Hardaway, France Hardaway, Galadriel Hardaway, Genifer Hardaway, Gladis Hardaway, Glori Hardaway, Harriette Hardaway, Ioanna Hardaway, Jeanann
Hardaway, Kittie Hardaway, Ladon Hardaway, Latesa Hardaway, Linn Hardaway, Loralie Hardaway, Mazie Hardaway, Mekisha Hardaway, Nivia Hardaway, Paulene Hardaway, Renette Hardaway, Saadia Hardaway,
Shakena Hardaway, Sheritta Hardaway, Stepahnie Hardaway, Taliah Hardaway, Teretha Hardaway, Tessy Hardaway, Wendelyn Hardaway, Zonia Hardaway, Ailsa Hardaway, Alonna Hardaway, Andres Hardaway, Asma
Hardaway, Austin Hardaway, Brandis Hardaway, Corinda Hardaway, Deetra Hardaway, Dyanne Hardaway, Eliabeth Hardaway, Evelynn Hardaway, Geana Hardaway, Geeta Hardaway, Georgeanne Hardaway, Glynnis
Hardaway, Hilari Hardaway, Hindy Hardaway, Jackquline Hardaway, Joaquina Hardaway, Jolynne Hardaway, Kirsta Hardaway, Latissa Hardaway, Lucile Hardaway, Makeisha Hardaway, Marli Hardaway, Meeghan
Hardaway, Michaelyn Hardaway, Monte Hardaway, Omeka Hardaway, Patra Hardaway, Ragen Hardaway, Reshma Hardaway, Ronnell Hardaway, Shakera Hardaway, Shantella Hardaway, Shar Hardaway, Shelagh Hardaway,
Sherisse Hardaway, Slyvia Hardaway, Stephenia Hardaway, Tarisha Hardaway, Tria Hardaway, Tynetta Hardaway, Veena Hardaway, Aleksandra Hardaway, Analia Hardaway, Aneshia Hardaway, Anesia Hardaway,
Anissia Hardaway, Caroll Hardaway, Celisa Hardaway, Chariti Hardaway, Charnette Hardaway, Cheria Hardaway, Dabney Hardaway, Davon Hardaway, Demita Hardaway, Denny Hardaway, Dwanda Hardaway, Erick
Hardaway, Fancy Hardaway, Felesha Hardaway, Genette Hardaway, Graziella Hardaway, Helaine Hardaway, Huong Hardaway, Idania Hardaway, Ilsa Hardaway, Iola Hardaway, Janifer Hardaway, Jaynie Hardaway,
Jil Hardaway, Joyann Hardaway, Kalina Hardaway, Kana Hardaway, Katheleen Hardaway, Kelie Hardaway, Kimbly Hardaway, Ladina Hardaway, Lamisha Hardaway, Landra Hardaway, Laretha Hardaway, Lashuna
Hardaway, Latorsha Hardaway, Lehua Hardaway, Libbie Hardaway, Loris Hardaway, Lovetta Hardaway, Marilin Hardaway, Marquis Hardaway, Medea Hardaway, Michaella Hardaway, Milly Hardaway, Montina
Hardaway, Muna Hardaway, Nickcole Hardaway, Nickia Hardaway, Quianna Hardaway, Retina Hardaway, Rubina Hardaway, Sallyann Hardaway, Sangeeta Hardaway, Shalinda Hardaway, Shatonya Hardaway, Tamasha
Hardaway, Tenise Hardaway, Terrilynn Hardaway, Teya Hardaway, Toma Hardaway, Trang Hardaway, Vienna Hardaway, Zenda Hardaway, Amisha Hardaway, Chenell Hardaway, Cherene Hardaway, Chimere Hardaway,
Darcell Hardaway, Desa Hardaway, Devi Hardaway, Dove Hardaway, Eilene Hardaway, Jarita Hardaway, Jenetta Hardaway, Johnie Hardaway, Kanani Hardaway, Kanya Hardaway, Katiria Hardaway, Kenyon Hardaway,
Khristi Hardaway, Kidada Hardaway, Kristien Hardaway, Laressa Hardaway, Lashonta Hardaway, Lataisha Hardaway, Leanda Hardaway, Leshonda Hardaway, Letia Hardaway, Lorenzo Hardaway, Makenzie Hardaway,
Manya Hardaway, Maribelle Hardaway, Maryjean Hardaway, Moniqua Hardaway, Myeisha Hardaway, Mystie Hardaway, Nakeesha Hardaway, Neeley Hardaway, Neil Hardaway, Nida Hardaway, Oletha Hardaway, Rebel
Hardaway, Rosaland Hardaway, Satonya Hardaway, Shanah Hardaway, Sherly Hardaway, Sherrilyn Hardaway, Shonia Hardaway, Shuna Hardaway, Shylah Hardaway, Tameisha Hardaway, Telecia Hardaway, Terrill
Hardaway, Torry Hardaway, Tracyann Hardaway, Undrea Hardaway, Usha Hardaway, Velicia Hardaway, Windie Hardaway, Yehudis Hardaway, Ameenah Hardaway, Amika Hardaway, Anupama Hardaway, Bethaney
Hardaway, Brookie Hardaway, Bryce Hardaway, Cherika Hardaway, Courtni Hardaway, Danyle Hardaway, Darilyn Hardaway, Darnita Hardaway, Eveline Hardaway, Gilbert Hardaway, Giovanni Hardaway, Jacquel
Hardaway, Jenai Hardaway, Jocelyne Hardaway, Jull Hardaway, Karene Hardaway, Karisha Hardaway, Kelvin Hardaway, Kenzie Hardaway, Kimblery Hardaway, Kyleen Hardaway, Laketta Hardaway, Larisha
Hardaway, Lashae Hardaway, Laveta Hardaway, Lezli Hardaway, Marlynn Hardaway, Melana Hardaway, Meribeth Hardaway, Onita Hardaway, Philicia Hardaway, Raychel Hardaway, Roschelle Hardaway, Rosilyn
Hardaway, Sandhya Hardaway, Selenia Hardaway, Seretha Hardaway, Shadia Hardaway, Shalise Hardaway, Sonnet Hardaway, Tacey Hardaway, Taurus Hardaway, Telly Hardaway, Tenita Hardaway, Tiawana Hardaway,
Timica Hardaway, Warren Hardaway, Yamile Hardaway, Yaminah Hardaway, Adel Hardaway, Aiyana Hardaway, Amybeth Hardaway, Angee Hardaway, Angeligue Hardaway, Areli Hardaway, Asheley Hardaway, Bina
Hardaway, Celines Hardaway, Charlet Hardaway, Charley Hardaway, Charonda Hardaway, Dameka Hardaway, Danne Hardaway, Deloise Hardaway, Denay Hardaway, Denene Hardaway, Donata Hardaway, Georgianne
Hardaway, Jamelia Hardaway, Jennetta Hardaway, Jesusa Hardaway, Johnda Hardaway, Juleen Hardaway, Khaliah Hardaway, Kimala Hardaway, Kimble Hardaway, Kimmarie Hardaway, Korene Hardaway, Lezley
Hardaway, Lindie Hardaway, Lorilyn Hardaway, Mayme Hardaway, Meeka Hardaway, Natlie Hardaway, Ranita Hardaway, Rasheen Hardaway, Rayetta Hardaway, Savina Hardaway, Scarlette Hardaway, Shaindy
Hardaway, Sheana Hardaway, Sheldon Hardaway, Shoni Hardaway, Stephan Hardaway, Taneeka Hardaway, Tanesia Hardaway, Tannya Hardaway, Tionna Hardaway, Tresia Hardaway, Tylisha Hardaway, Vonita
Hardaway, Vonnetta Hardaway, Wren Hardaway, Akiba Hardaway, Blaire Hardaway, Buffey Hardaway, Candita Hardaway, Charnelle Hardaway, Cynda Hardaway, Danesha Hardaway, Darbie Hardaway, Deah Hardaway,
Deette Hardaway, Devita Hardaway, Devonia Hardaway, Dianah Hardaway, Dustina Hardaway, Dylan Hardaway, Elli Hardaway, Eun Hardaway, Geneen Hardaway, Geni Hardaway, Jenevieve Hardaway, Julaine
Hardaway, Kaitlyn Hardaway, Lalania Hardaway, Latrise Hardaway, Mahalia Hardaway, Manon Hardaway, Marycatherine Hardaway, Nyasha Hardaway, Orlena Hardaway, Petina Hardaway, Precilla Hardaway, Raenell
Hardaway, Ramonica Hardaway, Reema Hardaway, Scotti Hardaway, Shakeya Hardaway, Shaylene Hardaway, Shenique Hardaway, Shimika Hardaway, Takiesha Hardaway, Tambi Hardaway, Tammatha Hardaway, Tauheedah
Hardaway, Timiko Hardaway, Tomisha Hardaway, Venesa Hardaway, Venise Hardaway, Akesha Hardaway, Anmarie Hardaway, Arnette Hardaway, Brennan Hardaway, Cayla Hardaway, Cecil Hardaway, Coreena Hardaway,
Danessa Hardaway, Day Hardaway, Denni Hardaway, Fernando Hardaway, Floretta Hardaway, Fred Hardaway, Gardenia Hardaway, Genevie Hardaway, Georgann Hardaway, Grasiela Hardaway, Jaimy Hardaway,
Jaquelyn Hardaway, Jerelyn Hardaway, Jinnifer Hardaway, Johnelle Hardaway, Jovonna Hardaway, Kady Hardaway, Keala Hardaway, Kesi Hardaway, Kyana Hardaway, Laketia Hardaway, Laquanta Hardaway, Latoy
Hardaway, Laurine Hardaway, Lynea Hardaway, Lynee Hardaway, Maegen Hardaway, Margarite Hardaway, Marijane Hardaway, Mellany Hardaway, Monia Hardaway, Monice Hardaway, Mylene Hardaway, Nakiya
Hardaway, Neila Hardaway, Oleta Hardaway, Queenie Hardaway, Quinetta Hardaway, Rekha Hardaway, Remi Hardaway, Rickey Hardaway, Roselynn Hardaway, Samanatha Hardaway, Shakila Hardaway, Shalandra
Hardaway, Shamira Hardaway, Shandel Hardaway, Shandrea Hardaway, Shinita Hardaway, Stayce Hardaway, Takela Hardaway, Talibah Hardaway, Tameria Hardaway, Taronda Hardaway, Telia Hardaway, Ticia
Hardaway, Tovah Hardaway, Venissa Hardaway, Verlinda Hardaway, Vittoria Hardaway, Alicyn Hardaway, Amylynn Hardaway, Anabell Hardaway, Anela Hardaway, Annice Hardaway, Aubry Hardaway, Coralie
Hardaway, Coryn Hardaway, Darlyn Hardaway, Dejuana Hardaway, Eleana Hardaway, Esmerelda Hardaway, Georganne Hardaway, Jacqlyn Hardaway, Jodine Hardaway, Joelyn Hardaway, Josey Hardaway, Keanna
Hardaway, Kendy Hardaway, Korine Hardaway, Kristianna Hardaway, Laneka Hardaway, Lannie Hardaway, Mathilda Hardaway, Myiesha Hardaway, Nessa Hardaway, Nikiya Hardaway, Penina Hardaway, Radha
Hardaway, Ralonda Hardaway, Shantele Hardaway, Sharlette Hardaway, Shelanda Hardaway, Shelda Hardaway, Shere Hardaway, Shuntel Hardaway, Stephaney Hardaway, Tabbitha Hardaway, Tamkia Hardaway, Teona
Hardaway, Terrisa Hardaway, Teryl Hardaway, Trellis Hardaway, Vanissa Hardaway, Abigale Hardaway, Bonnita Hardaway, Carren Hardaway, Clementine Hardaway, Coni Hardaway, Demonica Hardaway, Denis
Hardaway, Diantha Hardaway, Edana Hardaway, Elene Hardaway, Faithe Hardaway, Franklin Hardaway, Haidee Hardaway, Jaima Hardaway, Javonda Hardaway, Jayma Hardaway, Jeffifer Hardaway, Jennel Hardaway,
Jennfer Hardaway, Katryna Hardaway, Keta Hardaway, Kristyne Hardaway, Krystle Hardaway, Kymberlee Hardaway, Kymberli Hardaway, Lacole Hardaway, Laranda Hardaway, Leea Hardaway, Loryn Hardaway,
Lyndell Hardaway, Lyndy Hardaway, Mande Hardaway, Marlissa Hardaway, Marrissa Hardaway, Meko Hardaway, Miryam Hardaway, Neida Hardaway, Odilia Hardaway, Pamila Hardaway, Porscha Hardaway, Quentina
Hardaway, Riley Hardaway, Roselia Hardaway, Sharifah Hardaway, Sonyia Hardaway, Sugar Hardaway, Tamico Hardaway, Tanara Hardaway, Tiombe Hardaway, Verity Hardaway, Zoey Hardaway, Abril Hardaway, Aine
Hardaway, Aixa Hardaway, Ange Hardaway, Annah Hardaway, Avelina Hardaway, Chantil Hardaway, Charmel Hardaway, Cheralyn Hardaway, Chesley Hardaway, Colin Hardaway, Cornelius Hardaway, Cortni Hardaway,
Deserae Hardaway, Dru Hardaway, Emeline Hardaway, Iran Hardaway, Jesslyn Hardaway, Jovon Hardaway, Kalani Hardaway, Kaleen Hardaway, Kamilla Hardaway, Kemi Hardaway, Kendrick Hardaway, Lachele
Hardaway, Letesha Hardaway, Makita Hardaway, Maricia Hardaway, Mariya Hardaway, Marlise Hardaway, Nara Hardaway, Ngina Hardaway, Nikkol Hardaway, Okema Hardaway, Palma Hardaway, Priscillia Hardaway,
Ranell Hardaway, Rashmi Hardaway, Renessa Hardaway, Ricarda Hardaway, Rozlyn Hardaway, Shamia Hardaway, Sharissa Hardaway, Shianne Hardaway, Takeesha Hardaway, Tekia Hardaway, Tyhesha Hardaway,
Vanassa Hardaway, Vicenta Hardaway, Abagail Hardaway, Addy Hardaway, Akemi Hardaway, Analilia Hardaway, Andreanna Hardaway, Anh Hardaway, Annmargaret Hardaway, Caitlyn Hardaway, Carrin Hardaway,
Celestial Hardaway, Chameka Hardaway, Chandrika Hardaway, Charlsie Hardaway, Cherlynn Hardaway, Darien Hardaway, Darlean Hardaway, Devyn Hardaway, Dyane Hardaway, Eman Hardaway, Erna Hardaway,
Fontella Hardaway, Halie Hardaway, Ilyse Hardaway, Janece Hardaway, Jayla Hardaway, Jeannene Hardaway, Juwana Hardaway, Kachina Hardaway, Keleigh Hardaway, Kissie Hardaway, Krisi Hardaway, Kristyl
Hardaway, Krystie Hardaway, Kyesha Hardaway, Lizandra Hardaway, Lyda Hardaway, Majorie Hardaway, Marnita Hardaway, Marrisa Hardaway, Miguelina Hardaway, Nadege Hardaway, Nakiesha Hardaway, Neesha
Hardaway, Raeleen Hardaway, Rashana Hardaway, Renada Hardaway, Rupa Hardaway, Shaunette Hardaway, Sheetal Hardaway, Sherryann Hardaway, Tanecia Hardaway, Tashiba Hardaway, Tawney Hardaway, Telma
Hardaway, Tomeeka Hardaway, Xochil Hardaway, Yuki Hardaway, Ahna Hardaway, Aneesha Hardaway, Aphrodite Hardaway, Areatha Hardaway, Cam Hardaway, Chalene Hardaway, Charnetta Hardaway, Clarise
Hardaway, Cleta Hardaway, Danniel Hardaway, Deshana Hardaway, Eduardo Hardaway, Erminia Hardaway, Eron Hardaway, Hettie Hardaway, Janan Hardaway, Janda Hardaway, Jayda Hardaway, Jeanny Hardaway,
Katanya Hardaway, Kaylin Hardaway, Keeva Hardaway, Kimesha Hardaway, Kristeena Hardaway, Kristyna Hardaway, Lakeyshia Hardaway, Latesia Hardaway, Latoia Hardaway, Lavona Hardaway, Lonni Hardaway,
Loukisha Hardaway, Manika Hardaway, Marbella Hardaway, Marcina Hardaway, Marialuisa Hardaway, Mariellen Hardaway, Mariluz Hardaway, Mikia Hardaway, Morning Hardaway, Myndi Hardaway, Ronalda Hardaway,
Shanitra Hardaway, Sherea Hardaway, Shericka Hardaway, Sonny Hardaway, Suzane Hardaway, Terea Hardaway, Tersa Hardaway, Yarnell Hardaway, Yezenia Hardaway, Zona Hardaway, Alissia Hardaway, Anjenette
Hardaway, Antwan Hardaway, Bernette Hardaway, Cedar Hardaway, Chandria Hardaway, Cherina Hardaway, Chrisandra Hardaway, Christiann Hardaway, Cosandra Hardaway, Danea Hardaway, Demetress Hardaway,
Emmeline Hardaway, Ernesta Hardaway, Fabiana Hardaway, Fabienne Hardaway, Falecia Hardaway, Georgena Hardaway, Gerardo Hardaway, Gwynn Hardaway, Heavenly Hardaway, Ivett Hardaway, Jaya Hardaway,
Jessicia Hardaway, Jetaun Hardaway, Joane Hardaway, Kalliopi Hardaway, Kalynn Hardaway, Karmon Hardaway, Kischa Hardaway, Kita Hardaway, Lanea Hardaway, Laron Hardaway, Latonyia Hardaway, Lavone
Hardaway, Leasha Hardaway, Liann Hardaway, Madelyne Hardaway, Malissia Hardaway, Margit Hardaway, Marlowe Hardaway, Mashonda Hardaway, Meggen Hardaway, Merari Hardaway, Michille Hardaway, Missi
Hardaway, Nichoel Hardaway, Nikie Hardaway, Nyssa Hardaway, Pooja Hardaway, Purvi Hardaway, Ryane Hardaway, Sakeenah Hardaway, Sam Hardaway, Shaquan Hardaway, Sharmel Hardaway, Shlonda Hardaway,
Tamsen Hardaway, Tikia Hardaway, Tomicka Hardaway, Tremeka Hardaway, Wenonah Hardaway, Yavonne Hardaway, Adraine Hardaway, Aiysha Hardaway, Akila Hardaway, Almira Hardaway, Alnita Hardaway, Ambrosia
Hardaway, Aminta Hardaway, Anedra Hardaway, Angeliki Hardaway, Aymee Hardaway, Barrett Hardaway, Batina Hardaway, Brandey Hardaway, Brena Hardaway, Bryant Hardaway, Charidy Hardaway, Davona Hardaway,
Delissa Hardaway, Denea Hardaway, Denicia Hardaway, Deniese Hardaway, Desta Hardaway, Dimple Hardaway, Emelie Hardaway, Eustacia Hardaway, Farra Hardaway, Giulia Hardaway, Jannetta Hardaway, Jen
Hardaway, Jennice Hardaway, Kareemah Hardaway, Kaysie Hardaway, Latice Hardaway, Letanya Hardaway, Liticia Hardaway, Luna Hardaway, Madge Hardaway, Marielena Hardaway, Merrily Hardaway, Mitsy
Hardaway, Nachelle Hardaway, Nolita Hardaway, Palmira Hardaway, Parthenia Hardaway, Ramey Hardaway, Raymona Hardaway, Rhodesia Hardaway, Sanna Hardaway, Shakema Hardaway, Shamieka Hardaway, Shandria
Hardaway, Shaniece Hardaway, Sharronda Hardaway, Sheletha Hardaway, Shoshannah Hardaway, Suprina Hardaway, Tanyia Hardaway, Tarri Hardaway, Tequita Hardaway, Thressa Hardaway, Twala Hardaway,
Tyneshia Hardaway, Willena Hardaway, Aaryn Hardaway, Ailene Hardaway, Alfredo Hardaway, Alinda Hardaway, Avia Hardaway, Brandelyn Hardaway, Brandolyn Hardaway, Brieanna Hardaway, Brittanie Hardaway,
Celest Hardaway, Charman Hardaway, Chenise Hardaway, Colandra Hardaway, Damari Hardaway, Donni Hardaway, Dorice Hardaway, Dustine Hardaway, Elease Hardaway, Jamara Hardaway, Jameca Hardaway, Jessalyn
Hardaway, Joyanna Hardaway, Kalia Hardaway, Kennisha Hardaway, Kewanna Hardaway, Krystel Hardaway, Lashara Hardaway, Leslieann Hardaway, Letetia Hardaway, Letizia Hardaway, Lisanne Hardaway, Lovella
Hardaway, Maranatha Hardaway, Melisia Hardaway, Meriah Hardaway, Monaca Hardaway, Myrtis Hardaway, Nadean Hardaway, Naila Hardaway, Narissa Hardaway, Natale Hardaway, Nealy Hardaway, November
Hardaway, Petrice Hardaway, Renie Hardaway, Robecca Hardaway, Sarra Hardaway, Shakeena Hardaway, Shalunda Hardaway, Sharin Hardaway, Shatoya Hardaway, Soni Hardaway, Sulma Hardaway, Taffany Hardaway,
Tahara Hardaway, Taralynn Hardaway, Tarena Hardaway, Tatyana Hardaway, Tekeisha Hardaway, Timothea Hardaway, Tishana Hardaway, Toshua Hardaway, Tresea Hardaway, Tykisha Hardaway, Varonica Hardaway,
Yolinda Hardaway, Adra Hardaway, Amia Hardaway, Aquarius Hardaway, Baila Hardaway, Betheny Hardaway, Cady Hardaway, Calla Hardaway, Catrinia Hardaway, Chanika Hardaway, Chellie Hardaway, Chelsy
Hardaway, Christana Hardaway, Cicley Hardaway, Crescent Hardaway, Deronda Hardaway, Diamantina Hardaway, Dyani Hardaway, Elvina Hardaway, Emiley Hardaway, Haneefah Hardaway, Ivan Hardaway, Jamesetta
Hardaway, Janele Hardaway, Jolean Hardaway, Jory Hardaway, Julieanna Hardaway, Jyoti Hardaway, Kaela Hardaway, Katena Hardaway, Kaycie Hardaway, Kismet Hardaway, Kobi Hardaway, Lakechia Hardaway,
Lakesa Hardaway, Lalanya Hardaway, Latiesha Hardaway, Lauralyn Hardaway, Lekita Hardaway, Loletha Hardaway, Lonetta Hardaway, Lukisha Hardaway, Mele Hardaway, Nadra Hardaway, Natia Hardaway, Ngozi
Hardaway, Nija Hardaway, October Hardaway, Olinda Hardaway, Preeti Hardaway, Raine Hardaway, Ronie Hardaway, Sharlie Hardaway, Sharnetta Hardaway, Shontia Hardaway, Tamakia Hardaway, Taquita
Hardaway, Temeca Hardaway, Tenna Hardaway, Teresea Hardaway, Teryn Hardaway, Tessica Hardaway, Trayce Hardaway, Tujuana Hardaway, Alys Hardaway, Aris Hardaway, Asya Hardaway, Atina Hardaway, Ayonna
Hardaway, Cherl Hardaway, Corretta Hardaway, Cybill Hardaway, Gaylynn Hardaway, Heatherly Hardaway, Jacquelina Hardaway, Jeanean Hardaway, Jeniece Hardaway, Johnathan Hardaway, Krystine Hardaway,
Larua Hardaway, Lelah Hardaway, Lilla Hardaway, Linh Hardaway, Lisia Hardaway, Lucilla Hardaway, Malita Hardaway, Matisha Hardaway, Melika Hardaway, Merci Hardaway, Mishell Hardaway, Monae Hardaway,
Monee Hardaway, Montrice Hardaway, Najla Hardaway, Parrish Hardaway, Ramsey Hardaway, Raya Hardaway, Rejeana Hardaway, Roanna Hardaway, Roxan Hardaway, Shai Hardaway, Shanen Hardaway, Sharalyn
Hardaway, Shuntay Hardaway, Taheerah Hardaway, Takeysha Hardaway, Tamelia Hardaway, Tanyika Hardaway, Temperance Hardaway, Theodosia Hardaway, Thyra Hardaway, Tyria Hardaway, Urania Hardaway, Yvett
Hardaway, Angelynn Hardaway, Anitria Hardaway, Anny Hardaway, Arlisha Hardaway, Aya Hardaway, Brielle Hardaway, Careen Hardaway, Correne Hardaway, Dagmar Hardaway, Dalynn Hardaway, Dannetta Hardaway,
Delight Hardaway, Denetta Hardaway, Desaree Hardaway, Donyel Hardaway, Eisha Hardaway, Eleanora Hardaway, Felix Hardaway, Fransisca Hardaway, Jannel Hardaway, Jenia Hardaway, Jettie Hardaway, Jolina
Hardaway, Jowanna Hardaway, Junko Hardaway, Kaaren Hardaway, Kathey Hardaway, Keryn Hardaway, Ketina Hardaway, Korrine Hardaway, Kristalyn Hardaway, Laini Hardaway, Lamona Hardaway, Lanay Hardaway,
Leza Hardaway, Maddalena Hardaway, Mellody Hardaway, Merica Hardaway, Merita Hardaway, Nafeesa Hardaway, Nicholl Hardaway, Oneika Hardaway, Pamelyn Hardaway, Romonda Hardaway, Ronique Hardaway,
Rosslyn Hardaway, Rosy Hardaway, Sachiko Hardaway, Sarajane Hardaway, Satara Hardaway, Shamra Hardaway, Tamberly Hardaway, Tawona Hardaway, Tiasha Hardaway, Tine Hardaway, Trameka Hardaway, Tynia
Hardaway, Valentine Hardaway, Venisha Hardaway, Veronia Hardaway, Vinnie Hardaway, Yuri Hardaway, Aissa Hardaway, Aleda Hardaway, Amenda Hardaway, Andromeda Hardaway, Antigone Hardaway, Aren
Hardaway, Ariadne Hardaway, Bambie Hardaway, Belkis Hardaway, Bunnie Hardaway, Claudina Hardaway, Darcelle Hardaway, Dulcinea Hardaway, Enola Hardaway, Florida Hardaway, Francia Hardaway, Ginni
Hardaway, Grizelda Hardaway, Israel Hardaway, Jalynn Hardaway, Jamil Hardaway, Jenika Hardaway, Joeanna Hardaway, Katesha Hardaway, Katreena Hardaway, Keilani Hardaway, Kiah Hardaway, Kjersti
Hardaway, Korrin Hardaway, Leonda Hardaway, Liesa Hardaway, Lyna Hardaway, Manuelita Hardaway, Marka Hardaway, Markia Hardaway, Maronda Hardaway, Melondy Hardaway, Melonee Hardaway, Micheala
Hardaway, Misa Hardaway, Nikeisha Hardaway, Odalys Hardaway, Parul Hardaway, Petula Hardaway, Rainee Hardaway, Rainie Hardaway, Romi Hardaway, Rubie Hardaway, Shamone Hardaway, Shawnia Hardaway,
Sheritha Hardaway, Shery Hardaway, Tamula Hardaway, Tasheena Hardaway, Thao Hardaway, Thereasa Hardaway, Tjuana Hardaway, Tomesha Hardaway, Turquoise Hardaway, Tyiesha Hardaway, Vernee Hardaway,
Annastasia Hardaway, Ardra Hardaway, Bibi Hardaway, Blaine Hardaway, Bridie Hardaway, Carmelle Hardaway, Chalice Hardaway, Chanie Hardaway, Chauntelle Hardaway, Dareth Hardaway, Demitria Hardaway,
Denetria Hardaway, Didi Hardaway, Dimitria Hardaway, Dwight Hardaway, Elisabetta Hardaway, Ellisha Hardaway, Ileen Hardaway, Jamala Hardaway, Jayleen Hardaway, Jeny Hardaway, Jeraldine Hardaway,
Jowanda Hardaway, Joylene Hardaway, Kareena Hardaway, Karrah Hardaway, Kashonda Hardaway, Kendel Hardaway, Ketrina Hardaway, Lagena Hardaway, Lashann Hardaway, Lasheena Hardaway, Latish Hardaway,
Leroy Hardaway, Maryum Hardaway, Mende Hardaway, Mieka Hardaway, Nakea Hardaway, Niurka Hardaway, Philippa Hardaway, Pina Hardaway, Romanda Hardaway, Satrina Hardaway, Shaletha Hardaway, Shandrika
Hardaway, Shawonda Hardaway, Sherae Hardaway, Shernita Hardaway, Sheronica Hardaway, Silena Hardaway, Sora Hardaway, Stehanie Hardaway, Tanikka Hardaway, Tashi Hardaway, Tawyna Hardaway, Tenesia
Hardaway, Tikesha Hardaway, Trenell Hardaway, Tristina Hardaway, Vallie Hardaway, Verla Hardaway, Vinessa Hardaway, Walida Hardaway, Yolandra Hardaway, Aarti Hardaway, Aries Hardaway, Bayyinah
Hardaway, Bertie Hardaway, Blakely Hardaway, Cabrina Hardaway, Chandrea Hardaway, Cherrell Hardaway, Chiquitta Hardaway, Damarys Hardaway, Danett Hardaway, Dedee Hardaway, Demesha Hardaway, Denica
Hardaway, Derinda Hardaway, Deshanda Hardaway, Diondra Hardaway, Edgar Hardaway, Ewa Hardaway, Grabiela Hardaway, Jeanifer Hardaway, Jemma Hardaway, Jetta Hardaway, Jodelle Hardaway, Joeleen
Hardaway, Johari Hardaway, Kasaundra Hardaway, Kewana Hardaway, Kieran Hardaway, Kitina Hardaway, Kitrina Hardaway, Lacheryl Hardaway, Lakeishia Hardaway, Lasean Hardaway, Lashawne Hardaway, Lashica
Hardaway, Latanga Hardaway, Latreece Hardaway, Latysha Hardaway, Leonna Hardaway, Lilli Hardaway, Lititia Hardaway, Lorette Hardaway, Makala Hardaway, Mea Hardaway, Mendie Hardaway, Meosha Hardaway,
Milessa Hardaway, Nathasha Hardaway, Nefertari Hardaway, Nieves Hardaway, Odelia Hardaway, Phylis Hardaway, Phyliss Hardaway, Rami Hardaway, Shaena Hardaway, Shahara Hardaway, Shajuan Hardaway,
Shalane Hardaway, Shamaine Hardaway, Shaquanna Hardaway, Shauntelle Hardaway, Sheril Hardaway, Sherley Hardaway, Shuronda Hardaway, Sicily Hardaway, Sindi Hardaway, Sonni Hardaway, Tamantha Hardaway,
Tejal Hardaway, Tenley Hardaway, Tongela Hardaway, Trenita Hardaway, Vernica Hardaway, Ameena Hardaway, Chancey Hardaway, Chandi Hardaway, Charelle Hardaway, Cherisa Hardaway, Corby Hardaway,
Corianne Hardaway, Crystale Hardaway, Cyd Hardaway, Cyndie Hardaway, Dagny Hardaway, Daine Hardaway, Darin Hardaway, Debralee Hardaway, Delanda Hardaway, Donta Hardaway, Estefana Hardaway, Falesha
Hardaway, Fawna Hardaway, Galen Hardaway, Geanine Hardaway, Grazia Hardaway, Hala Hardaway, Jadie Hardaway, Jalyn Hardaway, Jamilia Hardaway, Jannice Hardaway, Jasmina Hardaway, Jehan Hardaway,
Joannah Hardaway, Joclyn Hardaway, Julietta Hardaway, Kody Hardaway, Krisa Hardaway, Lanaya Hardaway, Laray Hardaway, Larra Hardaway, Larrissa Hardaway, Leanora Hardaway, Lelania Hardaway, Lenetta
Hardaway, Lizett Hardaway, Lonnette Hardaway, Lorey Hardaway, Lynann Hardaway, Madalena Hardaway, Marlita Hardaway, Marrianne Hardaway, Maylene Hardaway, Merida Hardaway, Midori Hardaway, Nafeesah
Hardaway, Nekeshia Hardaway, Parris Hardaway, Payal Hardaway, Perlita Hardaway, Prairie Hardaway, Raeanna Hardaway, Rayette Hardaway, Sabriya Hardaway, Sadia Hardaway, Sequita Hardaway, Sharine
Hardaway, Sharlena Hardaway, Spencer Hardaway, Sugey Hardaway, Taheera Hardaway, Taiwan Hardaway, Tameki Hardaway, Tamme Hardaway, Tawonda Hardaway, Toscha Hardaway, Tyronza Hardaway, Valeska
Hardaway, Vannesa Hardaway, Vashon Hardaway, Viktoria Hardaway, Vonya Hardaway, Akita Hardaway, Andraya Hardaway, Anecia Hardaway, Annjeanette Hardaway, Antonya Hardaway, Brette Hardaway, Chase
Hardaway, Chena Hardaway, Cheris Hardaway, Chirstine Hardaway, Claude Hardaway, Clayton Hardaway, Clea Hardaway, Clorissa Hardaway, Consuello Hardaway, Cosette Hardaway, Darinda Hardaway, Dawnn
Hardaway, Delene Hardaway, Deneka Hardaway, Geisha Hardaway, Gwendelyn Hardaway, Hyacinth Hardaway, Ilia Hardaway, Jacqulin Hardaway, Jaynee Hardaway, Jennifr Hardaway, Kaysha Hardaway, Keiana
Hardaway, Kelsy Hardaway, Keziah Hardaway, Lakaisha Hardaway, Lashawndra Hardaway, Lennette Hardaway, Loran Hardaway, Margarete Hardaway, Marquise Hardaway, Marvel Hardaway, Maurica Hardaway, Mikell
Hardaway, Nya Hardaway, Odetta Hardaway, Raizy Hardaway, Samika Hardaway, Scotty Hardaway, Senita Hardaway, Shasha Hardaway, Shaton Hardaway, Sheilla Hardaway, Shital Hardaway, Shonika Hardaway,
Silver Hardaway, Tabathia Hardaway, Tabby Hardaway, Taniya Hardaway, Tawonna Hardaway, Teddy Hardaway, Tenicia Hardaway, Teodora Hardaway, Tessia Hardaway, Tinya Hardaway, Travia Hardaway, Trivia
Hardaway, Verda Hardaway, Veta Hardaway, Waneta Hardaway, Ysenia Hardaway, Yumiko Hardaway, Zully Hardaway, Adeana Hardaway, Adenike Hardaway, Afton Hardaway, Angeli Hardaway, Anicia Hardaway, Arron
Hardaway, Artie Hardaway, Ashlei Hardaway, Ayme Hardaway, Brieanne Hardaway, Cira Hardaway, Cynthea Hardaway, Damiana Hardaway, Darlynn Hardaway, Dennette Hardaway, Doreena Hardaway, Dorris Hardaway,
Dorsey Hardaway, Drew Hardaway, Drina Hardaway, Enrika Hardaway, Genean Hardaway, Glendora Hardaway, Gwendlyn Hardaway, Hedi Hardaway, Honesty Hardaway, Jackalyn Hardaway, Jillyn Hardaway, Jode
Hardaway, Joshlyn Hardaway, Joylyn Hardaway, Jullian Hardaway, Kajuana Hardaway, Kasee Hardaway, Kimberla Hardaway, Kjersten Hardaway, Kortni Hardaway, Latronda Hardaway, Leonie Hardaway, Lonette
Hardaway, Lonita Hardaway, Lynnelle Hardaway, Marenda Hardaway, Marybelle Hardaway, Paisley Hardaway, Patrena Hardaway, Rheannon Hardaway, Rochella Hardaway, Saleemah Hardaway, Samala Hardaway,
Shannie Hardaway, Shelitha Hardaway, Shemeika Hardaway, Shenia Hardaway, Shinika Hardaway, Shunte Hardaway, Sia Hardaway, Sinead Hardaway, Tal Hardaway, Tata Hardaway, Teela Hardaway, Tita Hardaway,
Verenice Hardaway, Wade Hardaway, Yesica Hardaway, Zaira Hardaway, Zuri Hardaway, Bobie Hardaway, Brendalee Hardaway, Carolanne Hardaway, Chevette Hardaway, Debera Hardaway, Denessa Hardaway, Desiray
Hardaway, Destinie Hardaway, Diania Hardaway, Edra Hardaway, Ekaterini Hardaway, Elane Hardaway, Eudora Hardaway, Evelyne Hardaway, Franci Hardaway, Gussie Hardaway, Herbert Hardaway, Ilse Hardaway,
Jaquay Hardaway, Jerra Hardaway, Juliene Hardaway, Kashia Hardaway, Kirstan Hardaway, Laconya Hardaway, Lajoyce Hardaway, Larraine Hardaway, Lera Hardaway, Loan Hardaway, Luwana Hardaway, Maire
Hardaway, Malene Hardaway, Marca Hardaway, Marcene Hardaway, Mathew Hardaway, Mayumi Hardaway, Nakeysha Hardaway, Orpha Hardaway, Quinta Hardaway, Robie Hardaway, Sarha Hardaway, Selah Hardaway,
Sergio Hardaway, Shalaine Hardaway, Shazia Hardaway, Shenae Hardaway, Shenica Hardaway, Shineka Hardaway, Sonnie Hardaway, Tashauna Hardaway, Tashica Hardaway, Tatjana Hardaway, Tecia Hardaway,
Therisa Hardaway, Tracina Hardaway, Tressia Hardaway, Vallery Hardaway, Ximena Hardaway, Aidee Hardaway, Alpa Hardaway, Ambur Hardaway, Annaliese Hardaway, Atasha Hardaway, Ave Hardaway, Barbaraann
Hardaway, Bekki Hardaway, Breena Hardaway, Catharina Hardaway, Chandelle Hardaway, Charmion Hardaway, Cherilynn Hardaway, Chesney Hardaway, Chirsty Hardaway, Cynethia Hardaway, Cyntia Hardaway, Dane
Hardaway, Dorethea Hardaway, Elyce Hardaway, Emery Hardaway, Garrett Hardaway, Gittel Hardaway, Gizelle Hardaway, Hinda Hardaway, Jaquelin Hardaway, Jenaya Hardaway, Jeneva Hardaway, Jenipher
Hardaway, Karessa Hardaway, Karita Hardaway, Kelita Hardaway, Kiyana Hardaway, Kora Hardaway, Laquanna Hardaway, Larkin Hardaway, Latangela Hardaway, Linsay Hardaway, Lisett Hardaway, Lizzy Hardaway,
Magdalen Hardaway, Marcea Hardaway, Mashawn Hardaway, Meiko Hardaway, Mekia Hardaway, Merisa Hardaway, Nikcole Hardaway, Norina Hardaway, Prima Hardaway, Rashada Hardaway, Rasheena Hardaway, Rozella
Hardaway, Selia Hardaway, Shaheen Hardaway, Shamon Hardaway, Shaya Hardaway, Shirly Hardaway, Shunna Hardaway, Shyanne Hardaway, Skylar Hardaway, Sukari Hardaway, Sylvana Hardaway, Tahlia Hardaway,
Taiesha Hardaway, Tamiki Hardaway, Tangala Hardaway, Taniqua Hardaway, Tanza Hardaway, Temesha Hardaway, Teonna Hardaway, Tiffane Hardaway, Torya Hardaway, Tremaine Hardaway, Tricha Hardaway, Vanesha
Hardaway, Yetunde Hardaway, Zarinah Hardaway, Amalie Hardaway, Atoya Hardaway, Bryony Hardaway, Caree Hardaway, Carron Hardaway, Celestia Hardaway, Charma Hardaway, Chelly Hardaway, Christon
Hardaway, Chrystle Hardaway, Conchetta Hardaway, Dan Hardaway, Deandria Hardaway, Donika Hardaway, Eulanda Hardaway, Eyvonne Hardaway, Gloriana Hardaway, Hyun Hardaway, Joslynn Hardaway, Kewanda
Hardaway, Laetitia Hardaway, Laketra Hardaway, Lakshmi Hardaway, Latawnya Hardaway, Lawren Hardaway, Leo Hardaway, Letonia Hardaway, Lichelle Hardaway, Lue Hardaway, Lynnmarie Hardaway, Mauricia
Hardaway, Nakkia Hardaway, Neka Hardaway, Parisa Hardaway, Quentin Hardaway, Ranada Hardaway, Rhianon Hardaway, Roselee Hardaway, Rozanne Hardaway, Rozetta Hardaway, Rozina Hardaway, Sada Hardaway,
Sala Hardaway, Samar Hardaway, Seandra Hardaway, Shaela Hardaway, Sharrell Hardaway, Shawneen Hardaway, Shefali Hardaway, Shekinah Hardaway, Sherriann Hardaway, Shuree Hardaway, Soyini Hardaway,
Teala Hardaway, Telitha Hardaway, Tempie Hardaway, Tirza Hardaway, Valori Hardaway, Veleda Hardaway, Vivianne Hardaway, Alexi Hardaway, Arnetra Hardaway, Azizi Hardaway, Belkys Hardaway, Cailin
Hardaway, Carmell Hardaway, Charlyne Hardaway, Courteney Hardaway, Danniell Hardaway, Danyiel Hardaway, Danylle Hardaway, Delita Hardaway, Deseri Hardaway, Donice Hardaway, Ebone Hardaway, Eleshia
Hardaway, Ellison Hardaway, Giuliana Hardaway, Harvest Hardaway, Hillari Hardaway, Ijeoma Hardaway, Janeene Hardaway, Kem Hardaway, Kya Hardaway, Laquandra Hardaway, Lashane Hardaway, Latana
Hardaway, Latoi Hardaway, Lenice Hardaway, Lethia Hardaway, Lindsi Hardaway, Lis Hardaway, Lorrine Hardaway, Marena Hardaway, Marlayna Hardaway, Mashanda Hardaway, Mercedez Hardaway, Merlene
Hardaway, Najwa Hardaway, Paraskevi Hardaway, Princella Hardaway, Raejean Hardaway, Raissa Hardaway, Ravin Hardaway, Reda Hardaway, Reisha Hardaway, Roland Hardaway, Sakia Hardaway, Salma Hardaway,
Shakela Hardaway, Sharica Hardaway, Shatisha Hardaway, Shauntee Hardaway, Sheriann Hardaway, Stori Hardaway, Sueellen Hardaway, Sylina Hardaway, Tametra Hardaway, Tanieka Hardaway, Tanikia Hardaway,
Tarisa Hardaway, Tiffony Hardaway, Tolanda Hardaway, Treana Hardaway, Tumeka Hardaway, Ursala Hardaway, Yumi Hardaway, Zoraya Hardaway, Alfie Hardaway, Aliyah Hardaway, Almeda Hardaway, Amarilys
Hardaway, Amberley Hardaway, Arabella Hardaway, Ardella Hardaway, Atonya Hardaway, Audree Hardaway, Benedetta Hardaway, Billiejean Hardaway, Carlynn Hardaway, Cherylynn Hardaway, Christabel Hardaway,
Clarrisa Hardaway, Colinda Hardaway, Cutina Hardaway, Dalyn Hardaway, Dametra Hardaway, Daphna Hardaway, Emili Hardaway, Fatema Hardaway, Glennda Hardaway, Izetta Hardaway, Janielle Hardaway, Jessamy
Hardaway, Jonique Hardaway, Karron Hardaway, Kathia Hardaway, Katrese Hardaway, Kendrah Hardaway, Kevia Hardaway, Khalia Hardaway, Kissa Hardaway, Kizzi Hardaway, Kutina Hardaway, Lanetra Hardaway,
Lashonya Hardaway, Latanja Hardaway, Latisia Hardaway, Latriece Hardaway, Lenny Hardaway, Leonarda Hardaway, Lewanna Hardaway, Lien Hardaway, Lyndia Hardaway, Mallissa Hardaway, Marlie Hardaway,
Marqueta Hardaway, Marvetta Hardaway, Mikayla Hardaway, Milicent Hardaway, Montana Hardaway, Morgana Hardaway, Nathania Hardaway, Neema Hardaway, Noriko Hardaway, Ondria Hardaway, Onna Hardaway,
Patrici Hardaway, Rachella Hardaway, Reatha Hardaway, Rise Hardaway, Rosabel Hardaway, Secret Hardaway, Shanetha Hardaway, Sharah Hardaway, Sharion Hardaway, Sherilynn Hardaway, Summar Hardaway, Tam
Hardaway, Tamanika Hardaway, Tarhonda Hardaway, Teddie Hardaway, Tennelle Hardaway, Tephanie Hardaway, Terryl Hardaway, Tiawanna Hardaway, Tischa Hardaway, Tonesha Hardaway, Amirah Hardaway, Amory
Hardaway, Anji Hardaway, Arasely Hardaway, Argentina Hardaway, Aviance Hardaway, Batsheva Hardaway, Briar Hardaway, Cena Hardaway, Chalon Hardaway, Chancy Hardaway, Chandler Hardaway, Chivon
Hardaway, Chonita Hardaway, Chrisa Hardaway, Darnelle Hardaway, Deanie Hardaway, Deeana Hardaway, Delise Hardaway, Elayna Hardaway, Errika Hardaway, Eurika Hardaway, Farrell Hardaway, Felcia
Hardaway, Isidra Hardaway, Jai Hardaway, Jamell Hardaway, Jamesha Hardaway, Jere Hardaway, Jodilyn Hardaway, Jovonne Hardaway, Jowana Hardaway, Juline Hardaway, Kameshia Hardaway, Katrisha Hardaway,
Keary Hardaway, Keyshia Hardaway, Kolette Hardaway, Lakecha Hardaway, Lakiya Hardaway, Larrisa Hardaway, Lasharn Hardaway, Lasundra Hardaway, Latica Hardaway, Latoiya Hardaway, Laurisa Hardaway,
Lucrezia Hardaway, Mariateresa Hardaway, Maryland Hardaway, Melicia Hardaway, Mieko Hardaway, Molina Hardaway, Mozella Hardaway, Nataly Hardaway, Nayda Hardaway, Olanda Hardaway, Onica Hardaway,
Ourania Hardaway, Quinette Hardaway, Rayma Hardaway, Rianne Hardaway, Sarika Hardaway, Shalona Hardaway, Shawntina Hardaway, Shellye Hardaway, Sheva Hardaway, Shironda Hardaway, Sonora Hardaway,
Sophronia Hardaway, Symantha Hardaway, Tahra Hardaway, Tearsa Hardaway, Terrika Hardaway, Tonyetta Hardaway, Treina Hardaway, Tzipora Hardaway, Violette Hardaway, Vivianna Hardaway, Yanet Hardaway,
Akira Hardaway, Amanada Hardaway, Atara Hardaway, Britten Hardaway, Camika Hardaway, Carlo Hardaway, Celicia Hardaway, Chanon Hardaway, Chanti Hardaway, Christeena Hardaway, Consandra Hardaway,
Correen Hardaway, Delayna Hardaway, Derenda Hardaway, Dlynn Hardaway, Donyetta Hardaway, Fawne Hardaway, Feliz Hardaway, Gisselle Hardaway, Glen Hardaway, Hong Hardaway, Hyla Hardaway, Jacques
Hardaway, Jasmyn Hardaway, Jonel Hardaway, Khaleelah Hardaway, Kista Hardaway, Lachell Hardaway, Lajoy Hardaway, Lakethia Hardaway, Lamia Hardaway, Lasheba Hardaway, Laurianne Hardaway, Lelani
Hardaway, Lonya Hardaway, Loucinda Hardaway, Maressa Hardaway, Mariesa Hardaway, Meshia Hardaway, Messina Hardaway, Paulett Hardaway, Rashon Hardaway, Rayshawn Hardaway, Rochanda Hardaway, Ronnica
Hardaway, Rorie Hardaway, Saunya Hardaway, Shakeitha Hardaway, Shalla Hardaway, Shalondra Hardaway, Shamecca Hardaway, Shavonn Hardaway, Sherril Hardaway, Shonell Hardaway, Sonita Hardaway, Svetlana
Hardaway, Takila Hardaway, Tala Hardaway, Tamecca Hardaway, Tiphany Hardaway, Torria Hardaway, Vanya Hardaway, Verdell Hardaway, Vonna Hardaway, Zsazsa Hardaway, Adonia Hardaway, Alease Hardaway,
Aleena Hardaway, Alishea Hardaway, Anetta Hardaway, Ania Hardaway, Anquinette Hardaway, Antonieta Hardaway, Archana Hardaway, Arely Hardaway, Artavia Hardaway, Arturo Hardaway, Batya Hardaway,
Bernardine Hardaway, Betti Hardaway, Breck Hardaway, Camara Hardaway, Cesar Hardaway, Chalise Hardaway, Charly Hardaway, Cherrise Hardaway, Chikita Hardaway, Cristiana Hardaway, Dayatra Hardaway,
Ericha Hardaway, Ermalinda Hardaway, Eulonda Hardaway, Fayth Hardaway, Halina Hardaway, Hunter Hardaway, Iveliz Hardaway, Ivie Hardaway, Jahna Hardaway, Jancy Hardaway, Janisha Hardaway, Jeaninne
Hardaway, Jeannemarie Hardaway, Jessamine Hardaway, Keeya Hardaway, Kelisha Hardaway, Kenyette Hardaway, Kyley Hardaway, Kyrie Hardaway, Ladora Hardaway, Laguana Hardaway, Larri Hardaway, Leyda
Hardaway, Louie Hardaway, Marykathryn Hardaway, Mekesha Hardaway, Meredeth Hardaway, Merridith Hardaway, Mittie Hardaway, Nakeitha Hardaway, Narda Hardaway, Nature Hardaway, Neeta Hardaway, Nitasha
Hardaway, Nuvia Hardaway, Paticia Hardaway, Promise Hardaway, Rabia Hardaway, Raquelle Hardaway, Richetta Hardaway, Shawneequa Hardaway, Shervon Hardaway, Siomara Hardaway, Starlyn Hardaway, Tahirih
Hardaway, Taia Hardaway, Talita Hardaway, Tamina Hardaway, Tamya Hardaway, Tanicka Hardaway, Tanyanika Hardaway, Tawan Hardaway, Tawanya Hardaway, Tiarra Hardaway, Timisha Hardaway, Tinita Hardaway,
Tonnia Hardaway, Toronda Hardaway, Toshika Hardaway, Triva Hardaway, Trula Hardaway, Tykesha Hardaway, Vaness Hardaway, Vennessa Hardaway, Wakeelah Hardaway, Wilhemina Hardaway, Yoshiko Hardaway,
Afia Hardaway, Afiya Hardaway, Allysa Hardaway, Alsha Hardaway, Amberlyn Hardaway, Arah Hardaway, Audri Hardaway, Bernardette Hardaway, Bernedette Hardaway, Binta Hardaway, Breeann Hardaway, Britni
Hardaway, Carnetta Hardaway, Cesilia Hardaway, Chelli Hardaway, Dawnna Hardaway, Desirie Hardaway, Fleur Hardaway, Gianina Hardaway, Hanh Hardaway, Ivelis Hardaway, Jalena Hardaway, Jenafer Hardaway,
Johann Hardaway, Joyanne Hardaway, Kamia Hardaway, Katerine Hardaway, Kecha Hardaway, Keir Hardaway, Kennita Hardaway, Kimra Hardaway, Kristiann Hardaway, Levita Hardaway, Lluvia Hardaway, Lorian
Hardaway, Lyndsy Hardaway, Mardell Hardaway, Maurissa Hardaway, Mischell Hardaway, Mistey Hardaway, Nataya Hardaway, Nickola Hardaway, Obdulia Hardaway, Preston Hardaway, Raynetta Hardaway, Rondell
Hardaway, Shakara Hardaway, Shalan Hardaway, Shanya Hardaway, Shawnika Hardaway, Shenetha Hardaway, Shilah Hardaway, Soo Hardaway, Tametha Hardaway, Tanicia Hardaway, Tanny Hardaway, Tennie Hardaway,
Tiffannie Hardaway, Torra Hardaway, Trinika Hardaway, Val Hardaway, Abeer Hardaway, Adah Hardaway, Aleece Hardaway, Andreya Hardaway, Annaliza Hardaway, Arrie Hardaway, Ayasha Hardaway, Berit
Hardaway, Birgit Hardaway, Bre Hardaway, Brigetta Hardaway, Cassundra Hardaway, Christyl Hardaway, Cristyn Hardaway, Danean Hardaway, Daralyn Hardaway, Davonne Hardaway, Deisha Hardaway, Denina
Hardaway, Dietrich Hardaway, Domini Hardaway, Erinne Hardaway, Evy Hardaway, Forrest Hardaway, Genetta Hardaway, Gioia Hardaway, Guenevere Hardaway, Gwenetta Hardaway, Gwenette Hardaway, Hesper
Hardaway, Ignacia Hardaway, Irmalinda Hardaway, Jacquita Hardaway, Jeanita Hardaway, Jodiann Hardaway, Kalin Hardaway, Kenny Hardaway, Keon Hardaway, Kimberlynn Hardaway, Kimela Hardaway, Kimm
Hardaway, Kiwanna Hardaway, Lacreshia Hardaway, Lakashia Hardaway, Latarshia Hardaway, Latrish Hardaway, Lauria Hardaway, Likisha Hardaway, Lindee Hardaway, Little Hardaway, Loreta Hardaway, Lutisha
Hardaway, Mahasin Hardaway, Maki Hardaway, Maxi Hardaway, Meesha Hardaway, Melitta Hardaway, Mery Hardaway, Mysty Hardaway, Quanisha Hardaway, Ranetta Hardaway, Rosia Hardaway, Sadonna Hardaway,
Sangita Hardaway, Sanora Hardaway, Shanina Hardaway, Sharonna Hardaway, Shavona Hardaway, Sherrica Hardaway, Storm Hardaway, Tabrina Hardaway, Tacha Hardaway, Taleshia Hardaway, Tamee Hardaway,
Tanetta Hardaway, Thanh Hardaway, Twanya Hardaway, Ulrica Hardaway, Yaffa Hardaway, Aarin Hardaway, Afi Hardaway, Aliscia Hardaway, Aneta Hardaway, Anetria Hardaway, Antoinett Hardaway, Athenia
Hardaway, Atisha Hardaway, Barb Hardaway, Brit Hardaway, Caley Hardaway, Camala Hardaway, Carena Hardaway, Carinne Hardaway, Carmine Hardaway, Charlott Hardaway, Chessa Hardaway, Chundra Hardaway,
Cotrina Hardaway, Drenda Hardaway, Elanor Hardaway, Eleftheria Hardaway, Erricka Hardaway, Feleshia Hardaway, Geanna Hardaway, Harley Hardaway, Honora Hardaway, Jakia Hardaway, Jannah Hardaway,
Jennene Hardaway, Jennett Hardaway, Jobeth Hardaway, Jordanna Hardaway, Juanetta Hardaway, Junie Hardaway, Kaira Hardaway, Kashana Hardaway, Katee Hardaway, Kateena Hardaway, Kelleigh Hardaway,
Kenyana Hardaway, Kerrilyn Hardaway, Kessa Hardaway, Kirk Hardaway, Lakitha Hardaway, Laneshia Hardaway, Lanissa Hardaway, Lashunta Hardaway, Leonia Hardaway, Lester Hardaway, Letina Hardaway,
Llesenia Hardaway, Loranda Hardaway, Lorilynn Hardaway, Lotus Hardaway, Luke Hardaway, Marisal Hardaway, Marvis Hardaway, Maurie Hardaway, Minnette Hardaway, Mishawn Hardaway, Natsha Hardaway, Ngoc
Hardaway, Nioka Hardaway, Particia Hardaway, Phillina Hardaway, Prisilla Hardaway, Raney Hardaway, Raphaela Hardaway, Rashaunda Hardaway, Raynelle Hardaway, Ross Hardaway, Rufina Hardaway, Sarahann
Hardaway, Shaconda Hardaway, Shanicka Hardaway, Shaunita Hardaway, Shekina Hardaway, Shelie Hardaway, Sherah Hardaway, Shermeka Hardaway, Shermika Hardaway, Sonnia Hardaway, Stephonie Hardaway,
Taffie Hardaway, Takima Hardaway, Talea Hardaway, Tamaya Hardaway, Taneika Hardaway, Tanina Hardaway, Tanisia Hardaway, Taysha Hardaway, Tekeshia Hardaway, Terie Hardaway, Therasa Hardaway, Thomasa
Hardaway, Tifini Hardaway, Tomie Hardaway, Tressy Hardaway, Trissa Hardaway, Tristy Hardaway, Tritia Hardaway, Twania Hardaway, Tyrene Hardaway, Uganda Hardaway, Wandy Hardaway, Wileen Hardaway,
Yumeka Hardaway, Zarina Hardaway, Adreana Hardaway, Alandra Hardaway, Alenda Hardaway, Amera Hardaway, Arlicia Hardaway, Artemis Hardaway, Avani Hardaway, Bena Hardaway, Candia Hardaway, Cathyjo
Hardaway, Chala Hardaway, Charitie Hardaway, Charlise Hardaway, Chasiti Hardaway, Chayla Hardaway, Chrisanne Hardaway, Christell Hardaway, Clarrissa Hardaway, Cortnee Hardaway, Dametria Hardaway,
Daneka Hardaway, Darlisa Hardaway, Delorse Hardaway, Deshone Hardaway, Dorette Hardaway, Esperansa Hardaway, Evagelia Hardaway, Feliciana Hardaway, Fikisha Hardaway, Freeda Hardaway, Frida Hardaway,
Gaila Hardaway, Genae Hardaway, Georgi Hardaway, Gitel Hardaway, Gricel Hardaway, Hilliary Hardaway, Ichelle Hardaway, Ishia Hardaway, Ivelise Hardaway, Jacci Hardaway, Jamesa Hardaway, Jammy
Hardaway, Jerrilynn Hardaway, Jillaine Hardaway, Jillann Hardaway, Jin Hardaway, Joda Hardaway, Jodeen Hardaway, Kalisa Hardaway, Karne Hardaway, Katura Hardaway, Kayci Hardaway, Keegan Hardaway,
Keenan Hardaway, Khia Hardaway, Koleen Hardaway, Krishana Hardaway, Krislyn Hardaway, Kurt Hardaway, Lafaye Hardaway, Lakeysa Hardaway, Lakresha Hardaway, Lalisa Hardaway, Lashonne Hardaway,
Leighanna Hardaway, Lesslie Hardaway, Lin Hardaway, Madlyn Hardaway, Malky Hardaway, Melenda Hardaway, Melisssa Hardaway, Mikita Hardaway, Monicia Hardaway, Monik Hardaway, Nakima Hardaway, Naquita
Hardaway, Natash Hardaway, Raphael Hardaway, Rashawna Hardaway, Reannon Hardaway, Reneta Hardaway, Reshanda Hardaway, Riana Hardaway, Rocky Hardaway, Rosamond Hardaway, Roshan Hardaway, Saleena
Hardaway, Schuyler Hardaway, Shaka Hardaway, Shakesha Hardaway, Shaleta Hardaway, Shanekia Hardaway, Surina Hardaway, Syrita Hardaway, Taj Hardaway, Tajuanna Hardaway, Tasheen Hardaway, Tiffine
Hardaway, Timmie Hardaway, Tomikia Hardaway, Trease Hardaway, Tyrhonda Hardaway, Wakisha Hardaway, Yvone Hardaway, Adalia Hardaway, Ahuva Hardaway, Angelicia Hardaway, Anina Hardaway, Annitra
Hardaway, Betsie Hardaway, Cachet Hardaway, Callista Hardaway, Camey Hardaway, Celita Hardaway, Cherissa Hardaway, Collin Hardaway, Daiana Hardaway, Equilla Hardaway, Evetta Hardaway, Eyvette
Hardaway, Feige Hardaway, Filicia Hardaway, Floyd Hardaway, Garnett Hardaway, Geoffrey Hardaway, Harper Hardaway, Ione Hardaway, Iyana Hardaway, Jametta Hardaway, Jeanee Hardaway, Jessalynn Hardaway,
Jonalyn Hardaway, Juanice Hardaway, Kaija Hardaway, Kalene Hardaway, Kassidy Hardaway, Katryn Hardaway, Kennethia Hardaway, Keosha Hardaway, Kresha Hardaway, Krisandra Hardaway, Labrina Hardaway,
Lachonda Hardaway, Lamara Hardaway, Lashika Hardaway, Lateka Hardaway, Laytoya Hardaway, Lourdez Hardaway, Lyndsi Hardaway, Markeita Hardaway, Marletta Hardaway, Marshella Hardaway, Martiza Hardaway,
Marylouise Hardaway, Miri Hardaway, Nani Hardaway, Payton Hardaway, Puja Hardaway, Qianna Hardaway, Raffaella Hardaway, Roanne Hardaway, Sadiqa Hardaway, Salvador Hardaway, Shabnam Hardaway, Shamar
Hardaway, Shamina Hardaway, Sharalee Hardaway, Shavette Hardaway, Sheilia Hardaway, Shekia Hardaway, Shelise Hardaway, Sherrilynn Hardaway, Sheyla Hardaway, Shy Hardaway, Suprena Hardaway, Tahitia
Hardaway, Talley Hardaway, Tamiya Hardaway, Tashena Hardaway, Tashona Hardaway, Tatisha Hardaway, Toia Hardaway, Tonji Hardaway, Triana Hardaway, Tymika Hardaway, Yanna Hardaway, Zainab Hardaway,
Amrita Hardaway, Aprel Hardaway, Atalie Hardaway, Aubre Hardaway, Auria Hardaway, Barbarita Hardaway, Bettyann Hardaway, Boni Hardaway, Brandace Hardaway, Bryanne Hardaway, Calleen Hardaway, Chakita
Hardaway, Chamaine Hardaway, Chamika Hardaway, Charlett Hardaway, Chereese Hardaway, Cherine Hardaway, Christalyn Hardaway, Conya Hardaway, Dalilah Hardaway, Dela Hardaway, Delanna Hardaway, Delinah
Hardaway, Dierdra Hardaway, Domitila Hardaway, Donnis Hardaway, Dore Hardaway, Elenor Hardaway, Elizabethann Hardaway, Elysha Hardaway, Erryn Hardaway, Evalyn Hardaway, Frannie Hardaway, Fredda
Hardaway, Ginnifer Hardaway, Hilaria Hardaway, Ilisa Hardaway, Ivanna Hardaway, Janeal Hardaway, Jeena Hardaway, Jeremie Hardaway, Jeriann Hardaway, Julina Hardaway, Karim Hardaway, Kenitha Hardaway,
Kery Hardaway, Korinna Hardaway, Lachrisha Hardaway, Lamonda Hardaway, Laneisha Hardaway, Lene Hardaway, Letita Hardaway, Lillia Hardaway, Lindsie Hardaway, Lissete Hardaway, Lynsay Hardaway, Magalie
Hardaway, Makini Hardaway, Manette Hardaway, Margareta Hardaway, Marlette Hardaway, Marsi Hardaway, Maryn Hardaway, Megon Hardaway, Melainie Hardaway, Meria Hardaway, Merrideth Hardaway, Moranda
Hardaway, Myah Hardaway, Myron Hardaway, Odalis Hardaway, Rakia Hardaway, Renesha Hardaway, Rhanda Hardaway, Richell Hardaway, Salinda Hardaway, Sammantha Hardaway, Shamonica Hardaway, Shaquanda
Hardaway, Sharmain Hardaway, Shereece Hardaway, Shron Hardaway, Tammika Hardaway, Terence Hardaway, Teresha Hardaway, Tiah Hardaway, Treniece Hardaway, Truly Hardaway, Venicia Hardaway, Virgil
Hardaway, Winnifred Hardaway, Yaritza Hardaway, Yarrow Hardaway, Zandria Hardaway, Alacia Hardaway, Albertha Hardaway, Andie Hardaway, Anuradha Hardaway, Artesia Hardaway, Beatrix Hardaway, Brandin
Hardaway, Brittni Hardaway, Carrey Hardaway, Catricia Hardaway, Chelise Hardaway, Christien Hardaway, Dalina Hardaway, Damia Hardaway, Deletha Hardaway, Eldora Hardaway, Eli Hardaway, Eufemia
Hardaway, Fabrienne Hardaway, Garland Hardaway, Genessa Hardaway, Ginnette Hardaway, Goldy Hardaway, Grant Hardaway, Greg Hardaway, Hasina Hardaway, Hermila Hardaway, Ima Hardaway, Jacquette
Hardaway, Jilian Hardaway, Jilleen Hardaway, Jolita Hardaway, Jorja Hardaway, Julita Hardaway, Juna Hardaway, Keicha Hardaway, Keshawn Hardaway, Ketty Hardaway, Kimmi Hardaway, Kizmet Hardaway, Koni
Hardaway, Kristiane Hardaway, Krystyn Hardaway, Laronica Hardaway, Lashonia Hardaway, Lateefa Hardaway, Latresia Hardaway, Lequisha Hardaway, Levi Hardaway, Lorenia Hardaway, Lubna Hardaway, Lucas
Hardaway, Marche Hardaway, Marlea Hardaway, Masha Hardaway, Miah Hardaway, Michala Hardaway, Michelyn Hardaway, Mistina Hardaway, Necola Hardaway, Netra Hardaway, Niomi Hardaway, Nira Hardaway,
Nonnie Hardaway, Phelicia Hardaway, Quanta Hardaway, Quita Hardaway, Raini Hardaway, Rayanna Hardaway, Roselind Hardaway, Roselinda Hardaway, Roshana Hardaway, Sagrario Hardaway, Sameka Hardaway,
Shaindel Hardaway, Sharetha Hardaway, Sharol Hardaway, Shawnese Hardaway, Shellia Hardaway, Sheray Hardaway, Sherian Hardaway, Shi Hardaway, Stepheni Hardaway, Subrena Hardaway, Tabithia Hardaway,
Taleen Hardaway, Taneha Hardaway, Tanysha Hardaway, Tekela Hardaway, Telesa Hardaway, Tima Hardaway, Tonni Hardaway, Ula Hardaway, Yitty Hardaway, Yona Hardaway, Yonna Hardaway, Yuko Hardaway, Zalika
Hardaway, Aarika Hardaway, Ajeenah Hardaway, Alegra Hardaway, Alichia Hardaway, Allecia Hardaway, Amor Hardaway, Aneatra Hardaway, Anjannette Hardaway, Anntionette Hardaway, Antonetta Hardaway,
Ardelia Hardaway, Atlanta Hardaway, Ayoka Hardaway, Breeze Hardaway, Casonya Hardaway, Cathern Hardaway, Chanise Hardaway, Charlina Hardaway, Charmon Hardaway, Chemika Hardaway, Cheresa Hardaway,
Cindra Hardaway, Claretha Hardaway, Coren Hardaway, Cyndia Hardaway, Cyntha Hardaway, Danialle Hardaway, Danyeal Hardaway, Davin Hardaway, Deanda Hardaway, Deaundra Hardaway, Dejah Hardaway, Demica
Hardaway, Demisha Hardaway, Dinita Hardaway, Donalyn Hardaway, Donesha Hardaway, Dotty Hardaway, Fidelia Hardaway, Golden Hardaway, Helana Hardaway, Hidi Hardaway, Ilena Hardaway, Jenniferann
Hardaway, Jolleen Hardaway, Jolynda Hardaway, Juanna Hardaway, Karlena Hardaway, Karlin Hardaway, Keandra Hardaway, Kionna Hardaway, Konni Hardaway, Lakeyta Hardaway, Lechelle Hardaway, Levon
Hardaway, Lilith Hardaway, Maliaka Hardaway, Mallie Hardaway, Margeaux Hardaway, Mariaisabel Hardaway, Marixa Hardaway, Memorie Hardaway, Meshawn Hardaway, Michole Hardaway, Myrian Hardaway, Neela
Hardaway, Nivea Hardaway, Peggi Hardaway, Porchia Hardaway, Racine Hardaway, Rashaun Hardaway, Reynalda Hardaway, Rosaleen Hardaway, Sabrine Hardaway, Samanda Hardaway, Shanitha Hardaway, Sharece
Hardaway, Sheretha Hardaway, Sherwanda Hardaway, Shonica Hardaway, Solveig Hardaway, Soyla Hardaway, Taka Hardaway, Taquilla Hardaway, Tedi Hardaway, Tema Hardaway, Terita Hardaway, Tippi Hardaway,
Tiziana Hardaway, Tomieka Hardaway, Tonga Hardaway, Trenace Hardaway, Tyeasha Hardaway, Tyresha Hardaway, Valda Hardaway, Vandy Hardaway, Victora Hardaway, Victory Hardaway, Waverly Hardaway, Adair
Hardaway, Agueda Hardaway, Aiko Hardaway, Akina Hardaway, Amaya Hardaway, Anthonette Hardaway, Ariann Hardaway, Averi Hardaway, Ayde Hardaway, Belissa Hardaway, Bernadett Hardaway, Berniece Hardaway,
Berry Hardaway, Bethsaida Hardaway, Bindu Hardaway, Bonniejean Hardaway, Brendy Hardaway, Cadence Hardaway, Camy Hardaway, Canisha Hardaway, Carilyn Hardaway, Caronda Hardaway, Chantrell Hardaway,
Chavela Hardaway, Chemeka Hardaway, Crystie Hardaway, Dafina Hardaway, Dangela Hardaway, Daphnee Hardaway, Darenda Hardaway, Darolyn Hardaway, Davelyn Hardaway, Desarie Hardaway, Devone Hardaway,
Dionisia Hardaway, Donnielle Hardaway, Ebonique Hardaway, English Hardaway, Eran Hardaway, Eryka Hardaway, Falisa Hardaway, Fareeda Hardaway, Herman Hardaway, Inda Hardaway, Janeice Hardaway, Janny
Hardaway, Jessicah Hardaway, Jodean Hardaway, Julius Hardaway, Kaisa Hardaway, Kanda Hardaway, Karel Hardaway, Karia Hardaway, Kimba Hardaway, Konya Hardaway, Kylah Hardaway, Kyrsten Hardaway,
Laretta Hardaway, Lasharon Hardaway, Latascha Hardaway, Leni Hardaway, Leontyne Hardaway, Linell Hardaway, Lovena Hardaway, Maghan Hardaway, Makeshia Hardaway, Malorie Hardaway, Marce Hardaway,
Maryetta Hardaway, Mechille Hardaway, Media Hardaway, Meira Hardaway, Miosotis Hardaway, Mirjana Hardaway, Mlissa Hardaway, Nekeya Hardaway, Nica Hardaway, Nikkisha Hardaway, Ortencia Hardaway,
Porche Hardaway, Raffaela Hardaway, Raleigh Hardaway, Reka Hardaway, Rotunda Hardaway, Sharline Hardaway, Sharlonda Hardaway, Shawnice Hardaway, Shone Hardaway, Shuntell Hardaway, Sneha Hardaway,
Stephanne Hardaway, Sundy Hardaway, Tabita Hardaway, Takara Hardaway, Takeia Hardaway, Talli Hardaway, Tanganika Hardaway, Taquana Hardaway, Thresea Hardaway, Tila Hardaway, Tim Hardaway, Tyan
Hardaway, Vaishali Hardaway, Xanthe Hardaway, Zenja Hardaway, Zerlina Hardaway, Aggie Hardaway, Aleathea Hardaway, Aleen Hardaway, Alexcia Hardaway, Aloma Hardaway, Angeleque Hardaway, Araina
Hardaway, Asucena Hardaway, Azadeh Hardaway, Brendan Hardaway, Candina Hardaway, Candiss Hardaway, Cantina Hardaway, Carlisha Hardaway, Charra Hardaway, Chaunta Hardaway, Dalonda Hardaway, Damica
Hardaway, Decarla Hardaway, Deneice Hardaway, Derhonda Hardaway, Derrica Hardaway, Dorea Hardaway, Elvera Hardaway, Estell Hardaway, Fae Hardaway, Graceann Hardaway, Grissel Hardaway, Hina Hardaway,
Jamielynn Hardaway, Jesscia Hardaway, Joley Hardaway, Junell Hardaway, Kalilah Hardaway, Kathe Hardaway, Katurah Hardaway, Kenni Hardaway, Keysa Hardaway, Kimie Hardaway, Lachana Hardaway, Ladell
Hardaway, Lanore Hardaway, Laramie Hardaway, Larrie Hardaway, Latonna Hardaway, Latori Hardaway, Livier Hardaway, Loletta Hardaway, Magdaline Hardaway, Malaina Hardaway, Mandalyn Hardaway, Marshawn
Hardaway, Marticia Hardaway, Maurisa Hardaway, Mellonie Hardaway, Nashira Hardaway, Orelia Hardaway, Quanna Hardaway, Raisa Hardaway, Rakisha Hardaway, Raymonda Hardaway, Remonia Hardaway, Robbye
Hardaway, Ronnita Hardaway, Roshaunda Hardaway, Sabre Hardaway, Sabreen Hardaway, Sharis Hardaway, Shawnae Hardaway, Sherrin Hardaway, Sindia Hardaway, Staphanie Hardaway, Stevi Hardaway, Tannisha
Hardaway, Tashunda Hardaway, Tekla Hardaway, Teralyn Hardaway, Tomecia Hardaway, Tonique Hardaway, Trilby Hardaway, Unika Hardaway, Vanisha Hardaway, Voula Hardaway, Waynetta Hardaway, Yasmina
Hardaway, Amesha Hardaway, Aniko Hardaway, Annica Hardaway, Antione Hardaway, Astria Hardaway, Camilia Hardaway, Caprina Hardaway, Carah Hardaway, Catonya Hardaway, Chalanda Hardaway, Chasty
Hardaway, Cherica Hardaway, Coy Hardaway, Crystalynn Hardaway, Dakisha Hardaway, Davine Hardaway, Dawnyel Hardaway, Demetrias Hardaway, Derica Hardaway, Dinamarie Hardaway, Dvora Hardaway, Elbony
Hardaway, Emanuela Hardaway, Gayleen Hardaway, Genevra Hardaway, Hadiyah Hardaway, Harolyn Hardaway, Hidie Hardaway, Indy Hardaway, Jaki Hardaway, Jalana Hardaway, Jalila Hardaway, Janiel Hardaway,
Jennah Hardaway, Jessia Hardaway, Jihan Hardaway, Jinnie Hardaway, Jonya Hardaway, Justa Hardaway, Justice Hardaway, Kameela Hardaway, Kameko Hardaway, Kaneisha Hardaway, Karletta Hardaway, Keneisha
Hardaway, Kimberl Hardaway, Kimyada Hardaway, Kirstyn Hardaway, Krisy Hardaway, Kyisha Hardaway, Labreeska Hardaway, Lacreasha Hardaway, Lajeana Hardaway, Lakiska Hardaway, Lanya Hardaway, Laren
Hardaway, Latash Hardaway, Lataunya Hardaway, Leiah Hardaway, Loralyn Hardaway, Lorre Hardaway, Malaka Hardaway, Malicia Hardaway, Marga Hardaway, Margaretann Hardaway, Markell Hardaway, Meriam
Hardaway, Micca Hardaway, Morningstar Hardaway, Neta Hardaway, Nigel Hardaway, Noah Hardaway, Nuria Hardaway, Oliver Hardaway, Osha Hardaway, Pierrette Hardaway, Quinita Hardaway, Quintessa Hardaway,
Rashan Hardaway, Ravonda Hardaway, Raylynn Hardaway, Reyes Hardaway, Rhondalyn Hardaway, Rosey Hardaway, Roshanna Hardaway, Ruthy Hardaway, Sasheen Hardaway, Shakeisha Hardaway, Shakirah Hardaway,
Shaneta Hardaway, Sharmeka Hardaway, Sharonica Hardaway, Shelba Hardaway, Shelinda Hardaway, Shell Hardaway, Sheranda Hardaway, Stachia Hardaway, Stefanee Hardaway, Sydnee Hardaway, Tamekka Hardaway,
Tanisa Hardaway, Telissa Hardaway, Terissa Hardaway, Tilda Hardaway, Tomoko Hardaway, Trinna Hardaway, Tulani Hardaway, Varina Hardaway, Verlene Hardaway, Vernisha Hardaway, Yasha Hardaway, Yvonnie
Hardaway, Albertine Hardaway, Angeletta Hardaway, Anni Hardaway, Arrica Hardaway, Billyjo Hardaway, Brocha Hardaway, Calie Hardaway, Cass Hardaway, Catie Hardaway, Chairty Hardaway, Chakakhan
Hardaway, Chassie Hardaway, Cherae Hardaway, Cheryn Hardaway, Chisa Hardaway, Chrystine Hardaway, Coronda Hardaway, Curtina Hardaway, Danuta Hardaway, Debie Hardaway, Delona Hardaway, Demia Hardaway,
Dosha Hardaway, Enrica Hardaway, Evone Hardaway, Fanchon Hardaway, Fina Hardaway, Gaetana Hardaway, Genevia Hardaway, Gera Hardaway, Gracy Hardaway, Hollyann Hardaway, Iasia Hardaway, Iyesha
Hardaway, Jackline Hardaway, Joanette Hardaway, Jonnette Hardaway, Kaamilya Hardaway, Karolina Hardaway, Kathlynn Hardaway, Kayle Hardaway, Kayse Hardaway, Kehaulani Hardaway, Kella Hardaway,
Kenyatte Hardaway, Kianga Hardaway, Kijuana Hardaway, Kinisha Hardaway, Kiwanda Hardaway, Kriss Hardaway, Lakea Hardaway, Lakeithia Hardaway, Lakeyia Hardaway, Latusha Hardaway, Leala Hardaway, Leina
Hardaway, Lucita Hardaway, Madia Hardaway, Mairead Hardaway, Makiba Hardaway, Marchella Hardaway, Marquel Hardaway, Martita Hardaway, Meco Hardaway, Melissaann Hardaway, Merilyn Hardaway, Minyon
Hardaway, Monnica Hardaway, Monnie Hardaway, Mora Hardaway, Netasha Hardaway, Nkechi Hardaway, Ocie Hardaway, Oni Hardaway, Patricie Hardaway, Quina Hardaway, Quintana Hardaway, Renu Hardaway, Rhetta
Hardaway, Rissa Hardaway, Rozalyn Hardaway, Salisa Hardaway, Seena Hardaway, Shaine Hardaway, Shandon Hardaway, Shanisha Hardaway, Shantai Hardaway, Sharisa Hardaway, Shary Hardaway, Shaylyn
Hardaway, Sheleta Hardaway, Shilonda Hardaway, Sojourner Hardaway, Syrena Hardaway, Taleah Hardaway, Tanekia Hardaway, Tatianna Hardaway, Tere Hardaway, Terez Hardaway, Teriann Hardaway, Tewana
Hardaway, Ticey Hardaway, Vanette Hardaway, Vidya Hardaway, Wonder Hardaway, Zoie Hardaway, Aeisha Hardaway, Alisande Hardaway, Allan Hardaway, Amantha Hardaway, Analee Hardaway, Ane Hardaway,
Artrice Hardaway, Audelia Hardaway, Ayelet Hardaway, Benji Hardaway, Benny Hardaway, Briann Hardaway, Brittan Hardaway, Calvina Hardaway, Cameka Hardaway, Cecilie Hardaway, Cecille Hardaway, Chanette
Hardaway, Chantia Hardaway, Charlese Hardaway, Charolett Hardaway, Chenille Hardaway, Chivonne Hardaway, Chudney Hardaway, Clemencia Hardaway, Courtny Hardaway, Cristalle Hardaway, Cynde Hardaway,
Daveda Hardaway, Derika Hardaway, Dezarae Hardaway, Dionicia Hardaway, Dunia Hardaway, Elspeth Hardaway, Fayette Hardaway, Glena Hardaway, Gordon Hardaway, Gretchin Hardaway, Halli Hardaway, Ilissa
Hardaway, Jauna Hardaway, Johnnette Hardaway, Joycelynn Hardaway, Junelle Hardaway, Kaydee Hardaway, Kerryn Hardaway, Koryn Hardaway, Kosha Hardaway, Kyoko Hardaway, Lala Hardaway, Lametra Hardaway,
Laresha Hardaway, Lauraann Hardaway, Lecretia Hardaway, Luv Hardaway, Lyndsie Hardaway, Malessa Hardaway, Marcine Hardaway, Mellissia Hardaway, Nakina Hardaway, Nashonda Hardaway, Neelam Hardaway,
Nikitia Hardaway, Orit Hardaway, Otis Hardaway, Pasqualina Hardaway, Porshia Hardaway, Rayshell Hardaway, Rico Hardaway, Rockell Hardaway, Roshon Hardaway, Savanna Hardaway, Shallan Hardaway,
Shashana Hardaway, Shawntia Hardaway, Sherrise Hardaway, Sofie Hardaway, Taji Hardaway, Talanda Hardaway, Tamella Hardaway, Tamsyn Hardaway, Taquisha Hardaway, Tashea Hardaway, Terisha Hardaway,
Tikita Hardaway, Timmy Hardaway, Trenese Hardaway, Valissa Hardaway, Valli Hardaway, Vergie Hardaway, Yancy Hardaway, Aina Hardaway, Albina Hardaway, Aletheia Hardaway, Alisson Hardaway, Allisyn
Hardaway, Amorita Hardaway, Anginette Hardaway, Anjana Hardaway, Anngela Hardaway, Ariela Hardaway, Arienne Hardaway, Arnisha Hardaway, Arvella Hardaway, Ayodele Hardaway, Badia Hardaway, Brooklynn
Hardaway, Candus Hardaway, Conswello Hardaway, Curtisha Hardaway, Cythina Hardaway, Dashia Hardaway, Dekisha Hardaway, Denyce Hardaway, Dilia Hardaway, Doreatha Hardaway, Dortha Hardaway, Ernesto
Hardaway, Fatisha Hardaway, Fauna Hardaway, Fifi Hardaway, Gordana Hardaway, Indya Hardaway, Jaina Hardaway, Janiene Hardaway, Jarah Hardaway, Jasmyne Hardaway, Jeananne Hardaway, Jeff Hardaway,
Jeronica Hardaway, Jessaca Hardaway, Joby Hardaway, Jodette Hardaway, Jolisa Hardaway, Juanda Hardaway, Juanette Hardaway, Kalie Hardaway, Kaneshia Hardaway, Karalynn Hardaway, Kavitha Hardaway,
Kelcy Hardaway, Kerrilynn Hardaway, Kierston Hardaway, Kineta Hardaway, Kiri Hardaway, Krisie Hardaway, Latorria Hardaway, Lazara Hardaway, Lekecia Hardaway, Lenell Hardaway, Lesle Hardaway, Levina
Hardaway, Liat Hardaway, Lindley Hardaway, Maiya Hardaway, Makela Hardaway, Mariane Hardaway, Marki Hardaway, Mashelle Hardaway, Maud Hardaway, Medora Hardaway, Megumi Hardaway, Milca Hardaway,
Missey Hardaway, Mistelle Hardaway, Nakeeta Hardaway, Neile Hardaway, Nicloe Hardaway, Nykisha Hardaway, Orly Hardaway, Patria Hardaway, Rama Hardaway, Ramonia Hardaway, Ranie Hardaway, Ranya
Hardaway, Refugio Hardaway, Rendi Hardaway, Schwanna Hardaway, Senia Hardaway, Shalaunda Hardaway, Shalia Hardaway, Shanieka Hardaway, Shantale Hardaway, Shatima Hardaway, Shawon Hardaway, Shelva
Hardaway, Shemeca Hardaway, Shemeeka Hardaway, Sher Hardaway, Sinda Hardaway, Skyler Hardaway, Stephaie Hardaway, Suzetta Hardaway, Suzzane Hardaway, Synethia Hardaway, Taesha Hardaway, Takina
Hardaway, Tanyetta Hardaway, Taran Hardaway, Tarasha Hardaway, Tarji Hardaway, Tauni Hardaway, Temeika Hardaway, Temisha Hardaway, Thania Hardaway, Thembi Hardaway, Thora Hardaway, Tiffin Hardaway,
Toney Hardaway, Trine Hardaway, Tyrina Hardaway, Uraina Hardaway, Valleri Hardaway, Vandana Hardaway, Vernie Hardaway, Vinetta Hardaway, Viveca Hardaway, Vonette Hardaway, Wynema Hardaway, Yasheka
Hardaway, Yevonne Hardaway, Yovana Hardaway, Zanita Hardaway, Zarah Hardaway, Zendre Hardaway, Abigayle Hardaway, Abraham Hardaway, Acquanetta Hardaway, Alika Hardaway, Alora Hardaway, Alvera
Hardaway, Amand Hardaway, Annabella Hardaway, Annelle Hardaway, Antia Hardaway, Arkisha Hardaway, Arlanda Hardaway, Aronda Hardaway, Ayo Hardaway, Bailey Hardaway, Banita Hardaway, Becka Hardaway,
Brannon Hardaway, Camden Hardaway, Cantrice Hardaway, Ceclia Hardaway, Chantella Hardaway, Charryse Hardaway, Cheramie Hardaway, Chessie Hardaway, Chiquetta Hardaway, Chrysti Hardaway, Cinzia
Hardaway, Cristian Hardaway, Curry Hardaway, Cybele Hardaway, Damion Hardaway, Darius Hardaway, Dawnyell Hardaway, Denora Hardaway, Dung Hardaway, Ebonye Hardaway, Elesa Hardaway, Ellery Hardaway,
Ellis Hardaway, Emile Hardaway, Eowyn Hardaway, Felicite Hardaway, Geniene Hardaway, Ginia Hardaway, Hadiya Hardaway, Hang Hardaway, Harlene Hardaway, Inita Hardaway, Jalaine Hardaway, Janne
Hardaway, Jari Hardaway, Jayci Hardaway, Jodilynn Hardaway, Jomarie Hardaway, Jyll Hardaway, Kabrina Hardaway, Karisma Hardaway, Kashina Hardaway, Katika Hardaway, Katye Hardaway, Keiona Hardaway,
Kemisha Hardaway, Kyli Hardaway, Lametria Hardaway, Lamica Hardaway, Larue Hardaway, Lasheika Hardaway, Latandra Hardaway, Launi Hardaway, Ligaya Hardaway, Lilibeth Hardaway, Linetta Hardaway, Linnet
Hardaway, Lorrain Hardaway, Loyce Hardaway, Lucresha Hardaway, Marguetta Hardaway, Matrice Hardaway, Meca Hardaway, Meda Hardaway, Meiling Hardaway, Melesa Hardaway, Michaelann Hardaway, Monick
Hardaway, Nonie Hardaway, Nyra Hardaway, Pepsi Hardaway, Porcha Hardaway, Rakeisha Hardaway, Reana Hardaway, Rebacca Hardaway, Rennee Hardaway, Roben Hardaway, Rondalyn Hardaway, Roseline Hardaway,
Salem Hardaway, Samella Hardaway, Sandar Hardaway, Santia Hardaway, Sarada Hardaway, Sayward Hardaway, Schelly Hardaway, Shamecka Hardaway, Sharmeen Hardaway, Shayleen Hardaway, Sheina Hardaway,
Sheleen Hardaway, Shenice Hardaway, Shiri Hardaway, Shonetta Hardaway, Suha Hardaway, Sulay Hardaway, Swanzetta Hardaway, Syndi Hardaway, Tanette Hardaway, Tarla Hardaway, Tiffanni Hardaway, Timaka
Hardaway, Tosheba Hardaway, Willene Hardaway, Wynn Hardaway, Yaisa Hardaway, Adrinne Hardaway, Aleisa Hardaway, Aleka Hardaway, Alka Hardaway, Anais Hardaway, Angy Hardaway, Annetra Hardaway,
Antinette Hardaway, Apryle Hardaway, Aretta Hardaway, Arnell Hardaway, Azusena Hardaway, Bianka Hardaway, Brandan Hardaway, Camron Hardaway, Capucine Hardaway, Carolene Hardaway, Catisha Hardaway,
Catrece Hardaway, Cendy Hardaway, Charmelle Hardaway, Charrisse Hardaway, Chrisha Hardaway, Cina Hardaway, Dajuana Hardaway, Dala Hardaway, Danial Hardaway, Daphyne Hardaway, Daya Hardaway, Delorise
Hardaway, Deneise Hardaway, Deshunda Hardaway, Desmond Hardaway, Desra Hardaway, Dinna Hardaway, Duchess Hardaway, Earnest Hardaway, Edda Hardaway, Eliane Hardaway, Elliott Hardaway, Erline Hardaway,
Feleica Hardaway, Florance Hardaway, Gayl Hardaway, Gaynelle Hardaway, Grisela Hardaway, Huma Hardaway, Jaimelyn Hardaway, Jamile Hardaway, Jeannifer Hardaway, Jolin Hardaway, Jonquil Hardaway, Jurea
Hardaway, Kaili Hardaway, Kanitra Hardaway, Katana Hardaway, Katoya Hardaway, Kenja Hardaway, Kennette Hardaway, Kenyotta Hardaway, Kertina Hardaway, Kiyomi Hardaway, Klarissa Hardaway, Lakeish
Hardaway, Lakena Hardaway, Lanett Hardaway, Lashante Hardaway, Lasonda Hardaway, Laurelle Hardaway, Lenda Hardaway, Leonette Hardaway, Madalene Hardaway, Madhavi Hardaway, Mariadelcarmen Hardaway,
Marine Hardaway, Marquerite Hardaway, Marylu Hardaway, Maury Hardaway, Merlyn Hardaway, Natilie Hardaway, Neco Hardaway, Nelissa Hardaway, Nocole Hardaway, Patrese Hardaway, Ranette Hardaway, Ranisha
Hardaway, Rhena Hardaway, Ronesha Hardaway, Royal Hardaway, Safiyyah Hardaway, Sahra Hardaway, Saima Hardaway, Sarahjane Hardaway, Senetra Hardaway, Serra Hardaway, Shallyn Hardaway, Shawni Hardaway,
Sherin Hardaway, Sherisa Hardaway, Sherrelle Hardaway, Shontina Hardaway, Shontrell Hardaway, Solana Hardaway, Tamalyn Hardaway, Tamesia Hardaway, Tametria Hardaway, Tamicia Hardaway, Taneya
Hardaway, Tango Hardaway, Taquila Hardaway, Tekelia Hardaway, Tianne Hardaway, Tonimarie Hardaway, Trea Hardaway, Tremayne Hardaway, Trichelle Hardaway, Tunesia Hardaway, Uma Hardaway, Von Hardaway,
Yanina Hardaway, Akela Hardaway, Aleesa Hardaway, Alonzo Hardaway, Annel Hardaway, Arisha Hardaway, Aylin Hardaway, Aynsley Hardaway, Bernina Hardaway, Billee Hardaway, Calisa Hardaway, Calisha
Hardaway, Canda Hardaway, Cantrell Hardaway, Carylon Hardaway, Casee Hardaway, Celica Hardaway, Chantilly Hardaway, Charnissa Hardaway, Chioma Hardaway, Chrystel Hardaway, Clotilde Hardaway, Coriann
Hardaway, Corisa Hardaway, Currie Hardaway, Cythnia Hardaway, Danay Hardaway, Danyella Hardaway, Darling Hardaway, Deloria Hardaway, Deone Hardaway, Devida Hardaway, Divya Hardaway, Dorean Hardaway,
Dorri Hardaway, Earlean Hardaway, Eletha Hardaway, Elfreda Hardaway, Elizabth Hardaway, Elna Hardaway, Elycia Hardaway, Elza Hardaway, Emmanuelle Hardaway, Felesia Hardaway, Galit Hardaway, Garnetta
Hardaway, Gwendolin Hardaway, Halee Hardaway, Haylie Hardaway, Jaala Hardaway, Jacilyn Hardaway, Jackquelyn Hardaway, Jamekia Hardaway, Jancie Hardaway, Jeany Hardaway, Jenness Hardaway, Johni
Hardaway, Kadee Hardaway, Kalinda Hardaway, Karmel Hardaway, Kashunda Hardaway, Kearston Hardaway, Keina Hardaway, Kely Hardaway, Kerissa Hardaway, Kiandra Hardaway, Kimley Hardaway, Krysia Hardaway,
Kyanna Hardaway, Lajuanda Hardaway, Lania Hardaway, Lanina Hardaway, Latanza Hardaway, Lauriann Hardaway, Lawan Hardaway, Laya Hardaway, Linna Hardaway, Lisbet Hardaway, Luzmaria Hardaway, Maeghan
Hardaway, Mali Hardaway | {"url":"https://www.instantcheckspy.com/lastname/Hardaway.php","timestamp":"2024-11-09T19:52:44Z","content_type":"text/html","content_length":"354808","record_id":"<urn:uuid:2b80cdbb-9fa8-45f7-a956-496d03c8944d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00239.warc.gz"} |
Jump Rope Length Calculator | Online Calculators
Jump Rope Length Calculator
You may need to calculate jumping rope length to assure the improvement of your cardiovascular health, agility, and coordination with jumping rope. You may also want to measure this length for skill
level to maximize effectiveness and prevent injury.
Simple enter your total height in inches into the input field of the calculator and get desired rope length in a single click.
It is the length of the rope that best suits your height, allow enables you efficient and comfortable jumping. The correct length can improve your performance and prevent from injuries.
Tool User Information
Input Action
Total Height (H) Enter your height in inches (e.g., 68 inches).
Click Calculate Displays the recommended rope length in inches.
Formula and Solved Calculation Example for Jump Rope Length
Calculate jump rope length with:
$\text{JRL} = H + 30$
Variable Description
JRL Jump Rope Length (inches)
H Your Total Height (inches)
Basic Calculation Example
Step Calculation
Height (H) 68 inches
Rope Length (JRL) $68 + 30 = 98$ inches
Answer: The recommended jump rope length is 98 inches.
Advanced Calculation (Based on Skill Level)
Skill Level Length Adjustment (in) Height (in) Jump Rope Length (JRL)
Beginner +30 68 98
Intermediate +20 68 88
Advanced +10 68 78
Jump Rope Size Chart
Height (inches) Recommended Rope Length (inches)
50-55 80-85
55-60 85-90
60-65 90-95
65-70 95-100
70-75 100-105
75-80 105-110
Benefits of Correct Jump Rope Measurement
• It enhances jumping efficiency
• It reduces injury risk
• It improves workout performance
• It offers personalized fit
• It increases comfort and control
Leave a Comment | {"url":"https://lengthcalculators.com/jump-rope-length-calculator/","timestamp":"2024-11-05T02:54:48Z","content_type":"text/html","content_length":"64692","record_id":"<urn:uuid:cf3b4143-bc0a-412e-b7bf-44d7493337f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00307.warc.gz"} |
PSC Computer Operator Exam Multiple Choice Question(MCQ) Quiz-2
This MCQ PSC computer quiz contains 10 multiple choice questions from the Fundamentals of Computer chapter. Each question has four options and one is the correct answer among them. These MCQ quizzes
are specially designed for PSC (Loksewa aayog) computer operator exam of both Assistant computer operator and computer operator.
PSC Computer Operator Exam Quiz-2
1. Charles Babbage invented :
1. ENIAC
2. Difference engine
3. Electronic computer
4. Punched card
2. The first electronics computer ENIAC is designed by?
1. Van-Neumann
2. Joseph M. Jacquard
3. J. Presper Eckert
4. All of the above
3. All of the following are examples of storage devices EXCEPT:
1. Hard disk
2. Printers
3. Floppy disk
4. CD
4. Demodulation is a process of :
1. Converting analog to digital signal
2. Converting digital signals
3. Multiplexing various solutions into one high-speed line signal
4. Performing data description
5. Which is not an operating system?
1. Linux
2. MS-DOS
3. MS-Word
4. Unix
6. The major functions of the operating system:
1. Memory management
2. Files management
3. Process management
4. All of the above
7. Spooling technique is associated with:
1. Operating system and scanner
2. Operating system and printer
3. Operating system and hard disk
4. Operating system and keyboard
8. GUI stands for:
1. Graphical Universal Interconnection
2. Graphical User Information
3. Graphical User Internet
4. Graphical User Interface
9. FAT stands for:
1. File for all type
2. Format all tab setting
3. File allocation table
4. File attributes type
10. To move to the bottom of a document press….
1. End key
2. Home key
3. Alt+end key
4. Ctrl+end key
In this blog, we have published various MCQ quizzes in Fundamental of Computer, Word Process, Excel, PowerPoint, Ms. Access, Networking, etc. Hope you all like it. If there is a mistake in any of the
questions please let us know by commenting on this post. Thanks!
Post a Comment | {"url":"https://www.unlimitededu.net/2021/01/psc-computer-operator-exam-quiz.html","timestamp":"2024-11-02T11:34:39Z","content_type":"application/xhtml+xml","content_length":"155168","record_id":"<urn:uuid:d85830a7-f377-43e7-8b33-77929e0d6850>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00882.warc.gz"} |
Drawing Multiple Plots with Matplotlib in Python - wellsr.com
Python’s Matplotlib library is one of the most widely used data visualization libraries. With Matplotlib, you can plot your data using all kinds of chart types, including line charts, bar charts, pie
charts and scatter plots.
Matplotlib lets you plot a single chart but it also allows you to draw multiple charts at once in the form of grids.
In this tutorial, we’ll demonstrate exactly how to draw multiple plots with the Matplotlib library.
Plotting a Single Plot
Before we show you how to plot multiple plots, let’s make sure we have the fundamentals down by walking through an example showing how to draw a single plot with Matplotlib. In this example, we’re
going to draw a line plot.
To draw plots with Matplotlib, use the pyplot submodule from the Matplotlib library.
Specifically, to draw a line plot, you need to call the plot() function from the pyplot module and pass it lists of values for your x and y axes.
The script below draws a line plot for the sine function. The input values consists of 50 equidistant points between -100 to 100.
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
x = np.linspace(-100, 100, 50)
y_sin = [np.sin(i) for i in x]
plt.plot(x, y_sin, 'bo-')
Note: The %matplotlib inline snippet only works with the Jupyter Notebook. If you’re not using Jupyter Notebook, just add plt.show() right after the point where we start making plots.
Plotting Multiple plots
Once you know how to do that, you’re ready to plot multiple plots. Again, Matplotlib allows you to plot multiple plots in the form of a grid. There are a couple of ways to do it:
1. Using the subplot() function
2. Using the subplots() function
Using the subplot() function
To draw multiple plots using the subplot() function from the pyplot module, you need to perform two steps:
1. First, you need to call the subplot() function with three parameters: (1) the number of rows for your grid, (2) the number of columns for your grid, and (3) the location or axis for plotting. For
instance, the subplot(2,3,1) tells the Python interpreter that the next plot should be plotted in a grid that contains 2 rows and 3 columns and the plot should appear at the first location in the
grid (row 1, column 1). The order of the plotting locations first goes from left to right, then top to bottom. This means means that the subplot(2,3,4) command will draw a plot in the second row
and first column of the grid.
2. After the subplot() command, simply call the corresponding function or chart type you want to plot using the the pyplot module. For instance, the script below makes line charts using the plot()
This script is going to make six line plots in a grid of two rows and three columns, using the subplot() function.
import matplotlib.pyplot as plt
import numpy as np
import math
%matplotlib inline
x = np.linspace(-100, 100, 50)
y_sin = [np.sin(i) for i in x]
y_cos = [np.cos(i) for i in x]
y_tan = [np.tan(i) for i in x]
y_log = [np.log(i) for i in x]
y_exp = [np.exp(i) for i in x]
y_sqr = [i*i for i in x]
plt.rcParams["figure.figsize"] = [12,8]
plt.plot(x, y_sin, 'bo-')
plt.plot(x, y_cos, 'rx-')
plt.plot(x, y_tan, 'g*-')
plt.plot(x, y_log, 'y<-')
plt.plot(x, y_exp, 'g')
plt.plot(x, y_sqr, 'r*-')
Code More, Distract Less: Support Our Ad-Free Site
You might have noticed we removed ads from our site - we hope this enhances your learning experience. To help sustain this, please take a look at our Python Developer Kit and our comprehensive cheat
sheets. Each purchase directly supports this site, ensuring we can continue to offer you quality, distraction-free tutorials.
Using the subplots() function
With the subplot() function, you need to set the location for every subsequent plot. The subplots() function eliminates this requirement.
You can set the the number of rows and columns for your grid at once using the subplots() function from the pyplot module. The number of rows and columns are passed as integer values to the nrows and
ncols attriubtes of the subplots() function. Depending on the number of rows and columns, the subplots() function returns a list of AxesSubplot objects.
For instance, in the script below, you call the subplots() method that creates a grid with 2 rows and 3 columns. The “axes” variable in the script below contains the list of “AxesSubplot” objects
which are printed on the console.
import matplotlib.pyplot as plt
import numpy as np
import math
x = np.linspace(-100, 100, 50)
y_sin = [np.sin(i) for i in x]
y_cos = [np.cos(i) for i in x]
y_tan = [np.tan(i) for i in x]
y_log = [np.log(i) for i in x]
y_exp = [np.exp(i) for i in x]
y_sqr = [i*i for i in x]
plt.rcParams["figure.figsize"] = [12,8]
fig, axes = plt.subplots(nrows= 2, ncols=3)
In the output, you can see a list of lists corresponding to the rows and columns of your grid. You can also see the empty axes. We’ve highlighted the list in yellow, along with the grid dimensions.
The next step is to draw plots in these empty charts. To do this, you have to select an item from the list of AxesSubplot objects and call the plot() function using that object.
For example, to draw a plot at the first row and first column in the grid, you need to access the AxesSubplot at index [0,0]. Notice the index numbers for the subplots begin at 0.
The script below draws six line plots in 2 rows and 3 columns using the subplots() function.
import matplotlib.pyplot as plt
import numpy as np
import math
x = np.linspace(-100, 100, 50)
y_sin = [np.sin(i) for i in x]
y_cos = [np.cos(i) for i in x]
y_tan = [np.tan(i) for i in x]
y_log = [np.log(i) for i in x]
y_exp = [np.exp(i) for i in x]
y_sqr = [i*i for i in x]
plt.rcParams["figure.figsize"] = [12,8]
fig, axes = plt.subplots(nrows= 2, ncols=3)
axes[0,0].plot(x, y_sin, 'bo-')
axes[0,1].plot(x, y_cos, 'rx-')
axes[0,2].plot(x, y_tan, 'g*-')
axes[1,0].plot(x, y_log, 'y<-')
axes[1,1].plot(x, y_exp, 'g')
axes[1,2].plot(x, y_sqr, 'r*-')
That’s all there is to it! We have a lot of tutorials showing how to visualize data with Python, including with Pandas and Seaborn. If you want more tutorials like these, go ahead and subscribe using
the form below and we’ll send them your way. | {"url":"https://wellsr.com/python/drawing-multiple-plots-with-matplotlib-in-python/","timestamp":"2024-11-12T22:21:16Z","content_type":"text/html","content_length":"42418","record_id":"<urn:uuid:826c7d31-9c2f-4fb1-81e0-5411c5aeb80f>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00406.warc.gz"} |
Two corners of a triangle have angles of pi / 3 and pi / 12 . If one side of the triangle has a length of 8 , what is the longest possible perimeter of the triangle? | HIX Tutor
Two corners of a triangle have angles of #pi / 3 # and # pi / 12 #. If one side of the triangle has a length of #8 #, what is the longest possible perimeter of the triangle?
Answer 1
Largest possible area of the triangle is 103.4256
Given are the two angles #(pi)/12# and #pi/3# and the length 8
#= pi - (((pi)/12) + pi/3) = ((7pi)/12#
I am assuming that length AB (1) is opposite the smallest angle.
Area#=( 8^2*sin(pi/3)*sin((7pi)/12))/(2*sin(pi/12))#
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To find the longest possible perimeter of the triangle, we need to consider the maximum lengths for the other two sides given the angles provided.
Let's label the triangle ABC, where angle A is π/3 and angle B is π/12.
1. Determine the length of the side opposite angle A (side BC) using the law of sines:
[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} ]
Given that angle A = π/3, we have:
[ \frac{\sin(\frac{\pi}{3})}{8} = \frac{\sin(\frac{\pi}{12})}{b} ]
[ b = \frac{8 \sin(\frac{\pi}{12})}{\sin(\frac{\pi}{3})} ]
2. Find the length of the side opposite angle B (side AC) using the law of sines:
[ \frac{\sin B}{b} = \frac{\sin A}{a} ]
[ \frac{\sin(\frac{\pi}{12})}{8} = \frac{\sin(\frac{\pi}{3})}{c} ]
[ c = \frac{8 \sin(\frac{\pi}{3})}{\sin(\frac{\pi}{12})} ]
3. Find the length of the side opposite angle C (side AB) using the law of cosines:
[ c^2 = a^2 + b^2 - 2ab\cos C ]
Given that angle C = π - (π/3 + π/12) = π - π/4 = 3π/4:
[ (8)^2 = a^2 + b^2 - 2ab\cos(\frac{3\pi}{4}) ]
[ 64 = a^2 + b^2 + \sqrt{2}ab ]
4. The perimeter of the triangle is given by ( P = a + b + c ).
Substitute the values of a, b, and c into the equation for the perimeter:
[ P = 8 + \frac{8 \sin(\frac{\pi}{12})}{\sin(\frac{\pi}{3})} + \frac{8 \sin(\frac{\pi}{3})}{\sin(\frac{\pi}{12})} ]
Simplify this expression to find the longest possible perimeter of the triangle.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
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
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/two-corners-of-a-triangle-have-angles-of-pi-3-and-pi-12-if-one-side-of-the-trian-8f9afa3908","timestamp":"2024-11-09T22:35:06Z","content_type":"text/html","content_length":"584014","record_id":"<urn:uuid:e1fc30d9-1ac5-467a-ae69-a5e6b3bd07a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00805.warc.gz"} |
Sowell and the Laffer curve
The basic principle of the Laffer curve is easily demonstrable:
• First, imagine a situation where the income tax rate is 0%. Income tax revenues are $0.
• Second, imagine a situation where income tax rate is 100%. Everyone opts out of earning an income, and revenues are again $0.
• Third, we know empirically that when the income tax rate is between these extremes, revenue is more than 0.
• Fourth, assume that the distribution is continuous. I don't think it has to be, but it's easier to picture.
From these, we know that there must be at least one maximum between 0% and 100%. Furthermore, we know that there must also be some subset of the range from 0%-100% tax rate where increasing taxes
decreases revenue. We can't say anthing about where that subset is, nor that there is only one such subset, just that there must be at least one.
“The Laffer Curve” refers specifically to a specific curve that Mr. Laffer drew, and his (studied, but not necessarily correct) opinions on where the US tax rate in the 1980s. That specific curve is
now outdated at best, and entirely irrelevent to any discussion not focusing on economics policy of the 1980s. When I refer to the Laffer Curve on this page, I am referring only to the general
Sowell makes the specific claim (Here, et al.) that for high income earners, at points in the past, tax revenue has risen after tax rates fell. He uses this economics arguement as the basis of
several political arguements which are not testable, so I am looking at the only at the truth of the specific economics claim.
I pulled income tax data from the IRS website, looking at aggregate US data. The raw excel files which I downloaded are available hereus_taxes_by_income.zip. Admittedly, it's a bit of a misnomer to
call them 'raw', as they are aggregate data. 1997 was the earliest that these were available in this format, and 2009 the most recent.
I pulled tax rate data from http://www.taxpolicycenter.org/taxfacts/displayafact.cfm?Docid=213. It may not be an unbiased source (I didn't bother to check) but this is factual information, and not
subject to skewing. Also, various other sources agree; this one just had an option to download in excel. From the attached caveats: “Perhaps most importantly, it ignores the large increase in
percentage of returns that were subject to this top rate.” I'll try to account for that, but no guarantees.
Constant dollar data is from http://oregonstate.edu/cla/polisci/individual-year-conversion-factor-tables. Again, I used it because excel download was an option. Pick the 1997 tables. | {"url":"http://wiki.fectin.com/doku.php/econ:laffer","timestamp":"2024-11-13T11:27:52Z","content_type":"application/xhtml+xml","content_length":"13348","record_id":"<urn:uuid:d5279126-8818-41df-9a0b-7e10a1dde2a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00789.warc.gz"} |
What is a Bar Graph, Types of Bar Graph - Data Handling
Bar Graph for Class 5 Maths
The bar graph is a visual representation of the data. Here students will learn about what is a bar graph. Bar graph images and how to draw a bar graph.
In this learning concept, the students will also learn to
• Classify the types of bar graphs.
• Evaluate the horizontal bar graph and vertical bar graph.
• Identify the advantage of bar graphs.
Each concept is explained to class 5 maths students using illustrations, examples, and mind maps. Students can assess their learning by solving the two printable worksheets given at the page’s end.
Download the bar graph worksheet for class 5 and check the solutions to the bar graph questions for class 5 provided in PDF format.
What Is a Bar Graph?
A bar graph is a graphical representation of data, by rectangular bars.
• Data can be represented by bars.
• In a bar graph each bar represents a number.
• The length of bars represents numerical value.
• In a bar graph, bars can be drawn vertically or horizontally.
Example:In a school, there are 15, 20, 45, and 39 students with yellow, red, blue, and green dresses, respectively. We can represent this data using a bar graph:
To make a bar graph, we need:
Data categories:Type of things in the data.
Data value:Numerical value of each category.
Scale:To draw the bar with numbers by scaling.
Title:We need to give an appropriate title to the graph.
• In a bar graph, there should be equal spacing between the bars.
Types of Bar Graph
Vertical Bar Graph
• In vertical bar graphs, the bars are drawn vertically to represent the data.
The number of children in five different batches of an educational institute is given below by a vertical bar graph.
Here, 1 unit length = 10 children.
From this graph, we can find
What is the number of students in Batch 3?
From the graph, the number of students in batch 3 is 40.
The maximum number of students present in which batch?
The maximum number of students present in batch 2, the number of students is 50.
Horizontal Bar Graph
• In the horizontal bar graph represents the data by the horizontal bars.
The data for the baking of cakes in a bakery from Monday to Saturday is shown below by a horizontal bar graph.
1 unit = 10 cakes
Question 1:
Maximum numbers of cakes baked on which day?
Question 2:
Minimum numbers of cakes baked on which day?
Minimum numbers of cakes baked on Monday.
How to Draw a Bar Graph?
Let us consider,
We have four different types of animals, such as cat, dog, rabbit, and hamster and the corresponding numbers are 40, 30, 10, and 70 respectively.
The bar graph becomes as given below:
Note: Scale: 1 unit=10 animals.
Transformation of a horizontal bar graph to a vertical bar graph:
From this graph, we can make the table
The following table shows number of visitors to park for the months January to March
Month January February March
Month 150 300 250
Also, from the graph we have
• The greatest number of visitors come to the park in February.
• Also, the maximum number of visitors in one month is 300.
• The number of visitors that come in March is 250.
• The minimum number of visitors comes in January.
• The number of visitors that come in January is 150.
Also, we can make the vertical graph by the similar data:
Advantages of bar graph
• Bar graph summaries the large set of data in simple visual form.
• Bar graph displays each category of data in the frequency distribution.
• Bar graph displays each category of data in the frequency distribution.
• The bar graph clarifies the trend of data better than the table.
• Bar graph helps in estimating the key values at a glance.
The disadvantage of bar graph
• Sometimes, the bar graph fails to reveal the patterns, cause, effects, etc.
• It can be easily manipulated to yield fake information.
Did you know?
Bar graph use to show the increase of daily cases of coivd-19 | {"url":"https://www.orchidsinternationalschool.com/maths-concepts/bar-graph","timestamp":"2024-11-07T01:15:58Z","content_type":"text/html","content_length":"1048945","record_id":"<urn:uuid:dcfbe6fc-6d9b-4e6f-be8a-1cdf4e4557dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00473.warc.gz"} |
Statistics - Simply Psychology
The field of statistics is concerned with collecting, analyzing, interpreting, and presenting data. Learn statistics and probability for free, in simple and easy steps starting from basic to advanced
• Scientific Method
• Variables
• P-value
Scientific Method
The scientific method is a step-by-step process used by researchers and scientists to determine if there is a relationship between two or more variables. Psychologists use this method to conduct
psychological research, gather data, process information, and describe behaviors.
Learn More: Steps of the Scientific Method
Variables apply to experimental investigations. The independent variable is the variable the experimenter manipulates or changes. The dependent variable is the variable being tested and measured in
an experiment, and is 'dependent' on the independent variable.
Learn More: Independent and Dependent Variables
When you perform a statistical test a p-value helps you determine the significance of your results in relation to the null hypothesis. A p-value less than 0.05 (typically ≤ 0.05) is statistically
Learn More: P-Value and Statistical Significance
A p-value less than 0.05 (typically ≤ 0.05) is statistically significant. It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the results have occurred by
random chance rather than a real effect. Therefore, we reject the null hypothesis and accept the alternative hypothesis.
However, it is important to note that the p-value is not the only factor that should be considered when interpreting the results of a hypothesis test. Other factors, such as effect size, should also
be considered.
Learn More: What A p-Value Tells You About Statistical Significance
A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. It is also known as a standard score because it allows the comparison
of scores on different variables by standardizing the distribution. The z-score is positive if the value lies above the mean and negative if it lies below the mean.
Learn More: Z-Score: Definition, Calculation, Formula, & Interpretation
The independent variable is the variable the experimenter manipulates or changes and is assumed to have a direct effect on the dependent variable. For example, allocating participants to either drug
or placebo conditions (independent variable) to measure any changes in the intensity of their anxiety (dependent variable).
Learn More: What are Independent and Dependent Variables?
Quantitative data is numerical information about quantities and qualitative data is descriptive and regards phenomena that can be observed but not measured, such as language.
Learn More: What’s the difference between qualitative and quantitative research?
Explore Statistics | {"url":"http://theltdfoundation.org/statistics.html","timestamp":"2024-11-03T18:34:28Z","content_type":"text/html","content_length":"152752","record_id":"<urn:uuid:f23fd3dc-0c32-42d1-aa6b-3a889dee0bff>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00424.warc.gz"} |
Span - (Intro to Mathematical Economics) - Vocab, Definition, Explanations | Fiveable
from class:
Intro to Mathematical Economics
In linear algebra, the span of a set of vectors is the collection of all possible linear combinations of those vectors. This concept is essential as it defines a vector space generated by the
vectors, showing all points that can be reached by scaling and adding them together. Understanding the span helps in determining the dimensionality of vector spaces and how they relate to each other.
congrats on reading the definition of Span. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The span of a single vector is a line through the origin in the direction of that vector.
2. If you have two non-parallel vectors in two dimensions, their span covers the entire two-dimensional plane.
3. The dimension of the span corresponds to the number of vectors in a linearly independent set that can be used to generate that span.
4. If the set of vectors is linearly dependent, the span remains unchanged by removing some of those vectors.
5. In three dimensions, three non-coplanar vectors can generate the entire three-dimensional space when combined.
Review Questions
• How does the concept of span relate to linear combinations and vector spaces?
□ The concept of span is directly tied to linear combinations, as it includes all possible combinations formed by scaling and adding vectors. By understanding how to create these combinations,
one can identify which points are reachable within a given vector space. This relationship highlights how spans define the boundaries and dimensions of vector spaces based on the vectors
• In what scenarios would removing a vector from a set still result in the same span?
□ Removing a vector from a set will not affect the span if that vector is linearly dependent on the others. This means that it can be expressed as a combination of the remaining vectors, so its
removal does not change the overall coverage of the span. Understanding this helps simplify calculations and determine the minimum necessary vectors to maintain a given span.
• Evaluate how understanding spans can impact solving systems of equations in economics.
□ Understanding spans plays a crucial role in solving systems of equations, especially when determining feasible solutions in economic models. By recognizing which combinations of variables
(vectors) create valid outcomes (spans), economists can analyze constraints and optimize resource allocation effectively. Additionally, this knowledge assists in understanding relationships
between multiple economic factors, allowing for more robust decision-making based on available data.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/introduction-to-mathematical-economics/span","timestamp":"2024-11-04T18:33:13Z","content_type":"text/html","content_length":"148141","record_id":"<urn:uuid:41381f3a-9652-4638-9255-b9b5fddccc58>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00223.warc.gz"} |
Cohomology of Groups and Algebraic K-theory
Advanced Lectures in Mathematics
Cohomology of Groups and Algebraic K-theory
Published: 1 March 2010
Publisher: International Press of Boston, Inc.
Language: English
526 pages
List Price: $97.50
Cohomology of groups is a fundamental tool in many subjects of modern mathematics. One important generalized cohomology theory is the algebraic K-theory. Indeed, algebraic K-groups of rings are
important invariants of the rings and have played important roles in algebra, topology, number theory, etc. This volume consists of expanded lecture notes from a 2007 seminar at Zhejiang University
in China, at which several leading experts presented introductions, to and surveys of, many aspects of cohomology of groups and algebraic K-theory, along with their broad applications. Two
foundational papers on algebraic K-theory by Daniel Quillen are also included.
Pub. Date ISBN-13 ISBN-10 Medium Binding Size, Etc. Status List Price
2010 Mar 9781571461445 1571461442 Print paperback 7” x 10” In Print US$97.50 | {"url":"https://intlpress.com/site/pub/pages/books/items/00000262/index.php","timestamp":"2024-11-11T07:09:11Z","content_type":"text/html","content_length":"8098","record_id":"<urn:uuid:99040ed9-6c51-4eab-89dc-e886186b8b48>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00618.warc.gz"} |
Trying to Understand Tries - 911 WeKnow
Trying to Understand Tries
In every installment of this series, we?ve tried to understand and dig deep into the tradeoffs of the things that we?re learning about.
When we were learning about data structures, we looked at the pros and cons of each structure, in an effort to make it easier and more obvious for us to see what types of problems that structure was
created to solve. Similarly, when we were learning about sorting algorithms, we focused a lot on the tradeoffs between space and time efficiency to help us understand when one algorithm might be the
better choice over another.
As it turns out, this is going to become more and more frequent as we start looking at even more complex structures and algorithms, some of which were invented as solutions to super specific
problems. Today?s data structure is, in fact, based on another structure that we?re already familiar with; however, it was created to solve a particular problem. More specifically, it was created as
a compromise between running time and space ? two things that we?re pretty familiar with in the context of Big O notation.
So, what is this mysterious structure that I keep talking about so vaguely but not actually naming? Time to dig in and find out!
Trying on tries
There are a handful of different ways to represent something as seemingly simple as a set of words. For example, a hash or dictionary is one that we?re probably familiar with, as is as hash table.
But there?s another structure that was created to solve the very problem of representing a set of words: a trie. The term ?trie? comes from the word retrieval, and is usually pronounced ?try?, to
distinguish it from other ?tree? structures.
However, a trie is basically a tree data structure, but it just has a few rules to follow in terms of how it is created and used.
A trie is a tree-like data structure whose nodes store the letters of an alphabet. By structuring the nodes in a particular way, words and strings can be retrieved from the structure by traversing
down a branch path of the tree.
Tries in the context of computer science are a relatively new thing. The first time that they were considered in computing was back in 1959, when a Frenchman named Ren de la Briandais suggested using
them. According to Donald Knuth?s research in The Art of Computer Programming:
Trie memory for computer searching was first recommended by Ren de la Briandais. He pointed out that we can save memory space at the expense of running time if we use a linked list for each node
vector, since most of the entries in the vectors tend to be empty.
The original idea behind using tries as a computing structure was that they could be a nice compromise between running time and memory. But we?ll come back to that in a bit. First, let?s take a step
back and try and understand what exactly this structure looks like to start.
We know that tries are often used to represent words in an alphabet. In the illustration shown here, we can start to get a sense of how exactly that representation works.
Each trie has an empty root node, with links (or references) to other nodes ? one for each possible alphabetic value.
The shape and the structure of a trie is always a set of linked nodes, connecting back to an empty root node. An important thing to note is that the number of child nodes in a trie depends completely
upon the total number of values possible. For example, if we are representing the English alphabet, then the total number of child nodes is directly connected to the total number of letters possible.
In the English alphabet, there are 26 letters, so the total number of child nodes will be 26.
Imagine, however, that we were creating a trie to hold words from the Khmer (Cambodian) alphabet, which is the longest known alphabet with 74 characters. In that case, the root node would contain 74
links to 74 other child nodes.
The size of a trie is directly correlated to the size of all the possible values that the trie could represent.
Okay, so a trie could be pretty small or big, depending on what it contains. But, so far, all we?ve talked about is the root node, which is empty. So where do the letters of different words live if
the root node doesn?t house them all?
The answer to that lies in the root node?s references to its children. Let?s take a closer look at what a single node in a trie looks like, and hopefully this will start to become more clear.
In the example shown here, we have a trie that has an empty root node, which has references to children nodes. If we look at the cross-section of one of these child nodes, we?ll notice that a single
node in a trie contains just two things:
1. A value, which might be null
2. An array of references to child nodes, all of which also might be null
Each node in a trie, including the root node itself, has only these two aspects to it. When a trie representing the English language is created, it consists of a single root node, whose value is
usually set to an empty string: “”.
That root node will also have an array that contains 26 references, all of which will point to null at first. As the trie grows, those pointers start to get filled up with references to other nodes
nodes, which we?ll see an example of pretty soon.
The way that those pointers or references are represented is particularly interesting. We know that each node contains an array of references/links to other nodes. What?s cool about this is that we
can use the array?s indexes to find specific references to nodes. For example, our root node will hold an array of indexes 0 through 25, since there are 26 possible slots for the 26 letters of the
alphabet. Since the alphabet is in order, we know that the reference to the node that will contain the letter A will live at index 0.
So, once we have a root node, where do we go from there? It?s time to try growing our trie!
Giving trie traversal a try
A trie with nothing more than a root node is simply no fun at all! So, let?s complicate things a bit further by playing with a trie that has some words in it, shall we?
In the trie shown below, we?re representing the nursery rhyme that starts off with something like ?Peter Piper picked a peck of pickled peppers?. I won?t try to make you remember the rest of it,
mostly because it is confusing and makes my head hurt.
Looking at our trie, we can see that we have an empty root node, as is typical for a trie structure. We also have six different words that we?re representing in this trie: Peter, piper, picked, peck,
pickled, and peppers.
To make this trie easier to look at, I?ve only drawn the references that actually have nodes in them; it?s important to remember that, even though they?re not illustrated here, every single node has
26 references to possible child nodes.
Notice how there are six different ?branches? to this trie, one for each word that?s being represented. We can also see that some words are sharing parent nodes. For example, all of the branches for
the words Peter, peck, and peppers share the nodes for p and for e. Similarly, the path to the word picked and pickled share the nodes p, i, c, and k.
So, what if we wanted to add the word pecked to this list of words represented by this trie? We?d need to do two things in order to make this happen:
1. First, we?d need to check that the word pecked doesn?t already exist in this trie.
2. Next, if we?ve traversed down the branch where this word ought to live and the words doesn?t exist yet, we?d insert a value into the node?s reference where the word should go. In this case, we?d
insert e and d at the correct references.
But how do we actually go about checking if the word exists? And how do we insert the letters into their correct places? This is easier to understand with a small trie as an example, so let?s look at
a trie that is empty, and try inserting something into it.
We know that we?ll have an empty root node, which will have a value of “”, and an array with 26 references in it, all of which will be empty (pointing to null) to start. Let?s say that we want to
insert the word “pie”, and give it a value of 5. Another way to think about it is that we have a hash that looks like this: { “pie”: 5 }.
We?ll work our way through the key, using each letter to build up our trie and add nodes as necessary.
We?ll first look for the pointer for p, since the first letter in our key “pie” is p. Since this trie doesn?t have anything in just yet, the reference at p in our root node will be null. So, we?ll
create a new node for p, and the root node now has an array with 25 empty slots, and 1 slot (at index 15) that contains a reference to a node.
Now we have a node at index 15, holding the value for p. But, our string is “pie”, so we?re not done yet. We?ll do the same thing for this node: check if there is a null pointer at the next letter of
the key: i. Since we encounter another null link for the reference at i, we?ll create another new node. Finally, we?re at the last character of our key: the e in “pie”. We create a new node for the
array reference to e, and inside of this third node that we?ve created, we?ll set our value: 5.
In the future, if we want to retrieve the value for the key “pie”, we?ll traverse down from one array to another, using the indices to go from the nodes p, to i, to e; when we get to the node at the
index for e, we?ll stop traversing, and retrieve the value from that node, which will be 5.
Let?s actually take a look at what searching through our newly-built trie would look like!
In the illustration shown here, if we search for the key “pie”, we traverse down each node?s array, and look to see if there is a value for the branch path: p-i-e. If it does have a value, we can
simply return it. This is sometimes referred to as a search hit, since we were able to find a value for the key.
But what if we search for something that doesn?t exist in our trie? What if we search for the word “pi”, which we haven?t added as a key with a value? Well, we?ll go from the root node to the node at
index p, and then we?ll go from the node at p to the node at index i. When we get to this point, we?ll see if the node at the branch path p-i has a value. In this case, it doesn?t have a value; it?s
pointing at null. So, we can be sure that the key “pi” doesn?t exist in our trie as a string with a value. This is often referred to as a search miss, since we could not find a value for the key.
Finally, there?s one other action that we might want to do to our trie: delete things! How can we remove a key and its value from our trie structure? To illustrate this, I?ve added another word to
our trie. We now have both the keys “pie” and “pies”, each with their own values. Let?s say we want to remove the key “pies” from our trie.
In order to do this, we?d need to take two steps:
1. First, we need to find the node that contains the value for that key, and set its value to null. This means traversing down and finding the last letter of the word “pies”, and then resetting the
value of the last node from 12 to null.
2. Second, we need to check the node?s references and see if all of its pointers to other nodes are also null. If all of them are empty, that means that there are no other words/branches below this
one, and they can all be removed. However, if there are pointers for other nodes that do have values, we don?t want to delete the node that we?ve just set to null.
This last check is particularly important in order to not remove longer strings when we remove substrings of a word. But other than that single check, there?s nothing more to it!
Trying our hand at tries
When I was first learning about tries, they reminded me a lot of hash tables, which we learned about earlier in this series. In fact, the more that I read about tries and how to build and search
through them, the more I wondered what the tradeoffs between the two structures actually were.
As it turns out, both tries and hash tables are reminscient of one another because they both use arrays under the hood. However, hash tables use arrays combined with linked lists, whereas tries use
arrays combined with pointers/references.
There are quite a few minor differences between both of these two structures, but the most obvious difference between hash tables and tries is that a trie has no need for a hash function, because
every key can be represented in order (alphabetically), and is uniquely retrievable since every branch path to a string?s value will be unique to that key. The side effect of this is that there are
no collisions to deal with, and thus a relying on the index of an array is enough, and a hashing function is unnecessary.
However, unlike hash tables, the downside of a trie is that is takes up a lot of memory and space with empty (null) pointers. We can imagine how a large trie would start grow in size, and with each
node that was added, an entire array containing 26 null pointers would have to be initialized as well. For longer words, those empty references would probably never get filled up; for example,
imagine we had a key ?Honorificabilitudinitatibus?, with some value. That?s super long word, and we?re probably not going to be adding any other sub-branches to that word in the trie; that?s a bunch
of empty pointers for each letter of that word that are taking up space, but not really ever being used!
Hopefully though, we?re not going to use the word ?Honorificabilitudinitatibus? as a string.
There are some great benefits to using tries, however. For starters, the bulk of the work in creating a trie happens early on. This makes sense if we think about it, because when we?re first adding
nodes, we have to do some heavy lifting of allocating memory for an array each time. But, as the trie grows in size, we have to do less work each time to add a value, since it?s likely that we?ve
already initialized nodes and their values and references. Adding ?intermediate nodes? becomes a lot easier since the branches of the trie have already been built up.
Another fact in the ?pro column? for tries is that each time we add a word?s letter, we know that we?ll only ever have to look at 26 possible indexes in a node?s array, since there are only 26
possible letters in the English alphabet. Even though 26 seems like a lot, for our computers, it?s really not that much space. However, the fact that we are sure that each array will only ever
contain 26 references is a huge benefit, because this number will never change in the context of our trie! It is a constant value.
On that note, let?s look quickly at the Big O time complexity of a trie data structure. The amount of time it takes to create a trie is tied directly to how many words/keys the trie contains, and how
long those keys could potentially be. The worst-case runtime for creating a trie is a combination of m, the length of the longest key in the trie, and n, the total number of keys in the trie. Thus,
the worst case runtime of creating a trie is O(mn).
The time complexity of searching, inserting, and deleting from a trie depends on the length of the word a that?s being searched for, inserted, or deleted, and the number of total words, n, making the
runtime of these operations O(an). Of course, for the longest word in the trie, inserting, searching, and deleting will take more time and memory than for the shortest word in the trie.
So, now that we know all the inner working of tries, there?s one question that?s still left to answer: where are tries used? Well, the truth is that they?re rarely used exclusively; usually, they?re
used in combination with another structure, or in the context of an algorithm. But perhaps the coolest example of how tries can be leveraged for their form and function is for autocomplete features,
like the one used in search engines like Google.
Now that we know how tries function, we can imagine how typing two letters into a search box would retrieve a subset of a much larger trie structure. Another powerful aspect of this is that tries
make it easy to search for a subset of elements, since, similar to binary search trees, each time we traverse down a branch of a tree, we are cutting out the number of other nodes we need to look at!
It?s worth mentioning that search engines probably have more complexity to their tries, since they will return certain terms based on how popular they are, and likely have some additional logic to
determine the weight associated with certain terms in their trie structures. But, under the hood, they probably are using tries to make this magic happen!
Tries are also used for matching algorithms and implementing things like spellcheckers, and can also be used for imlementing versions of radix sort, too.
I suppose that if we trie hard enough, we?ll see that tries are all around us! (Sorry, I just couldn?t resist the pun)
Tries often show up in white boarding or technical interview questions, often in some variation of a question like ?search for a string or substring from this sentence?. Given their unique ability to
retrieve elements in constant time, they are often a great tool to use, and luckily, many people have written about them. If you want some helpful resources, here are a few good places to start.
1. Digit-based sorting and data structures, Professor Avrim Blum
2. Lecture Notes on Tries, Professor Frank Pfenning
3. Algorithms: Tries, Robert Sedgewick and Kevin Wayne
4. Tries, Brilliant Learning
5. Tries, Daniel Ellard
6. Tries, Harvard CS50 | {"url":"https://911weknow.com/trying-to-understand-tries","timestamp":"2024-11-03T15:12:48Z","content_type":"text/html","content_length":"60011","record_id":"<urn:uuid:acb31236-f459-413d-953e-71dbe593e84b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00076.warc.gz"} |
LM 27.4 Cosmology Collection
27.4 Cosmology by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
27.4 Cosmology
The Big Bang
Section 19.5 presented the evidence, discovered by Hubble, that the universe is expanding in the aftermath of the Big Bang: when we observe the light from distant galaxies, it is always
Doppler-shifted toward the red end of the spectrum, indicating that no matter what direction we look in the sky, everything is rushing away from us. This seems to go against the modern attitude,
originated by Copernicus, that we and our planet do not occupy a special place in the universe. Why is everything rushing away from our planet in particular? But general relativity shows that this
anti-Copernican conclusion is wrong. General relativity describes space not as a rigidly defined background but as something that can curve and stretch, like a sheet of rubber. We imagine all the
galaxies as existing on the surface of such a sheet, which then expands uniformly. The space between the galaxies (but not the galaxies themselves) grows at a steady rate, so that any observer,
inhabiting any galaxy, will see every other galaxy as receding. There is therefore no privileged or special location in the universe.
We might think that there would be another kind of special place, which would be the one at which the Big Bang happened. Maybe someone has put a brass plaque there? But general relativity doesn't
describe the Big Bang as an explosion that suddenly occurred in a preexisting background of time and space. According to general relativity, space itself came into existence at the Big Bang, and the
hot, dense matter of the early universe was uniformly distributed everywhere. The Big Bang happened everywhere at once.
Observations show that the universe is very uniform on large scales, and for ease of calculation, the first physical models of the expanding universe were constructed with perfect uniformity. In
these models, the Big Bang was a singularity. This singularity can't even be included as an event in spacetime, so that time itself only exists after the Big Bang. A Big Bang singularity also creates
an even more acute version of the black hole information paradox. Whereas matter and information disappear into a black hole singularity, stuff pops out of a Big Bang singularity, and there is no
physical principle that could predict what it would be.
As with black holes, there was considerable skepticism about whether the existence of an initial singularity in these models was an arifact of the unrealistically perfect uniformity assumed in the
models. Perhaps in the real universe, extrapolation of all the paths of the galaxies backward in time would show them missing each other by millions of light-years. But in 1972 Stephen Hawking proved
a variant on the Penrose singularity theorem that applied to Big Bang singularities. By the Hawking singularity theorem, the level of uniformity we see in the present-day universe is more than
sufficient to prove that a Big Bang singularity must have existed.
The cosmic censorship hypothesis
It might not be too much of a philosophical jolt to imagine that information was spontaneously created in the Big Bang. Setting up the initial conditions of the entire universe is traditionally the
prerogative of God, not the laws of physics. But there is nothing fundamental in general relativity that forbids the existence of other singularities that act like the Big Bang, being information
producers rather than information consumers. As John Earman of the University of Pittsburgh puts it, anything could pop out of such a singularity, including green slime or your lost socks. This would
eliminate any hope of finding a universal set of laws of physics that would be able to make a prediction given any initial situation.
That would be such a devastating defeat for the enterprise of physics that in 1969 Penrose proposed an alternative, humorously named the “cosmic censorship hypothesis,” which states that every
singularity in our universe, other than the Big Bang, is hidden behind an event horizon. Therefore if green slime spontaneously pops out of one, there is limited impact on the predictive ability of
physics, since the slime can never have any causal effect on the outside world. A singularity that is not modestly cloaked behind an event horizon is referred to as a naked singularity. Nobody has
yet been able to prove the cosmic censorship hypothesis.
The advent of high-precision cosmology
We expect that if there is matter in the universe, it should have gravitational fields, and in the rubber-sheet analogy this should be represented as a curvature of the sheet. Instead of a flat
sheet, we can have a spherical balloon, so that cosmological expansion is like inflating it with more and more air. It is also possible to have negative curvature, as in figure e on p. 797. All three
of these are valid, possible cosmologies according to relativity. The positive-curvature type happens if the average density of matter in the universe is above a certain critical level, the
negative-curvature one if the density is below that value.
density. Historically, it has been very difficult to do this, even to within an order of magnitude. Most of the matter in the universe probably doesn't emit light, making it difficult to detect.
Astronomical distance scales are also very poorly calibrated against absolute units such as the SI.
Instead, we measure the universe's curvature, and infer the density of matter from that. It turns out that we can do this by observing the cosmic microwave background (CMB) radiation, which is the
light left over from the brightly glowing early universe, which was dense and hot. As the universe has expanded, light waves that were in flight have expanded their wavelengths along with it. This
afterglow of the big bang was originally visible light, but after billions of years of expansion it has shifted into the microwave radio part of the electromagnetic spectrum. The CMB is not perfectly
uniform, and this turns out to give us a way to measure the universe's curvature. Since the CMB was emitted when the universe was only about 400,000 years old, any vibrations or disturbances in the
hot hydrogen and helium gas that filled space in that era would only have had time to travel a certain distance, limited by the speed of sound. We therefore expect that no feature in the CMB should
be bigger than a certain known size. In a universe with negative spatial curvature, the sum of the interior angles of a triangle is less than the Euclidean value of 180 degrees. Therefore if we
observe a variation in the CMB over some angle, the distance between two points on the sky is actually greater than would have been inferred from Euclidean geometry. The opposite happens if the
curvature is positive.
This observation was done by the 1989-1993 COBE probe, and its 2001-2009 successor, the Wilkinson Microwave Anisotropy Probe. The result is that the angular sizes are almost exactly equal to what
they should be according to Euclidean geometry. We therefore infer that the universe is very close to having zero average spatial curvature on the cosmological scale, and this tells us that its
average density must be within about 0.5% of the critical value. The years since COBE and WMAP mark the advent of an era in which cosmology has gone from being a field of estimates and rough guesses
to a high-precision science.
If one is inclined to be skeptical about the seemingly precise answers to the mysteries of the cosmos, there are consistency checks that can be carried out. In the bad old days of low-precision
cosmology, estimates of the age of the universe ranged from 10 billion to 20 billion years, and the low end was inconsistent with the age of the oldest star clusters. This was believed to be a
problem either for observational cosmology or for the astrophysical models used to estimate the ages of the clusters: “You can't be older than your ma.” Current data have shown that the low estimates
of the age were incorrect, so consistency is restored. (The best figure for the age of the universe is currently `13.8+-0.1` billion years.)
Dark energy and dark matter
Not everything works out so smoothly, however. One surprise, discussed in section 10.6, is that the universe's expansion is not currently slowing down, as had been expected due to the gravitational
attraction of all the matter in it. Instead, it is currently speeding up. This is attributed to a variable in Einstein's equations, long assumed to be zero, which represents a universal gravitational
repulsion of space itself, occurring even when there is no matter present. The current name for this is “dark energy,” although the fancy name is just a label for our ignorance about what causes it.
Another surprise comes from attempts to model the formation of the elements during the era shortly after the Big Bang, before the formation of the first stars (section 26.4.10). The observed relative
abundances of hydrogen, helium, and deuterium (^2H) cannot be reconciled with the density of low-velocity matter inferred from the observational data. If the inferred mass density were entirely due
to normal matter (i.e., matter whose mass consisted mostly of protons and neutrons), then nuclear reactions in the dense early universe should have proceeded relatively efficiently, leading to a much
higher ratio of helium to hydrogen, and a much lower abundance of deuterium. The conclusion is that most of the matter in the universe must be made of an unknown type of exotic matter, known as “dark
matter.” We are in the ironic position of knowing that precisely 96% of the universe is something other than atoms, but knowing nothing about what that something is. As of 2013, there have been
several experiments that have been carried out to attempt the direct detection of dark matter particles. These are carried out at the bottom of mineshafts to eliminate background radiation. Early
claims of success appear to have been statistical flukes, and the most sensitive experiments have not detected anything.^2
Homework Problems
`sqrt` A computerized answer check is available online.
`int` A problem that requires calculus.
`***` A difficult problem.
1. Prove, as claimed in the caption of figure a on p. 795, that `S-180°=4(s-180°)`, where `S` is the sum of the angles of the large equilateral triangle and `s` is the corresponding sum for one of
the four small ones.
2. If a two-dimensional being lived on the surface of a cone, would it say that its space was curved, or not?
3. (a) Verify that the equation `1-gh"/"c^2` for the gravitational Doppler shift and gravitational time dilation has units that make sense. (b) Does this equation satisfy the correspondence
. (a) Calculate the Doppler shift to be expected in the Pound-Rebka experiment described on p. 800. (b) In the 1978 Iijima mountain-valley experiment (p. 652), analysis was complicated by the clock's
sensitivity to pressure, humidity, and temperature. A cleaner version of the experiment was done in 2005 by hobbyist Tom Van Baak. He put his kids and three of his atomic clocks in a minivan and
drove from Bellevue, Washington to a lodge on Mount Rainier, 1340 meters higher in elevation. At home, he compared the clocks to others that had stayed at his house. Verify that the effect shown in
the graph is as predicted by general relativity.
5. The International Space Station orbits at an altitude of about 350 km and a speed of about 8000 m/s relative to the ground. Compare the gravitational and kinematic time dilations. Over all, does
time run faster on the ISS than on the ground, or more slowly?
6. Section 27.3 presented a Newtonian estimate of how compact an object would have to be in order to be a black hole. Although this estimate is not really right, it turns out to give the right answer
to within about a factor of 2. To roughly what size would the earth have to be compressed in order to become a black hole?
7. Clock A sits on a desk. Clock B is tossed up in the air from the same height as the desk and then comes back down. Compare the elapsed times. (hint:tossed-clock)
8. The angular defect `d` of a triangle (measured in radians) is defined as `s-pi`, where `s` is the sum of the interior angles. The angular defect is proportional to the area `A` of the triangle.
Consider the geometry measured by a two-dimensional being who lives on the surface of a sphere of radius `R`. First find some triangle on the sphere whose area and angular defect are easy to
calculate. Then determine the general equation for `d` in terms of `A` and `R`. `sqrt`
Exercise 27: Misconceptions about relativity
The following is a list of common misconceptions about relativity. The class will be split up into random groups, and each group will cooperate on developing an explanation of the misconception, and
then the groups will present their explanations to the class. There may be multiple rounds, with students assigned to different randomly chosen groups in successive rounds.
1. How can light have momentum if it has zero mass?
2. What does the world look like in a frame of reference moving at `c`?
3. Alice observes Betty coming toward her from the left at `c"/"2`, and Carol from the right at `c"/"2`. Therefore Betty is moving at the speed of light relative to Carol.
4. Are relativistic effects such as length contraction and time dilation real, or do they just seem to be that way?
5. Special relativity only matters if you've moving close to the speed of light.
6. Special relativity says that everything is relative.
7. There is a common misconception that relativistic length contraction is what we would actually see. Refute this by drawing a spacetime diagram for an object approaching an observer, and tracing
rays of light emitted from the object's front and back that both reach the observer's eye at the same time.
8. When you travel close to the speed of light, your time slows down.
9. Is a light wave's wavelength relativistically length contracted by a factor of gamma?
10. Accelerate a baseball to ultrarelativistic speeds. Does it become a black hole?
11. Where did the Big Bang happen?
12. The universe can't be infinite in size, because it's only had a finite amount of time to expand from the point where the Big Bang happened.
27.4 Cosmology by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license. | {"url":"https://www.vcalc.com/collection/?uuid=1eda9e43-f145-11e9-8682-bc764e2038f2","timestamp":"2024-11-07T00:02:58Z","content_type":"text/html","content_length":"61463","record_id":"<urn:uuid:a8408978-9023-48de-8248-2ff39b80ae2f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00404.warc.gz"} |
Linear Equations in Two Variables Class 9 Notes Maths Chapter 8
CBSE Class 9 Maths Notes Chapter 8 Linear Equations in Two Variables Pdf free download is part of Class 9 Maths Notes for Quick Revision. Here we have given NCERT Class 9 Maths Notes Chapter 8 Linear
Equations in Two Variables.
CBSE Class 9 Maths Notes Chapter 8 Linear Equations in Two Variables
1. Linear Equations: Any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero is called a linear equation in two variables.
The solution of a linear equation is not affected when
□ the same number is added to (or subtracted from) both the sides of the equation.
□ you multiply or divide both the sides of the equation by the same non-zero number.
2. A linear equation in two variables has infinitely many solutions.
3. The graph of every linear equation in two variables is a straight line.
4. x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.
5. The graph of x = a is a straight line parallel to the y-axis.
6. The graph of y = a is a straight line parallel to the x-axis.
7. An equation of the type y = mx represents a line passing through the origin.
8. Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the linear
9. Graph of a Linear Equation in Two Variables: We know that a linear equation in two variables has infinitely many solutions. We write the solutions as a pair of values and plot, these points on a
graph paper and join them to get a line.
NCERT Notes for Class 9 Maths
We hope the given CBSE Class 9 Maths Notes Chapter 8 Linear Equations in Two Variables Pdf free download will help you. If you have any query regarding NCERT Class 9 Maths Notes Chapter 8 Linear
Equations in Two Variables, drop a comment below and we will get back to you at the earliest. | {"url":"https://www.learncbse.in/linear-equations-in-two-variables-class-9-notes/","timestamp":"2024-11-13T12:23:42Z","content_type":"text/html","content_length":"141791","record_id":"<urn:uuid:c1232a84-11b7-4d15-aa84-2d78f1e436c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00762.warc.gz"} |
Your home loan calculator might be establishing you up for a shock. Buying a property?
Buying home is exciting. It is additionally perhaps one of the most essential decisions that are financial make. Selecting a home loan to fund your brand-new house is simply as crucial as seeking the
right house.
The right is had by you to regulate the method. Have a look at our other blog sites on homebuying subjects, and join the conversation on Facebook and Twitter making use of #ShopMortgage.
Secrets are fun — in movies, publications, and TV, that is. Secrets involving finances? Not really much. If you’re considering purchasing a true house, finding out just how much you really can
afford to pay may feel just like re re solving a puzzle.
Many individuals move to mortgage calculators to resolve that secret. Home financing calculator translates a property cost or loan quantity in to the matching payment that is monthly. While home
financing calculator may be a tool that is great crunch some complicated figures and obtain a ballpark estimate of one’s payment per month, many calculators won’t provide a whole image of all of
the costs. That’s why you may be establishing your self up for a shock in the event that you just depend on a home loan calculator without making your adjustments that are own.
Buying a home?
Subscribe to our 2-week Get Homebuyer boot camp that is ready. We’ll take you step-by-step through the whole homebuying procedure.
Exactly just How a home loan calculator works
A home loan is that loan which allows one to borrow funds buying a house and spend the loan back in monthly obligations. The mathematical formula for calculating the monthly obligations for a offered
home mortgage quantity is pretty complicated. That is where a home loan calculator will come in. The math is done by a mortgage calculator for you personally.
Home loan calculators are superb for quickly finding out of the payment that is monthly a specific house cost or loan amount — there’s no need certainly to attempt to perform some mathematics by
hand. But there’s two difficulties with home loan calculators.
Problem 1: numerous home loan calculators just determine the main and interest re re payment.
Principal could be the quantity you borrowed and need to pay straight back, and interest is exactly what the lending company prices for lending you the cash. Principal and interest make up the almost
all a mortgage payment that is monthly.
But, major and interest aren’t the only expenses you’ll pay each thirty days.
You can afford to spend on a home, you may be significantly underestimating how much you’ll have to pay each month if you’re using a mortgage calculator to decide how much. That’s a surprise
you don’t desire.
To ensure you’re generating decisions utilizing the right figures, do your research to learn just how much you are likely to spend every month for homeowner’s insurance coverage, home taxes, and
home loan insurance coverage. Include those month-to-month quantities to your principal and interest re re payment from your own home loan calculator to learn simply how much you are likely to pay
money for your total payment per month.
If you’re considering buying an apartment or a house in a residential area having a homeowner’s association (HOA), you’ll want to calculate and add condo/HOA dues, also. Although month-to-month
condo or HOA dues are compensated individually from your own month-to-month mortgage repayment, these are typically element of your general housing that is monthly. These dues can differ commonly and
impact the house cost you really can afford. As an example, a $200,000 condo with lots of amenities and $500 condo that is monthly might have the exact same general month-to-month price as a $300,000
single-family house with no condo or HOA dues.
How will you calculate these other expenses?
If you’re simply getting started off with your homebuying procedure, all you have to for the time being is really a rough estimate to assist you figure out how much you really can afford to fund a
property. While you move ahead and gather extra information, you’ll be in a position to make more accurate quotes.
Problem 2: Mortgage calculators are just just like the information you let them have.
A home loan calculator makes use of your inputs and a typical formula to determine a payment that is monthly. Some calculators earn some presumptions for your needs, while some allow you to control
most of the inputs. The main element facets that determine the principal that is monthly interest re re payment will be the loan quantity, the size of the loan (referred to as loan term),
additionally the interest.
selecting an interest that is realistic to make use of with a home loan calculator is crucial. The attention price makes a huge difference between your home loan repayments. For instance, a $200,000,
30-year, fixed-rate loan at four % interest has a month-to-month principal and interest re re re payment of $955. The loan that is same five % interest includes a payment per month of $1,074.
The attention prices that loan providers promote on the net are definitely not the prices you shall be capable of getting. Promoted prices usually assume you have actually a exemplary credit history
and can produce a deposit with a minimum of 20 %.
Utilize our device to explore the factors that are different impact the interest lenders are able to provide you with and acquire a feeling of the product range of prices you could expect. Ensure you
use a interest that is realistic in the home loan calculator so that you obtain a good estimate regarding the month-to-month principal and interest payment.
Three types of expenses
Many home loan calculators concentrate only regarding the month-to-month principal and interest re re payment. Discover the 3 different types of expenses you’ll pay whenever purchasing a property. | {"url":"http://kanzlei-heindl.com/2020/12/31/your-home-loan-calculator-might-be-establishing-6/","timestamp":"2024-11-07T01:18:00Z","content_type":"application/xhtml+xml","content_length":"28521","record_id":"<urn:uuid:cd46ae7a-0c40-4490-9858-4d9874d17c93>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00567.warc.gz"} |
vector position
Finding vector positions
To find any vector position from a starting point
Use any vector
Example 1
If you are shown a diagram as:
Think of it in terms of separate vectors.
Example 2
If you are shown a diagram as:
Think of it in terms of separate vectors.
More Info | {"url":"https://mammothmemory.net/maths/pythagoras-and-trigonometry/vectors/finding-vector-positions.html","timestamp":"2024-11-12T07:08:18Z","content_type":"text/html","content_length":"37506","record_id":"<urn:uuid:5e944743-432d-4cfd-b8a5-14682714d450>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00793.warc.gz"} |
asters i
Launched in 2016, MITx MicroMasters programs open up graduate-level MIT courses MicroMasters offers five fantastic programs: ~Statistics and Data Science
Elever med MITx MicroMasters legitimation i SCM kan sedan ansöka om MIT för University of the Pacific: s MS in Data Science-program ger kandidaterna den
The program comprises four online courses and a virtually proctored exam EdX, the platform used for MIT and UC San Diego MicroMasters, also has an impressive number of free programs including
Microsoft's Introduction to Data Science and Data Science and Machine Learning Essentials. They have many certificate programs for very reasonable fees, starting at $25 for a Microsoft certificate.
Credits: The Big Data MicroMasters program certificate represents 25% or 12 units of the Master of Data Science program, which requires a total of 48 units of coursework to complete. Who should take
that program: People who already have a sound understanding of data science and are familiar with at least one programming language on an intermediate level and want to have an introduction into the
Big Data area. In my opinion: A Micro-Masters or any Certification is useful only if you have a job and you want to keep your hands warm with programming, data processing and the concepts of Data
Science. MIT’s Department of Economics and the Abdul Latif Jameel Poverty Action Lab (J-PAL) designed the MicroMasters® program credential in Data, Economics, and Development Policy (DEDP) to equip
learners with the practical skills and theoretical knowledge to tackle some of the most pressing challenges facing developing countries and the world’s poor.
Since it's not officially a masters program, I wonder … Successful graduates of the MIT Micromasters in Statistics and Data Science program have demonstrated proficiency with the fundamental elements
of data Master the skills needed to solve complex challenges with data, from probability and statistics to data analysis and machine learning. This program consists of University's Master of Data
Science accepts students with MIT MicroMasters credentials Institute of Technology (MIT) as RMIT will enable students who are MicroMasters programs are a series of graduate level courses designed to
advance your The Data Science MicroMasters Program provides a professional, Now, Tsinghua is the first pathway university for MIT MicroMasters program in of Science in Engineering degree, Data
Science and Information Technology, expertise demand through the MIT MicroMasters program in Statistics and Data Science as a concentration in the MCPHS MBA in Healthcare Management. Recently, the
MITx MicroMasters Track in Statistics and Data Science held its that is) MIT OpenCourseWare website are missing written notes or homework Statistics and Data Science MicroMasters — MIT @ edX; CS109
Data Science — Harvard; Python for Data Science and Machine Learning Bootcamp — Udemy To enroll in the MicroMasters track or to learn more about this program and how it integrates with MIT's new
blended Master's degree, go to MITx's MicroMasters This may include courses on general statistical theory or machine learning, program with courses originating from different academic departments at
MIT. MicroMasters programs are a series of online graduate level courses offered by universities In its early stage MIT offered the MicroMasters as a pilot within its supply chain management program,
consulting industry leaders. Big Da degree for students from MIT's Data, Economics, and Development Policy ( DEDP), Statistics and Data Science (SDS) and Finance MicroMasters programs. Feb 25, 2020
of taking the Micromasters Program in Statistics and Data Science as my to up scale into data analytics (https://micromasters.mit.edu/ds/). New from MIT Anthropology - Qualitative Research Methods:
Data Coding and day to register for the Statistics and Data Science MITx MicroMasters Program?
Statistics and Data Science MicroMasters is a series of graduate level courses that MIT provides through edX platform. The program comprises four online courses and a virtually proctored exam
Got certified from MITx on Statistics and Data Science MicroMasters program, which includes: Four graduate level courses on Statistics, Probability, Data Analysis and Machine Learning delivered by
MIT professors. From probability and statistics to data analysis and machine learning, master the skills needed to solve complex challenges with data. Find out more and enro This MicroMasters program
encompasses two sides of data science learning: the mathematical and the applied. Mathematical courses cover probability, statistics, and machine learning.
Department of Geological Sciences - University of Texas at . Although precise effectiveness estimates depend on the assumed data and parameters, there are clear trends M.I.T. 18.03 Ordinary Di
erential Equations MICROMASTER 430 Parameter List 6SE6400-5AF00-0BP0 5 Parameters MICROMASTER 430 This
The IDSS mission to advance data science education through the MicroMasters Program led to a pilot collaboration with Aporta, a social impact group developing Peru’s next generation of data
scientists .
Learn data science methods and tools, get hands-on training in data analysis and machine learning, and find opportunities in a growing field. Watch our latest informational webinar. Not only is there
a huge demand, but there is a significant shortage of qualified data scientists with 39% of the most rigorous data science positions requiring a degree higher than a bachelor’s. This MicroMasters®
program in Statistics and Data Science (SDS) was developed by MITx and the MIT Institute for Data, Systems, and Society (IDSS) . The MicroMasters Program in Statistics and Data Science credential
enables learners to receive academic credit to universities around the world, making the credential a pathway to a master’s degree. The amount of credit, and the conditions for receiving it, depends
upon each institution which are all listed below.
25 ölbryggerier i prag
MITx MicroMasters® Program in Statistics and Data Science | Learner Testimonials. Watch later. Statistics and Data Science MicroMasters Program from MIT. This program consists of a total of five
master’s level courses in order to learn the foundations of machine learning, data science and statistics.
A new elective course in the MITx MicroMasters Program in Statistics and Data Science (SDS) offers an increased focus on applying data science to complex, real-world problems. Data Analysis:
Statistical Modeling and Computation in Applications launches in Spring 2021, and is open for enrollment now.
Sista ansokningsdag gymnasiet 2021
rosa farg betydelseanmälan om svenskt medborgarskap för barnstadsbiblioteket tranaspeter harjungpuberteten vad händer i kroppenstarbreeze mikael nermark
MIT Faculty Director, MicroMasters, Statistics and Data Science Karene Chu received her Ph.D. in mathematics from the University of Toronto in 2012. Since then she has been a postdoctoral fellow
first at the University of Toronto and the Fields Institute, and then at MIT, with research focus on knot theory and quantum invariants.
Employers recognize it for providing deep learning in specific career fields. MicroMasters in Statistics and Data Science. Learn data science methods and tools, get hands-on training in data analysis
and machine learning, and find opportunities in a growing field. Watch our latest informational webinar. The Statistics and Data Science Center is an MIT-wide focal point for advancing research and
education programs related to The MITx MicroMasters Program in Statistics and Data Science will help online learners develop their skills in the booming field of data science. The program offers
learners an MIT-quality, professional credential, while also providing an academic pathway to pursue a PhD at MIT or a master’s degree elsewhere. | {"url":"https://hurmanblirrikjedz.web.app/81758/97123.html","timestamp":"2024-11-11T14:24:50Z","content_type":"text/html","content_length":"13219","record_id":"<urn:uuid:06bde5b4-2e53-4970-804f-aa5de1f910f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00364.warc.gz"} |
What is Math?
I provide detailed explanations in math and related fields.
• Added on December 05 2023
How to use Math?
• Step 1 ： Click the open gpts about Math button above, or the link below.
• Step 2 ： Follow some prompt about Math words that pop up, and then operate.
• Step 3 ： You can feed some about Math data to better serve your project.
• Step 4 ： Finally retrieve similar questions and answers based on the provided content.
FAQ from Math?
Arithmetic mean, or average, is a calculated central value of a set of numbers. It is computed by adding up the given numbers and dividing the sum by the number of values in the set.
A prime number is a whole number which is greater than one and can only be divided evenly by itself and one. Prime numbers are those that are not divisible by any other number.
The basic formula for finding the area of a triangle is A = 0.5 x b x h, where b is the base and h is the height of the triangle. The base and the height must be in the same unit of measurement. | {"url":"http://gpts.la/gpts-store/math","timestamp":"2024-11-04T14:07:29Z","content_type":"text/html","content_length":"81162","record_id":"<urn:uuid:88b76745-4d75-4570-a80d-d40998fb83ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00002.warc.gz"} |
Filling a curve with the number of objects with Blender Geometry Nodes
Let’s fill the curve with the required number of objects so that they occupy the entire length of the curve without gaps, adjusting their scale if necessary.
Open an empty blend-file. Add a curve (shift + a – Curve – Bezier). Append the Geometry Nodes modifier to it and initialize the initial node tree.
Let’s fill the curve, for example, with spheres.
Add an Instance On Points (shift + a – Instance – Instance On Points) node to the main branch of the node tree – this node is responsible for placing the mesh at each point of the curve.
Add a UV Sphere node (shift + a – Mesh Primitives – UV Sphere) and link its Mesh output with the Instance input of the Instance On Points node. Thus, at each point of the curve, we placed a UV Sphere
Our curve consists of only two points. Let’s increase the number of points on the curve using the Resample Curve node (shift + a – Curve – Resample Curve), adding it to the very beginning of the main
branch of the node tree.
Now there are 10 points on our curve, and in each of them there is a UV Sphere mesh instance.
Let’s make the scale of the sphere independently adjusts to the number of points on the curve so that the spheres do not intersect.
To get the scaling factor for the spheres, we need to divide the length of the curve by the number of its points. This will give us the distance between two points on the curve – the desired size of
the mesh.
Separate the number of points on the curve to the single node. Add an Integer node (shift + a – Input – Integer), set its value to 10, and link its Integer output with the Count input of the Resample
Curve node.
We can get the curve length using the Curve Length node (shift + a – Curve – Curve Length). Add it to the node tree. Link its Curve input with the Geometry output of the Geometry Output node. From
its Length output, we can now take the length of our curve.
Add a Math node (shift + a – Utilities – Math) and switch it to the Divide mode. Link its upper Value input with the Length output of the Curve Length node, and its lower input with the Integer
output of the Integer node. Thus, we divide the curve length by the number of its points.
Link the Value output of the Math node with the Scale input of the Instance On Points node, to set the scale coefficient depending on curve length and number of its points.
As we can see, the spheres have decreased, but clearly not enough.
Since the size of the sphere is given by the radius, and when filling the curve, the full sphere – its diameter is used, the scaling factor that we got must be divided by 2.
Add another Math node (shift + a – Utilities – Math) in Divide mode. Insert it between the existing Divide node and the Instance On Points node. The value of the lower Value input, which remains
free, set to 2.
The spheres are now scaled beauty to fill the entire curve. By changing the value in the Integer node, we can easily control the number of spheres.
However, with a few spheres number, some gaps appears. The artifact appears due to the fact that we filled the curve based on the number of points on it, but it is necessary to fill in the gaps
between the points. Since the spheres placed on two extreme points on the curve will half crawl out.
The number of gaps between points is always one less than the number of points. Therefore, to achieve final accuracy, we just need to subtract 1 from the number of points when calculating the scale
factor for spheres.
Add another Math node (shift + a – Utilities – Math) and switch it to the Subtract mode. Place it between the Integer node and the first Divide node.
Set to 1 the value of the lower Value input.
The spheres are now scaled exactly along the length of the curve.
It should be noted that for closed curves, the last action is not required. This happens because in a closed curve, the last point always coincides with the first.
For closed curves, the subtraction node can be “muted” – excluded from the calculation, by selecting it and pressing the “m” key.
0 Comment
Inline Feedbacks
View all comments | {"url":"https://b3d.interplanety.org/en/filling-a-curve-with-the-number-of-objects-with-blender-geometry-nodes/","timestamp":"2024-11-07T09:09:33Z","content_type":"text/html","content_length":"217969","record_id":"<urn:uuid:a6deb9f0-44d3-41dd-8b16-8b2c9d42f552>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00408.warc.gz"} |
Sensors and Regional Gradient Observability of Hyperbolic Systems
Sensors and Regional Gradient Observability of Hyperbolic Systems ()
1. Introduction
For a distributed parameter system evolving on a spatial domain
In this paper we present an extension of the above results on regional gradient observability to hyperbolic systems evolving on a spatial domain
Here, we consider the problem of regional gradient observability of hyperbolic systems and we establish condition that allows the reconstruction of the initial gradient on such a subregion. And the
paper is organized as follows.
The second section is devoted to definitions and characterizations of this notion for hyperbolic systems. In the third section we establish a relation between regional gradient observability and
sensors structure. The fourth section is focused on regional reconstruction of the initial gradient. In the last section we give a numerical approach, extending the Hilbert Uniqueness Method
developed by J.L. Lions [5], and illustrations with efficient simulations.
2. Regional Gradient Observability
Consider the system described by the hyperbolic equation
Equation (1) has a unique solution
Suppose that measurements on system (1) are given by an output function:
Let us recall that a sensor is defined by a couple
Let us define the observability operator
which is linear and bounded with its adjoint denoted by
while their adjoints are denoted by
2.1. Definition 2.1
The system (1) together with the output (2) is said to be exactly (resp. approximately) gradient observable if
Such a system will be said exactly (resp. approximately) G-observable.
For a positive Lebesgue measure subset
while their adjoints, denoted by
We finally introduce the operator
2.2. Definition 2.2
1) The system (1) together with the output Equation (2) is said to be exactly regionally gradient observable or exactly G-observable on
2) The system (1) together with the output equation (2) is said to be approximately regionally gradient observable or approximately G-observable on
The notion of regional G-observability on
2.3. Proposition 2.3
1) The system (1) together with the output Equation (2) is exactly G-observable on
a) For all
2) The system (1) together with the output Equation (2) is approximately G-observable on
2.4. Proof
1) a) let us consider the operator
Since the system is exactly G-observable on
b) Let
since the system (1) is exactly G-observable on
Let put
Conversely, let
such that
2) Let
Conversely, let
2.5. Remark 2.4
1) If a system is exactly (resp. approximately) G-observable on
2) There exist systems which are not G-observable on the whole domain but may be G-observable on some subregion.
2.6. Example 2.5
The operator
Measurements are given by the output function
Let the subregion
Then the initial state gradient to be observed is
We have the result.
2.7. Proposition 2.6
The gradient
2.8. Proof
To prove that
we have
This gives
On the other hand
Indeed, suppose that
Since for
but for
witch gives,
But for
3. Gradient Strategic Sensors
The purpose of this section is to establish a link between regional gradient observability and the sensors structure.
Let us consider the system (1) observed by
3.1. Definition 3.1
A sensor
We assume that the operator
3.2. Proposition 3.2
If the sequence of sensors
is the row vector the elements of which are
3.3. Proof
The proof is developed in the case zone sensors.
The sequence of sensors
Suppose that the sequence of sensors
and let
assume that
Integrating on
but we have
Using the fact that
then we obtain
from (5), (6) and (7) we obtain
this gives
3.4. Remark 3.3
1) The above proposition implies that the required number of sensors is greater than or equal to the largest multiplicity of eigenvalues.
2) By infinitesimally deforming of the domain, the multiplicity of the eigenvalues can be reduced to one [9,10]. Consequently, the regional G-observability on the subregion
4. Regional Gradient Reconstruction
In this section, we give an approach which allows the reconstruction of the initial state gradient on
has a unique solution
We consider the zone sensor case where the system (1) is observed by the output function
The reverse system given by
has a unique solution
We denote the solution
Let consider the operator
and consider the retrograde system which has a unique solution
We denote the solution
4.1. Proposition 4.1
If the sensor
4.2. Proof
1) Let us show first that if the system (1) is G-observable, then (10) defines a norm on
Consider a basis
The set
and since the sensor
and from
2) Let denote by
for the first term, we obtain
Using Green formula for the second term, we obtain
and with the boundary conditions, we obtain
Using Cauchy-Schwartz inequality, we have,
which proves that
4.3. Remark 4.2
The previous approach can be established with similar techniques when the output is defined by means of internal or boundary pointwise sensors.
5. Numerical Approach
In this section we give a numerical approach which leads to explicit formulas for
5.1. Proposition 5.1
If the sensor
5.2. Proof
In the previous section, it has been seen that the regional reconstruction of the initial state gradient on
And solving Equation (13) turns up to minimize
After development and when
On the other hand, we have
we obtain
The minimization of (13) is equivalent to solve the two following problems
which solutions are,
Now, let
then, we obtain
With these developments, according to (18) and (19), we obtain.
We replace that in the relation (16) and (17), we obtain
We consider a truncation up to order
We define a final error
The good choice of
Step 1: Data: The region
Step 2: Choose a low truncation order
Step 3: Computation of
Step 4: If
Step 5:
5.3. Remark 5.2
6. Simulations
6.1. Example
In this section we develop a numerical example that leads to results related to the choice of the subregion, the sensor location and the initial state gradient.
Measurements are given by the output function
The previous system is G-observable on
We denote that numerically an irrational number does not exist but it can be considered as irrational if truncation number exceeds the desired precision.
The coefficients
Applying the previous algorithm, using the formulae (20), (21) we respectively obtain the Figures 1 and 2 for
The estimated gradient is obtained with error
6.2. Simulating Conjectures
Now we show numerically how the error grows with respect to the subregion area. It means that the larger the region is, the greater the error is. The obtained results are presented in Table 1.
Figure 1. Iheb2.eps: initial state gradient Ñy^0 (continuous line) and estimate initial state gradient bar (Ñ)y^0 (dashed line).
Figure 2. Nouha2.eps: initial speed gradient Ñy^1 (continuous line) and estimate initial speed gradient bar (Ñ)y^1 (dashed line).
Figure 3. Iheb1.eps: initial state gradient Ñy^0 (continuous line) and estimate initial state gradient bar (Ñ)y^0 (dashed line).
Figure 4. Nouha1.eps: initial speed gradient Ñy^1 (continuous line) and estimate initial speed gradient bar (Ñ)y^1 (dashed line).
Table 1. Evolution error with respect to the area of the subregion.
Table 2. Evolution error with respect to the initial state gradient amplitude.
Also how both the error decreases with respect to the amplitude Table 2.
7. Conclusion
Gradient Observability on a subregion interior to the spatial evolution domain of hyperbolic system is considered. A relation between this notion and the sensors structure is established and
numerical approach for its reconstruction is given. This allows the computation of the initial state gradient without the knowledge of the system state. Illustrations by numerical simulations show
the efficiency of the approach. Interesting questions remain open, the case where the subregion | {"url":"https://scirp.org/journal/paperinformation?paperid=17578","timestamp":"2024-11-06T08:19:16Z","content_type":"application/xhtml+xml","content_length":"159610","record_id":"<urn:uuid:904e8d8f-8073-454e-803f-24ac14bea458>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00445.warc.gz"} |
Discussion on Project Euler #227: The Chase Challenge
• Imagine a circle and person1 and person2 who have the dice and are sitting exactly opposite to each other on the left and write side of the circle. The probability of person1 rolling 1 and
person2 rolling m, is equal to the probability of person1 rolling m and person2 rolling 1. That means the probability of each moving one step in the upper half of the circle (and therefore
reducing the distance by two units) is equal to the probability of them doing the same thing in the lower half. And the same is true for other types of move (no passing and one to the right or
left). Imho, the average is infinite. | {"url":"https://www.hackerrank.com/contests/projecteuler/challenges/euler227/forum/comments/561221","timestamp":"2024-11-03T04:38:01Z","content_type":"text/html","content_length":"786248","record_id":"<urn:uuid:f6fba2ce-19bd-457e-a4b0-0cec065f5930>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00742.warc.gz"} |
7.4 Conservative Forces and Potential Energy
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
• Define conservative force, potential energy, and mechanical energy
• Explain the potential energy of a spring in terms of its compression when Hooke’s law applies
• Use the work-energy theorem to show how having only conservative forces leads to conservation of mechanical energy
The information presented in this section supports the following AP® learning objectives and science practices:
• 4.C.1.1 The student is able to calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the
total energy. (S.P. 1.4, 2.1, 2.2)
• 4.C.2.1 The student is able to make predictions about the changes in the mechanical energy of a system when a component of an external force acts parallel or antiparallel to the direction of the
displacement of the center of mass. (S.P. 6.4)
• 5.B.1.1 The student is able to set up a representation or model showing that a single object can only have kinetic energy and use information about that object to calculate its kinetic energy.
(S.P. 1.4, 2.2)
• 5.B.1.2 The student is able to translate between a representation of a single object, which can only have kinetic energy, and a system that includes the object, which may have both kinetic and
potential energies. (S.P. 1.5)
• 5.B.3.1 The student is able to describe and make qualitative and/or quantitative predictions about everyday examples of systems with internal potential energy. (S.P. 2.2, 6.4, 7.2)
• 5.B.3.2 The student is able to make quantitative calculations of the internal potential energy of a system from a description or diagram of that system. (S.P. 1.4, 2.2)
• 5.B.3.3 The student is able to apply mathematical reasoning to create a description of the internal potential energy of a system from a description or diagram of the objects and interactions in
that system. (S.P. 1.4, 2.2)
Potential Energy and Conservative Forces
Potential Energy and Conservative Forces
Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force is one, like the gravitational force, for which work done by or against it depends only on
the starting and ending points of a motion and not on the path taken. We can define a potential energy $(PE)(PE) size 12{ \( "PE" \) } {}$ for any conservative force, just as we did for the
gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in the spring. We treat these springs as ideal, in
that we assume there is no friction and no production of thermal energy. This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring.
Indeed, the reason that the spring has this characteristic is that its force is conservative. That is, a conservative force results in stored or potential energy. Gravitational potential energy is
one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.
Potential Energy and Conservative Forces
Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable.
A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken.
We can define a potential energy $(PE)(PE) size 12{ \( "PE" \) } {}$ for any conservative force. The work done against a conservative force to reach a final configuration depends on the
configuration, not the path followed, and is the potential energy added.
Real-World Connections: Energy of a Bowling Ball
How much energy does a bowling ball have? Just think about it for a minute.
If you are thinking that you need more information, you’re right. If we can measure the ball’s velocity, then determining its kinetic energy is simple. Note that this requires defining a reference
frame in which to measure the velocity. Determining the ball’s potential energy also requires more information. You need to know its height above the ground, which requires a reference frame to the
ground. Without the ground—in other words, Earth—the ball does not classically have potential energy. Potential energy comes from the interaction between the ball and the ground. Another way of
thinking about this is to compare the ball’s potential energy on Earth and on the Moon. A bowling ball a certain height above Earth is going to have more potential energy than the same bowling ball
the same height above the surface of the Moon, because Earth has greater mass than the Moon and therefore exerts more gravity on the ball. Thus, potential energy requires a system of at least two
objects, or an object with an internal structure of at least two parts.
Potential Energy of a Spring
Potential Energy of a Spring
First, let us obtain an expression for the potential energy stored in a spring ($PEsPEs size 12{"PE" rSub { size 8{s} } } {}$). We calculate the work done to stretch or compress a spring that obeys
Hooke’s law. Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force $FF size 12{F} {}$ on the spring and the resulting deformation $ΔLΔL size 12{ΔL} {}$ are
proportional, $F=kΔLF=kΔL size 12{F=kΔL} {}$ (see Figure 7.11). For our spring, we will replace $ΔLΔL$ (the amount of deformation produced by a force $FF$) by the distance $xx$ that the spring is
stretched or compressed along its length. So the force needed to stretch the spring has magnitude $F = kxF = kx size 12{ ital "F = kx"} {}$, where $kk size 12{k} {}$ is the spring’s force constant.
The force increases linearly from 0 at the start to $kxkx size 12{ ital "kx"} {}$ in the fully stretched position. The average force is $kx/2kx/2$. Thus, the work done in stretching or compressing
the spring is $Ws=Fd=kx2x=12kx2Ws=Fd=kx2x=12kx2 size 12{W rSub { size 8{s} } = ital "Fd"= left ( { { ital "kx"} over {2} } right )""x= { {1} over {2} } ital "kx" rSup { size 8{2} } } {}$.
Alternatively, we noted in Kinetic Energy and the Work-Energy Theorem that the area under a graph of $FF size 12{F} {}$ vs. $xx size 12{x} {}$ is the work done by the force. In Figure 7.11(c) we see
that this area is also $12kx212kx2 size 12{ { {1} over {2} } ital "kx" rSup { size 8{2} } } {}$. We therefore define the potential energy of a spring, $PEsPEs size 12{"PE" rSub { size 8{s} } } {}$,
to be
7.48 $PEs=12kx2,PEs=12kx2, size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {}$
where $kk size 12{k} {}$ is the spring’s force constant and $xx size 12{x} {}$ is the displacement from its undeformed position. The potential energy represents the work done on the spring and the
energy stored in it as a result of stretching or compressing it a distance $xx size 12{x} {}$. The potential energy of the spring $PEsPEs size 12{"PE" rSub { size 8{s} } } {}$ does not depend on the
path taken; it depends only on the stretch or squeeze $xx size 12{x} {}$ in the final configuration.
The equation $PEs=12kx2PEs=12kx2 size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {}$ has general validity beyond the special case for which it was derived. Potential
energy can be stored in any elastic medium by deforming it. Indeed, the general definition of potential energy is energy due to position, shape, or configuration. For shape or position deformations,
stored energy is $PEs=12kx2PEs=12kx2 size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {}$, where $kk size 12{k} {}$ is the force constant of the particular system and
$xx size 12{x} {}$ is its deformation. Another example is seen in Figure 7.12 for a guitar string.
Conservation of Mechanical Energy
Conservation of Mechanical Energy
Let us now consider what form the work-energy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that
the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is
7.49 $W net = 1 2 mv 2 − 1 2 mv 0 2 = Δ KE. W net = 1 2 mv 2 − 1 2 mv 0 2 = Δ KE. size 12{W rSub { size 8{"net"} } = { {1} over {2} } ital "mv" rSup { size 8{2} } - { {1} over {2} } ital "mv" rSub {
size 8{0} rSup { size 8{2} } } =Δ"KE" "." } {}$
If only conservative forces act, then
7.50 $Wnet=Wc,Wnet=Wc, size 12{W rSub { size 8{"net"} } =W rSub { size 8{c} } } {}$
where $WcWc$ is the total work done by all conservative forces. Thus,
7.51 $Wc=ΔKE.Wc=ΔKE. size 12{W rSub { size 8{c} } =Δ"KE"} {}$
Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, $Wc=−ΔPEWc=−ΔPE size 12{W rSub { size 8{c} } = +- D"PE"} {}$.
7.52 $− Δ PE = Δ KE − Δ PE = Δ KE size 12{ - Δ"PE"=Δ"KE"} {}$
7.53 $ΔKE+ΔPE=0.ΔKE+ΔPE=0. size 12{Δ"KE"+Δ"PE"=0} {}$
This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,
7.54 $KE + PE = constant or KE i + PE i = KE f + PE f } (conservative forces only), KE + PE = constant or KE i + PE i = KE f + PE f } (conservative forces only),$
where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy principle. Remember
that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy, $
(KE+PE)(KE+PE) size 12{ \( "KE"+"PE" \) } {}$. In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between
$KEKE size 12{"KE"} {}$ and the various types of $PEPE size 12{"PE"} {}$, with the total energy remaining constant.
The internal energy of a system is the sum of the kinetic energies of all of its elements, plus the potential energy of all interactions due to conservative forces between all of the elements.
Real-World Connections
Consider a wind-up toy, such as a toy car. It uses a spring system to store energy. The amount of energy stored depends only on how many times it is wound, not how quickly or slowly the winding
happens. Similarly, a dart gun using compressed air stores energy in its internal structure. In this case, the energy stored inside depends only on how many times it is pumped, not how quickly or
slowly the pumping is done. The total energy put into the system, whether through winding or pumping, is equal to the total energy conserved in the system, minus any energy loss in the system—such as
air leaks in the dart gun. Since the internal energy of the system is conserved, you can calculate the amount of stored energy by measuring the kinetic energy of the system, the moving car or dart,
when the potential energy is released.
Example 7.8 Using Conservation of Mechanical Energy to Calculate the Speed of a Toy Car
A 0.100-kg toy car is propelled by a compressed spring, as shown in Figure 7.13. The car follows a track that rises 0.180 m above the starting point. The spring is compressed 4.00 cm and has a force
constant of 250.0 N/m. Assuming work done by friction to be negligible, find (a) how fast the car is going before it starts up the slope and (b) how fast it is going at the top of the slope.
The spring force and the gravitational force are conservative forces, so conservation of mechanical energy can be used. Thus,
7.55 $KE i + PE i = KE f + PE f KE i + PE i = KE f + PE f size 12{"KE""" lSub { size 8{i} } +"PE" rSub { size 8{i} } ="KE" rSub { size 8{f} } +"PE" rSub { size 8{f} } } {}$
7.56 $12 mvi2 + mgh i +12 kxi2 =12 mvf2+ mgh f + 12 kxf2, 12 mvi2 + mgh i +12 kxi2 =12 mvf2+ mgh f + 12 kxf2,$
where $hh size 12{h} {}$ is the height (vertical position) and $xx size 12{x} {}$ is the compression of the spring. This general statement looks complex but becomes much simpler when we start
considering specific situations. First, we must identify the initial and final conditions in a problem; then, we enter them into the last equation to solve for an unknown.
Solution for (a)
This part of the problem is limited to conditions just before the car is released and just after it leaves the spring. Take the initial height to be zero, so that both $hihi size 12{h rSub { size 8
{i} } } {}$ and $hfhf size 12{h rSub { size 8{f} } } {}$ are zero. Furthermore, the initial speed $vivi size 12{v rSub { size 8{i} } } {}$ is zero and the final compression of the spring $xfxf size
12{x rSub { size 8{f} } } {}$ is zero, and so several terms in the conservation of mechanical energy equation are zero and it simplifies to
7.57 $1 2 kx i 2 = 1 2 mv f 2 . 1 2 kx i 2 = 1 2 mv f 2 .$
In other words, the initial potential energy in the spring is converted completely to kinetic energy in the absence of friction. Solving for the final speed and entering known values yields
7.58 v f = k m x i = 250 .0 N/m 0.100 kg ( 0.0400 m ) = 2.00 m/s. v f = k m x i = 250 .0 N/m 0.100 kg ( 0.0400 m ) = 2.00 m/s. alignl { stack { size 12{v rSub { size 8{f} } = sqrt { { {k} over {m} }
} x rSub { size 8{i} } } {} # " "= sqrt { { {"250" "." 0" N/m"} over {0 "." "100 kg"} } } \( 0 "." "0400"" m" \) {} # " "=2 "." "00"" m/s" "." {} } } {}
Solution for (b)
One method of finding the speed at the top of the slope is to consider conditions just before the car is released and just after it reaches the top of the slope, completely ignoring everything in
between. Doing the same type of analysis to find which terms are zero, the conservation of mechanical energy becomes
7.59 $12 kxi 2 =1 2 mvf 2 +mghf.12 kxi 2 =1 2 mvf 2 +mghf. size 12{ { {1} over {2} } ital "kx" rSub { size 8{i} rSup { size 8{2} } } = { {1} over {2} } ital "mv" rSub { size 8{f} rSup { size 8{2} } }
+ ital "mgh" rSub { size 8{f} } } {}$
This form of the equation means that the spring’s initial potential energy is converted partly to gravitational potential energy and partly to kinetic energy. The final speed at the top of the slope
will be less than at the bottom. Solving for $vfvf size 12{v rSub { size 8{f} } } {}$ and substituting known values gives
7.60 v f = kx i 2 m − 2 gh f = 250.0 N/m 0.100 kg ( 0.0400 m ) 2 − 2 ( 9.80 m/s 2 ) ( 0.180 m ) = 0.687 m/s. v f = kx i 2 m − 2 gh f = 250.0 N/m 0.100 kg ( 0.0400 m ) 2 − 2 ( 9.80 m/s 2 ) ( 0.180 m )
= 0.687 m/s. alignl { stack { size 12{v rSub { size 8{f} } = sqrt { { { ital "kx" rSub { size 8{i} rSup { size 8{2} } } } over {m} } - 2 ital "gh" rSub { size 8{f} } } } {} # " "= sqrt { left ( {
{"250" "." 0" N/m"} over {0 "." "100 kg"} } right )"" \( 0 "." "0400"" m" \) rSup { size 8{2} } - 2 \( 9 "." "80"" m/s" rSup { size 8{2} } \) \( 0 "." "180"" m" \) } {} # " "=0 "." "687"" m/s" "."
{} } } {}
Another way to solve this problem is to realize that the car’s kinetic energy before it goes up the slope is converted partly to potential energy—that is, to take the final conditions in part (a) to
be the initial conditions in part (b).
Applying the Science Practices: Potential Energy in a Spring
Suppose you are running an experiment in which two 250 g carts connected by a spring (with spring constant 120 N/m) are run into a solid block, and the compression of the spring is measured. In one
run of this experiment, the spring was measured to compress from its rest length of 5.0 cm to a minimum length of 2.0 cm. What was the potential energy stored in this system?
Note that the change in length of the spring is 3.0 cm. Hence we can apply Equation 7.42 to find that the potential energy is PE = (1/2)(120 N/m)(0.030 m)^2 = 0.0541 J.
Note that, for conservative forces, we do not directly calculate the work they do; rather, we consider their effects through their corresponding potential energies, just as we did in Example 7.8.
Note also that we do not consider details of the path taken—only the starting and ending points are important—as long as the path is not impossible. This assumption is usually a tremendous
simplification because the path may be complicated and forces may vary along the way. | {"url":"https://texasgateway.org/resource/74-conservative-forces-and-potential-energy?binder_id=78541","timestamp":"2024-11-07T19:59:12Z","content_type":"text/html","content_length":"122565","record_id":"<urn:uuid:4b5fe949-943d-4944-99d3-2773d6b4c0e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00840.warc.gz"} |
Data Sources
Curves are comprised of an X data vector and a Y data vector. The X and Y vectors can be read from a data file, defined as mathematical expressions, or entered as values. The X and Y vectors of a
curve do not have to come from the same source. For instance, the data source for the X vector of a curve can be an ASCII file and the source for the Y vector of the same curve can be defined by an
expression such as sqrt(x).
File as a Data Source
If you select File as the data source, use the file browser to select data files for the X and Y vectors. From the list view, select the subcases, types, requests, and components.
Figure 1.
Math as a Data Source
If Math is selected as the source, the Expression Builder is displayed, allowing you to define the vector mathematically.
Math as a source allows you to use additional curve vectors to plot a curve. In addition to the traditional X and Y vectors, you can use U and V vectors to perform operations on a curve at plotting
time. As a result, only one curve is generated in the session. This is helpful any time that you want to create a curve and immediately apply a function (for example, a filter) to that curve.
Figure 2.
Expression Builder
Curves can be defined mathematically using the Expression Builder.
The Expression Builder is displayed when Math is selected as the data source for a vector.
The Expression Builder contains the math functions necessary to build an expression. Use the in-app Entity Editor to select parameters as arguments for the selected Templex functions.
Direct Insert
Select a Templex function from the list and use Direct Insert to place it directly into the vector field.
Use In Expression
Select a function from the list and select Use in Expression to build a large expression, placing the function in the expression field.
You can evaluate and check the function values in a tabular form.
Once the expression is constructed, click Apply to place the expression in the vector field and plot the curve.
In the preferences file, use *RegisterExternalFunction() and *RegisterTemplexFunction() to register external functions or Templex functions in HyperGraph. You can insert any of the functions into
the current expression by selecting the function name from the list or by typing the name directly into the equation.
See the List of Standard Functions topic in the Math Reference for a detailed description of each function and its purpose.
External Functions
In addition to the built-in math functions and operators, external C-programs can also be called from within a math expression.
Calling an external function enables you to process plot data from within the program using your own set of specialized programs. For example, the external program can be a customized filter for
manipulating plot data or a program that passes plot data to another application for processing.
External programs must be registered in the preferences file before they can be called from within a math expression. Registering an external program associates the program file with a function
name. External programs can be registered in your own preferences file or in the global preferences file, making it available to everyone on the network.
See the Altair IPC for more information on writing and calling external C-programs from within a math expression.
See the *RegisterExternalFunction() statement in Preference Files help for more information on registering external functions in the preferences.mvw file.
Freezing Vectors
When a vector is defined by an expression, the program automatically recalculates the vector each time the expression is altered, updating the curve. If an expression contains a reference to
another curve and the referenced curve changes, the program recalculates the vector and updates the curve containing the reference.
Vectors can be frozen so that the program does not recalculate the curve. When a vector is frozen, it is no longer dependent on a referenced curve, so changes made to other curves are not
reflected in the frozen vector. Vectors can be unfrozen, making them once again subject to change. The X and Y vectors can be frozen independently of each other or together, freezing the entire
curve. Frozen vectors are saved as data point values in session files.
Values as a Data Source
If Values is selected as the source, a table is displayed. Enter data point values to the table directly.
Figure 3. | {"url":"https://help.altair.com/hwdesktop/hwx/topics/hypergraph/define_curves_data_sources_r.htm","timestamp":"2024-11-12T23:18:12Z","content_type":"application/xhtml+xml","content_length":"74424","record_id":"<urn:uuid:6565aa71-476e-48e5-a60d-8febc88c6dad>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00680.warc.gz"} |
Did you know...
Arranging a Wikipedia selection for schools in the developing world without internet was an initiative by SOS Children. SOS Children is the world's largest charity giving orphaned and abandoned
children the chance of family life.
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we
added, we return to the number we started with. Subtraction is denoted by a minus sign in infix notation, in contrast to the use of the plus sign for addition.
Since subtraction is not a commutative operator, the two operands are named. The traditional names for the parts of the formula
c − b = a
are minuend (c) − subtrahend (b) = difference (a).
Subtraction is used to model four related processes:
1. From a given collection, take away (subtract) a given number of objects. For example, 5 apples minus 2 apples leaves 3 apples.
2. From a given measurement, take away a quantity measured in the same units. If I weigh 200 pounds, and lose 10 pounds, then I weigh 200 − 10 = 190 pounds.
3. Compare two like quantities to find the difference between them. For example, the difference between $800 and $600 is $800 − $600 = $200. Also known as comparative subtraction.
4. To find the distance between two locations at a fixed distance from starting point. For example if, on a given highway, you see a mileage marker that says 150 miles and later see a mileage marker
that says 160 miles, you have traveled 160 − 150 = 10 miles.
In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the additive inverse. We can view 7 − 3 = 4 as the sum of two terms: 7 and -3. This
perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative or commutative—in fact, it is anticommutative and
left-associative—but addition of signed numbers is both.
Basic subtraction: integers
Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled
mathematically by addition:
a + b = c.
From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:
c − b = a.
Now, a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1.
This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended.
To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is
still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction.
The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). From 3, it takes 4 steps to the left to get to −1:
3 − 4 = −1.
Subtraction as addition
There are some cases where subtraction as a separate operation becomes problematic. For example, 3 − (−2) (i.e. subtract −2 from 3) is not immediately obvious from either a natural number view or a
number line view, because it is not immediately clear what it means to move −2 steps to the left or to take away −2 apples. One solution is to view subtraction as addition of signed numbers. Extra
minus signs simply denote additive inversion. Then we have 3 − (−2) = 3 + 2 = 5. This also helps to keep the ring of integers "simple" by avoiding the introduction of "new" operators such as
subtraction. Ordinarily a ring only has two operations defined on it; in the case of the integers, these are addition and multiplication. A ring already has the concept of additive inverses, but it
does not have any notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring axioms to subtraction without needing to prove anything.
Algorithms for subtraction
There are various algorithms for subtraction, and they differ in their suitability for various applications. A number of methods are adapted to hand calculation; for example, when making change, no
actual subtraction is performed, but rather the change-maker counts forward.
For machine calculation, the method of complements is preferred, whereby the subtraction is replaced by an addition in a modular arithmetic.
The teaching of subtraction in schools
Methods used to teach subtraction to elementary school varies from country to country, and within a country, different methods are in fashion at different times. In what is, in the U.S., called
traditional mathematics, a specific process is taught to students at the end of the 1st year or during the 2nd year for use with multi-digit whole numbers, and is extended in either the fourth or
fifth grade to include decimal representations of fractional numbers.
Some American schools currently teach a method of subtraction using borrowing and a system of markings called crutches. Although a method of borrowing had been known and published in textbooks prior,
apparently the crutches are the invention of William A. Brownell who used them in a study in November 1937. This system caught on rapidly, displacing the other methods of subtraction in use in
America at that time.
Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid
memory), which vary by country.
Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of subtrahend:
s[j] s[j−1] ... s[1]
from minuend
m[k] m[k−1] ... m[1],
where each s[i] and m[i] is a digit, proceeds by writing down m[1] − s[1], m[2] − s[2], and so forth, as long as s[i] does not exceed m[i]. Otherwise, m[i] is increased by 10 and some other digit is
modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit m[i+1] by one (or continuing the borrow leftwards until there is a non-zero digit from
which to borrow). The European method corrects by increasing the subtrahend digit s[i+1] by one.
Example: 704 − 512. The minuend is 704, the subtrahend is 512. The minuend digits are m[3] = 7, m[2] = 0 and m[1] = 4. The subtrahend digits are s[3] = 5, s[2] = 1 and s[1] = 2. Beginning at the
one's place, 4 is not less than 2 so the difference 2 is written down in the result's one place. In the ten's place, 0 is less than 1, so the 0 is increased to 10, and the difference with 1, which is
9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced
by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192.
The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundred's digit by one. A small mark is made near or below this digit (depending on the school). Then the
subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundred's place.
There is an additional subtlety in that the student always employs a mental subtraction table in the American method. The Austrian method often encourages the student to mentally use the addition
table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it,
makes 7.
When subtracting two numbers with units, they must have the same unit. In most cases the difference will have the same unit as the original numbers. One exception is when subtracting two numbers with
percentage as unit. In this case, the difference will have percentage points as unit. | {"url":"https://www.valeriodistefano.com/en/wp/s/Subtraction.htm","timestamp":"2024-11-09T02:52:09Z","content_type":"text/html","content_length":"88769","record_id":"<urn:uuid:81afb91f-5232-4064-a8ea-e62b46726788>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00557.warc.gz"} |
Aggregation Functions | Tecton
Version: 0.7
Tecton's Aggregation Engine supports the following aggregations out of the box.
Support for custom aggregation functions is coming soon.
An aggregation function that returns, for a materialization time window, the approximate number of distinct row values for a column, per entity value (such as a user_id value). Null values are
Not currently supported with:
• Tecton on Snowflake
• Serverless Feature Retrieval with Athena
Input column types
• Tecton on Spark: String, Int32, Int64
Output column types
Import this aggregation with from tecton.aggregation_functions import approx_count_distinct.
Then, define an Aggregation object using function=approx_count_distinct(precision), where precision is an integer >= 4 and <= 16, in a Batch or a Stream Feature View.
The precision parameter controls the accuracy of the approximation. A higher precision yields lower error at the cost of more storage; the impact on performance (i.e. speed) is negligible. The
storage cost (in both the offline and online store) is proportional to 2^precision. The standard error of the approximation is 1.04 / sqrt(2^precision). Here are the standard errors for several
different values of precision:
Precision Standard Error
4 26.0%
6 13.0%
8 6.5%
10 3.3%
12 1.6%
14 0.8%
16 0.4%
The default value of precision is 8. We recommend using the default precision unless extreme accuracy is important.
In general, the approx_count_distinct aggregation might not return the exact correct value. However, the aggregation is typically able to return the exact correct value for low-cardinality data (i.e.
data with at most several hundred distinct elements), as long as the maximum precision (16) is used.
This aggregation uses the HyperLogLog algorithm.
Aggregation(column="address", function=approx_count_distinct(), time_window=timedelta(days=1)) to use the default value of precision=8
Aggregation(column="address", function=approx_count_distinct(precision=10), time_window=timedelta(days=1))
approx_percentile(percentile, precision)​
An aggregation function that returns, for a materialization time window, a value that is approximately equal to the specified percentile, per entity value (such as a user_id value). Null values are
excluded. For Float32 and Float64 input columns, NaNs, positive infinity, and negative infinity are also excluded.
Not currently supported with:
• Tecton on Snowflake
• Serverless Feature Retrieval with Athena
Input column types
• Tecton on Spark: Float32, Float64, Int32, Int64
Output column types
Import this aggregation with from tecton.aggregation_functions import approx_percentile.
Then, define an Aggregation object, using function=approx_percentile(percentile, precision), where percentile is a float >= 0.0 and <= 1.0 and precision is an integer >= 20 and <= 500, in a Batch
Feature View or a Stream Feature View.
The precision parameter controls the accuracy of the approximation. A higher precision yields lower error at the cost of more storage; the impact on performance (i.e. speed) is negligible.
Specifically, the error rate of the estimate is inversely proportional to precision, and the storage cost is proportional to precision. The default value of precision is 100. We recommend using the
default precision unless extreme accuracy is important.
This aggregation uses the t-Digest algorithm.
This aggregation is not fully deterministic. Its final estimate depends on the order in which input data is processed. Therefore, for example, it is possible for get_historical_features to return
different results when run twice, as Spark could shuffle the input data differently. Similarly, the feature server may return different results than the offline store, as there is no guarantee that
the input data is processed in the exact same order. In practice, getting different results is rare, and when it does happen, the differences are extremely small.
This aggregation is computationally intensive. As a result, running get_historical_features with from_source=True can be slow. If possible, we recommend waiting for offline materialization to finish
and using from_source=False.
Aggregation(column="count", function=approx_percentile(percentile=0.5), time_window=timedelta(days=1)) to get the 50th percentile with the default value of precision=100
Aggregation(column="count", function=approx_percentile(percentile=0.99, precision=500), time_window=timedelta(days=1)) to get the 99th percentile with extreme precision
An aggregation function that returns, for a materialization time window, the number of row values for a column, per entity value (such as a user_id value). Null values are excluded.
Input column types
• Tecton on Spark: All types
• Tecton on Snowflake: All types
Output column types
To use this aggregation, define an Aggregation object, using function="count", in a Batch Feature View or a Stream Feature View.
Aggregation(column="transaction_id", function="count", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the first N distinct row values for a column, per entity value (such as a user_id value).
For example, if the first 2 distinct row values for a column, in the materialization time window, are 10 and 20, then the function returns [10,20].
The output sequence is in ascending order based on timestamp.
Not currently supported with:
• Tecton on Snowflake
• Serverless Feature Retrieval with Athena
Input column types
Output column type
Import this aggregation with from tecton.aggregation_functions import first_distinct.
Then, define an Aggregation object, using function=first_distinct(n), where n is an integer > 0 and <= 1000, in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function=first_distinct(2), time_window=timedelta(days=1)).
An aggregation function that returns, for a materialization time window, the first N row values for a column, per entity value (such as a user_id value).
For example, if the first 2 row values for a column, in the materialization time window, are 10 and 20, then the function returns [10,20].
The output sequence is in ascending order based on the timestamp.
Not currently supported with:
• Serverless Feature Retrieval with Athena
Input column types
Output column type
Import this aggregation with from tecton.aggregation_functions import first.
Then, define an Aggregation object, using function=first(n), where n is an integer > 0 and <= 1000, in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function=first(2), time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the last N distinct row values for a column, per entity value (such as a user_id value).
For example, if the last 2 distinct row values for a column, in the materialization time window, are 10 and 20, then the function returns [10,20].
The output sequence is in ascending order based on the timestamp.
Not currently supported with:
• Tecton on Snowflake
• Serverless Feature Retrieval with Athena
Input column types
Output column type
Import this aggregation with from tecton.aggregation_functions import last_distinct.
Then, define an Aggregation object, using function=last_distinct(n), where n is an integer > 0 and <= 1000, in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function=last_distinct(2), time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the last row value for a column, per entity value (such as a user_id value).
Not currently supported with:
• Tecton on Snowflake
• Serverless Feature Retrieval with Athena
Input column types
• Int64, Int32, Float64, Bool, String
Output column type
• Int64, Float64, Bool, String
To use this aggregation, define an Aggregation object, using function="last", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="last", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the last N row values for a column, per entity value (such as a user_id value).
For example, if the last 2 row values for a column, in the materialization time window, are 10 and 20, then the function returns [10,20].
The output sequence is in ascending order based on the timestamp.
Not currently supported with:
• Serverless Feature Retrieval with Athena
Input column types
Output column type
Import this aggregation with from tecton.aggregation_functions import last.
Then, define an Aggregation object using function=last(n), where n is an integer > 0 and <= 1000, in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function=last(2), time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the maximum of the row values for a column, per entity value (such as a user_id value).
Input column types
• Int64, Int32, Float64, String
Output column type
To use this aggregation, define an Aggregation object, using function="max", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="max", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the mean of the row values for a column, per entity value (such as a user_id value).
Input column types
Output column type
To use this aggregation, define an Aggregation object, using function="mean", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="mean", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the minimum of the row values for a column, per entity value (such as a user_id value).
Input column types
• Int64, Int32, Float64, String
Output column type
To use this aggregation, define an Aggregation object, using function="min", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="min", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the standard deviation of the row values for a column around the population mean, per entity value (such as a user_id value).
Input column types
Output column type
To use this aggregation, define an Aggregation object, using function="stddev_pop", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="stddev_pop", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the standard deviation of the row values for a column around the sample mean, per entity value (such as a user_id value).
Input column types
Output column type
To use this aggregation, define an Aggregation object, using function="stddev_samp", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="stddev_samp", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the sum of the row values for a column, per entity value (such as a user_id value).
Input column types
Output column type
To use this aggregation, define an Aggregation object, using function="sum", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="sum", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the variance of the row values for a column around the population mean, per entity value (such as a user_id value).
Input column types
Output column type
To use this aggregation, define an Aggregation object, using function="var_pop", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="var_pop", time_window=timedelta(days=1))
An aggregation function that returns, for a materialization time window, the variance of the row values for a column around the sample mean, per entity value (such as a user_id value).
Input column types
Output column type
To use this aggregation, define an Aggregation object, using function="var_samp", in a Batch Feature View or a Stream Feature View.
Aggregation(column="amt", function="var_samp", time_window=timedelta(days=1)) | {"url":"https://docs.tecton.ai/docs/0.7/defining-features/feature-views/aggregation-engine/aggregation-functions","timestamp":"2024-11-08T15:48:28Z","content_type":"text/html","content_length":"69781","record_id":"<urn:uuid:7b25b8ac-cc0b-44dd-8da6-386139d0c1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00160.warc.gz"} |
Bortolomiol, S., Lurkin, V., and Bierlaire, M. (2021)
Benders decomposition for choice-based optimization problems with discrete upper-level variables
21st Swiss Transport Research Conference, ,
In this work, we consider a class of choice-based optimization problems in which the decision variables of the supplier are discrete. First, we compare this class of problems with a more general
formulation which admits both continuous and discrete variables. A feasible solution of the problem with both continuous and discrete upper-level variables can be found by solving a problem with only
discrete variables by discretizing all continuous variables of the original formulation. An appropriate discretization of all continuous variables can guarantee a good approximation of the solution
of the original problem. A computational analysis shows that the discrete formulation is faster than the continuous-discrete formulation for complex problems. Then, we show that in the discrete
formulation, in which all utility functions of the customers can be expressed as parameters of the supplier's optimization problem, the lower-level optimization problem of the customer is a
continuous knapsack problem. This property is used to derive a Benders decomposition algorithm for the choice-based optimization problem with discrete variables. | {"url":"https://transp-or.epfl.ch/php/abstract.php?type=4&id=BorLurBieBon_STRC2021","timestamp":"2024-11-06T05:50:26Z","content_type":"application/xhtml+xml","content_length":"2379","record_id":"<urn:uuid:3140ec75-9771-466d-af0a-92d11605e442>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00765.warc.gz"} |
all-electron Gaussian basis for actinides
• From: Georg Schreckenbach <h.g.schreckenbach \\at// dl.ac.uk>
• Organization: CLRC Daresbury Laboratory
• Subject: all-electron Gaussian basis for actinides
• Date: Fri, 24 Mar 2000 15:11:45 +0000
Dear everybody,
does anyone know of ALL-ELECTRON Gaussian basis sets for
actinide elements, published or unpublished? I'd be particularly
interested in the early actinides, Th thru' Am, say.
I have already searched the CCL archives and the EMSL basis set
repository but with no success. I know, of course, that one would
normally use ECPs or related methods for these heavy elements.
(... and there are a couple of ECP basis sets in the EMSL library ...)
I also know that the ADF program provides all-electron
SLATER basis sets for these elements.
Would it, perhaps, be possible to use the basis sets that were
developed for four-component relativistic methods? If so, would
they have to be modified? (I suppose so ... e.g. take the large
component only?) Where are they to be found?
I'll summarize as usual.
Thank you, and best regards, | {"url":"https://server.ccl.net/chemistry/resources/messages/2000/03/24.007-dir/index.html","timestamp":"2024-11-07T03:24:07Z","content_type":"text/html","content_length":"2763","record_id":"<urn:uuid:1a075d0f-98e4-4a6b-85f8-1d9d5f965f6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00289.warc.gz"} |
Lesson: Comparing 2-digit numbers | Oak National Academy
Switch to our new maths teaching resources
Slide decks, worksheets, quizzes and lesson planning guidance designed for your classroom.
Lesson details
Key learning points
1. In this lesson, we will represent 2-digit numbers, comparing the tens and ones in 2-digit numbers and identifying which number is greater or less.
This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.
5 Questions
Which number is greater? 24 or 42?
Which number is worth less? 45 or 54?
Look at the two numbers below. Tick the THREE statements that are true.
The numbers are equal to each other
Look at the number cards below. What is the smallest 2-digit number you can make?
Look at the number cards below. What is the greatest 2-digit number you can make?
5 Questions
Look at the 2-digit numbers below. Which sign is needed in the blank space?
Look at the 2-digit numbers below. Which sign is needed is the blank space?
Look at the 2-digit numbers below. Which sign is needed is the blank space?
Which of the number sentences below is incorrect?
Which of the number sentences below is incorrect? | {"url":"https://www.thenational.academy/teachers/lessons/comparing-2-digit-numbers-c4t38e","timestamp":"2024-11-05T08:40:15Z","content_type":"text/html","content_length":"285686","record_id":"<urn:uuid:f6ac6cb6-0c47-4646-9bcd-dd6d5134d266>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00611.warc.gz"} |
Phase functions
The note describes a method to compute the phase function from the differential equations for some simple cases.
Let \(\phi(x(t))\) be the phase function. Then,
\[\frac{d\phi}{dt} = \frac{d\phi}{dx}.\frac{dx}{dt}=1\]
However, we know, from the definition of the differential equations,
\[\frac{dx}{dt} = f(x)\]
\[\frac{d\phi}{dx}.\frac{dx}{dt} = \frac{d\phi}{dx}.f(x) = 1\]
This helps us get the following equation to compute phase function.
\[\frac{d\phi}{dx}.f(x) = 1\]
Where \(\frac{d\phi}{dx}\) is the PRC.
\[PRC = \frac{1}{f(x(t))} = \frac{1}{\dot{\zeta}(t)}\]
Where \(\zeta (t)\) is the limit cycle solution of the differential equation!
subscribe via RSS | {"url":"https://www.dreamingsailor.me/pd/2021/03/06/isochrone2","timestamp":"2024-11-09T04:01:48Z","content_type":"text/html","content_length":"12218","record_id":"<urn:uuid:246ff65d-ff62-4074-856c-dbfb42a74be7>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00595.warc.gz"} |
Angular harmonic oscillator
Simple harmonic motion can also be angular. In this case, the restoring torque required for producing SHM is directly proportional to the angular displacement and is directed towards the mean
Consider a wire suspended vertically from a rigid support. Let some weight be suspended from the lower end of the wire. When the wire is twisted through an angle θ from the mean position, a restoring
torque acts on it tending to return it to the mean position. Here restoring torque is proportional to angular displacement θ.
Hence r = -C θ ???..(1)
where C is called torque constant.
It is equal to the moment of the couple required to produce unit angular displacement. Its unit is N m rad−1.
The negative sign shows that torque is acting in the opposite direction to the angular displacement. This is the case of angular simple harmonic motion.
Examples : Torsional pendulum, balance wheel of a watch.
But τ = I α ...(2)
where τ is torque, I is the moment of inertia and α is angular acceleration
∴ Angular acceleration,
Α = τ /I = - C θ / I
This is similar to a = −ω^2 y
Replacing y by θ, and a by α we get
α = −ω^2θ = - (C/I) θ
ω = rt(C/I)
Period of SHM T = 2π rt(I/C)
Frequency n = 1/T = 1/2 π rt(C/I)
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th 12th std standard Class Physics sciense Higher secondary school College Notes : Angular harmonic oscillator | | {"url":"https://www.brainkart.com/article/Angular-harmonic-oscillator_3138/","timestamp":"2024-11-07T19:32:42Z","content_type":"text/html","content_length":"33619","record_id":"<urn:uuid:d394937d-bf64-495b-b865-f8e76e25f074>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00428.warc.gz"} |
Prerequisites: Queue
Heaps are data structures that can pop the maximum or minimum value or push a value very efficiently. Heaps are special binary trees which have the property that: the value of each node is greater
than the values of all its children (a max heap) or the value of each node is less than the values of all its children (a min heap). Priority queue's are most efficiently implemented as heaps. This
guarantees that the maximum or minimum element is the root node.
Heaps store their data level by level in a binary tree. This allows us to store heaps in an array. The root index is 0. For every node with index i, the left index can be found by using the formula
2*i+1 and the right index can be found by using the formula 2*i + 2. The parent of a node can be found by integer division as (i-1)/2.
root = 0
leftChild = index * 2 + 1
rightChild = index * 2 + 2
parent = (index - 1)/2
Indexes of a heap
Example Heap:
A heap has two operations: push and pop. Pushing an element into a heap adds it into the heap while ensuring that the properties of the heap still hold. Popping removes an element from the top of the
heap and the heap needs to ensure that the properties of the heap still hold.
Operation Resize Push Pop Heapify
Time Complexity O(n) O(log n) O(log n) O(n)
We will implement a max heap in Java and in our class we need to store the elements in the heap and the size of the heap.
public class Heap {
public int[] arr;
public int size;
public Heap(int startSize) {
arr = new int[startSize];
size = 0;
When the heap reaches capacity, we need to resize it to be able to contain more elements.
public void resize() {
int[] newArr = new int[arr.length * 2];
for (int i = 0; i < size; i++) {
newArr[i] = arr[i];
arr = newArr;
Swap will switch two nodes in the heap.
public void swap(int a, int b) {
int tmp = arr[a];
arr[a] = arr[b];
arr[b] = tmp;
Pushes the number x into the priority queue. We can do this by adding it to the bottom of the heap and then keep swapping it upwards if it is greater than the parent.
Add 9 to the end of the heap.
The parent of 9 is 3 and smaller so we can swap the two.
The parent of 9 is 8 and smaller so we can swap the two.
Since the parent of 9 is 10 and greater than 9 then we can stop. We can also see that the heap structure is still valid.
public void push(int x) {
if (size >= arr.length) {
// Insert to the end of the heap.
arr[size] = x;
int idx = size - 1;
int parent = (idx - 1) / 2;
// Push the node up until the parent is larger.
while (idx > 0 && arr[parent] < arr[idx]) {
swap(parent, idx);
idx = parent;
parent = (idx - 1) / 2;
Popping removes the greatest element in the heap. Since the root is guaranteed to be the greatest element as a property of a heap, we remove it and return that element. After removing the root, we
replace it with the element at the bottom of the heap and we can keep swapping it with its children until the heap property is satisfied.
Let's do an example of where we pop from a heap. We want to remove the root of the heap and replace it with the last element of the heap.
We switch the node and replace it with the last element in the heap.
The largest child is the left child of 8 and it is larger than the current node, so we swap them.
The largest child is the right child of 7 and it it larger than the current node, so we swap the nodes again.
When the current node is larger than both children or the current node is at the bottom, then we can stop.
public void bubbleDown(int idx) {
while (idx < size) {
int left = idx * 2 + 1;
int right = idx * 2 + 2;
// If both child exists.
if (left < size && right < size) {
// If left child is larger than right child and current node.
if (arr[left] > arr[right] && arr[left] > arr[idx]) {
swap(left, idx);
idx = left;
// If right child is larger or equal than left child and current node.
else if (arr[right] >= arr[left] && arr[right] > arr[idx]) {
swap(right, idx);
idx = right;
// If no children, stop.
else {
// If there is only a left child.
else if (left < size) {
swap(left, idx);
idx = left;
// If there is only a right child.
else if (right < size) {
swap(right, idx);
idx = right;
else {
public int pop() {
if (size == 0) {
return 0;
// Swap root and last element of heap.
int ret = arr[0];
arr[0] = arr[size - 1];
// Push the root down until parent is greater than children
return ret;
Heapify takes an array of N elements and transforms it into a heap in the same array. The runtime of heapify is suprisingly O(n) and the proof of that is beyond the scope of this book.
public void heapify(int arr[]) {
this.arr = arr;
// Reach height of tree.
for (int i = 0; i < Math.floor(arr.length / 2.0); i++) {
// Iterate through array.
1. Implement a min heap in Java.
2. Write a function that checks if an array is a heap.
3. Prove that heapify is O(n). | {"url":"https://thecshandbook.com/Heap","timestamp":"2024-11-13T08:06:05Z","content_type":"text/html","content_length":"14393","record_id":"<urn:uuid:817fa51c-0beb-48bf-b207-11c2fdd6a70c>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00491.warc.gz"} |
Pseudorandom Number Generation using Cellular Automata - Rule 30
A pseudorandom number generator produces numbers deterministically but they seem aperiodic (random) most of the time for most use-cases. The generator accepts a seed value (ideally a true random
number) and starts producing the sequence as a function of this seed and/or a previous number of the sequence. These are Pseudorandom (not truly random) because if seed value is known they can be
determined algorithmically. True random numbers are hardware generated or generated from blood volume pulse, atmospheric pressure, thermal noise, quantum phenomenon, etc.
There are lots of techniques to generate Pseudorandom numbers, namely: Blum Blum Shub algorithm, Middle-square method, Lagged Fibonacci generator, etc. Today we dive deep into Rule 30 that uses a
controversial science called Cellular Automaton. This method passes many standard tests for randomness and was used in Mathematica for generating random integers.
Cellular Automaton
Before we dive into Rule 30, we will spend some time understanding Cellular Automaton. A Cellular Automaton is a discrete model consisting of a regular grid, of any dimension, with each cell of the
grid having a finite number of states and a neighborhood definition. There are rules that determine how these cells interact and transition into the next generation (state). The rules are mostly
mathematical/programmable functions that depend on the current state of the cell and its neighborhood.
In the above Cellular Automaton, each cell has 2 finite states 0 (shown in red), 1 (shown in black). Each cell transitions into the next generation by XORing the state values of its 8 neighbors. The
first generation (initial state) of the grid is allocated at random and the state transitions, of the entire grid, is as below
Cellular Automata was originally conceptualized in the 1940s by Stanislaw Ulam and John von Neumann; it finds its application in computer science, mathematics, physics, complexity science,
theoretical biology and microstructure modeling. In the 1980s, Stephen Wolfram did a systematic study of one-dimensional cellular automata (also called elementary cellular automata) on which Rule 30
is based.
Rule 30
Rule 30 is an elementary (one-dimensional) cellular automaton where each cell has two possible states 0 (shown in red) and 1 (shown in black). The neighborhood of a cell is its two immediate
neighbors, one on its left and other on right. The next state (generation) of the cell depends on its current state and the state of its neighbors; the transition rules are as illustrated below
The above transition rules could be simplified as left XOR (central OR right).
We visualize Rule 30 in a 2-dimensional grid where each row represents one generation (state). The next generation (state) of the cells is computed and populated in the row below. Each row contains a
finite number of cells which “wraps around” at the end.
The above pattern emerges from an initial state (row 0) in a single cell with state 1 (shown as black) surrounded by cells with state 0 (red). The next generation (as seen in row 1) is computed using
the rule chart mentioned above. The vertical axis represents time and any horizontal cross-section of the image represents the state of all the cells in the array at a specific point in the pattern’s
As the pattern evolves, frequent red triangles of varying sizes pop up but the structure as a whole has no recognizable pattern. The above snapshot of the grid was taken at a random point of time and
we could observe chaos and aperiodicity. This property is exploited to generate pseudorandom numbers.
Pseudorandom Number Generation
As established earlier, Rule 30 is exhibits aperiodic and chaotic behavior and hence it produces complex, seemingly random patterns from simple, well-defined rules. To generate random numbers from
using Rule 30 we use the center column and pick a batch of n random bits and form the required n bit random number from it. The next random number is built using the next n bits from the column.
If we always start from the first row, the sequence of the numbers we generate will always be predictable - which is not what we want. To make things pseudorandom, we take a random seed value (ex:
current timestamp) and skip that number of bits and then pick batches of n and build random numbers.
The pseudorandom numbers generated using Rule 30 are not cryptographically secure but are suitable for simulation as long as we do not use bad seed like 0.
One major advantage of using Rule 30 to generate pseudorandom numbers is that we could generate multiple random numbers in parallel by picking multiple columns to batch n bits each at random. A
sample 8-bit random integer sequence generated using this method with seed 0 is 220, 197, 147, 174, 117, 97, 149, 171, 240, 241, etc.
The seed value could also be used as the initial state (row 0) for Rule 30 and random numbers are then simply the n bits batches picked from the center column starting from row 0. This approach is
more efficient but is heavily dependent on the quality of seed value, as a bad seed value could make things extremely predictable. A demonstration of this approach could be found on Wolfram Cloud
Demonstration Page.
Rule 30 in the real world
Rule 30 is also seen in nature, on the shell of code snail species Conus textile. The Cambridge North railway station is decorated with architectural panels displaying the evolution of Rule 30.
If you found Rule 30 interesting I urge you to write your own simulation of using p5 library; you could keep it generic enough to so that the program could generate patterns for different rules like
90, 110, 117, etc. The patterns generated using these rules are quite interesting. If you want, you could things to the next level and extend rule to work in 3 dimensions and see how patterns evolve.
I believe programming is fun when it is visual.
It is exciting when two seemingly unrelated fields, Cellular Automata and Cryptography, come together and create something wonderful. Although this algorithm is not widely used anymore, because of
more efficient algorithms, it urges us to be creative in using Cellular Automata in more ways than one. This article is first in the series of Cellular Automata, so stay tuned and watch this space
for more. | {"url":"http://edge.arpitbhayani.me/blogs/rule-30-cellular-automata","timestamp":"2024-11-05T21:40:48Z","content_type":"text/html","content_length":"27423","record_id":"<urn:uuid:4d92172d-38fc-4671-8d62-fe4e0199542a>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00613.warc.gz"} |
A Cylinder Has 3 Faces. - Delight Mix
A Cylinder Has 3 Faces.
The concept of a cylinder having three faces is quite intriguing, as traditionally cylinders are considered to have two faces – the circular top and bottom surfaces, also known as bases, and the
curved lateral surface connecting the two bases. However, when discussing a cylinder having three faces, it could be interpreted in a few different ways, and in this article, we will explore various
aspects related to cylinders and their faces.
Understanding the Basics of a Cylinder
Before delving into the idea of a cylinder having three faces, let’s brush up on the basic definition of a cylinder in geometry. A cylinder is a three-dimensional geometric shape that consists of two
parallel circular bases connected by a curved surface. The bases are congruent and lie in parallel planes. The distance between the centers of the two bases is the height of the cylinder.
Parts of a Cylinder:
1. Bases: The top and bottom circular surfaces of the cylinder.
2. Lateral Surface: The curved surface that connects the bases.
3. Axis: The line segment connecting the centers of the two bases.
4. Height: The distance between the bases along the axis.
5. Radius: The distance from the center of a base to its edge, defining the size of the cylinder.
6. Volume: The amount of space inside the cylinder, calculated as the base area multiplied by the height.
7. Surface Area: The total area covered by the cylinder, including both bases and the lateral surface.
Interpreting the Idea of Three Faces in a Cylinder
When we talk about a cylinder having three faces, it can be viewed from the following perspectives:
1. Two Circular Faces and One Rectangular Face:
In a conceptual sense, some may refer to the two bases (circles) and the rectangular lateral face when mentioning three faces of a cylinder. While mathematically, the lateral face is a curved
surface, in simpler terms, it can be visualized as a flattened 2D rectangular face when the cylinder is unwrapped.
1. Top Base, Bottom Base, and Side Face:
Another interpretation could be considering each base as a face along with the side face (lateral surface). This interpretation aligns with the notion of counting each distinct part of the cylinder
surface as a face, making it three in total.
Visualizing Faces of a Cylinder
To better visualize the faces of a cylinder, you can think of a canned drink container. The top circular lid, the bottom circular lid, and the main body of the can that wraps around constitute the
three faces when viewed from a simplistic perspective.
Frequently Asked Questions (FAQs) About Cylinders:
1. How is the volume of a cylinder calculated?
The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius of the base and h is the height of the cylinder.
1. What is the formula for the surface area of a cylinder?
The formula for the surface area of a cylinder is SA = 2πrh + 2πr², where r is the radius and h is the height of the cylinder.
1. Can a cylinder have different base shapes other than circles?
No, by definition, a cylinder has circular bases. If the base shape is different, it would fall under a different geometric category.
1. Are all prisms considered cylinders?
No, not all prisms are cylinders. For a shape to be classified as a cylinder, it must have circular bases connected by a curved surface.
1. What is the difference between a cylinder and a cone?
A cylinder has two parallel circular bases, whereas a cone has a circular base connected to a single vertex, forming a tapering structure.
In conclusion, while a cylinder conventionally has two faces, the concept of it having three faces can be viewed from a broader perspective considering the various components that make up its
structure. Whether you envision it as two circular faces and a rectangular face or as the top base, bottom base, and side face, understanding the fundamental properties and characteristics of
cylinders is essential in geometry and real-world applications. Keep exploring geometric shapes and their properties to deepen your understanding of the mathematical world! | {"url":"https://delightmix.com/a-cylinder-has-3-faces/","timestamp":"2024-11-09T15:43:51Z","content_type":"text/html","content_length":"99866","record_id":"<urn:uuid:c077fa81-edf4-423a-a869-a68ddbad4084>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00428.warc.gz"} |
Suppression of Irregular Frequency Effect in Hydrodynamic Problems and Free-Surface Singularity Treatment
Multibody operations are routinely performed in offshore activities, for example, the floating liquefied natural gas (FLNG) and liquefied natural gas carrier (LNGC) side-by-side offloading case. To
understand the phenomenon occurring inside the gap is of growing interest to the offshore industry. One important issue is the existence of the irregular frequency effect. The effect can be confused
with the physical resonance. Thus, it needs to be removed. An extensive survey of the previous approaches to the irregular frequency problem has been undertaken. The matrix formulated in the boundary
integral equations will become nearly singular for some frequencies. The existence of numerical round-off errors will make the matrix still solvable by a direct solver, however, it will result in
unreasonably large values in some aspects of the solution, namely, the irregular frequency effect. The removal of the irregular effect is important especially for multibody hydrodynamic analysis in
identifying the physical resonances caused by the configuration of floaters. This paper will mainly discuss the lid method on the internal free surface. To reach a higher accuracy, the singularity
resulting from the Green function needs special care. Each term in the wave Green function will be evaluated using the corresponding analysis methods. Specifically, an analytical integral method is
proposed to treat the log singularity. Finally, results with and without irregular frequency removal will be shown to demonstrate the effectiveness of our proposed method.
Boundary-value problems, Damping, Errors, Integral equations, Resonance, Shapes, Waves, Ocean engineering, Liquefied natural gas, Hydrodynamics, Fluid structure interaction, Green's function methods
Multibody operations are routinely performed in offshore activities. This scenario is economically efficient especially in remote offshore locations. One classical example of such kind is the
side-by-side case of floating liquefied natural gas (FLNG) and liquefied natural gas carrier (LNGC). The integrated concept has moved the land-based factory offshore. FLNG will produce and process
the natural gas and off-load it to the close-by LNGC. This scenario will reduce the cost and the negative environmental impact to the local area. It has great potential in future offshore gas
However, when two huge floaters are in close proximity, the hydrodynamic interactions will become complex and large. It is necessary to understand the complex physical phenomenon happening between
the two vessels, make accurate predictions, and take proper measurement when operating in practice. Therefore, the multibody hydrodynamics analysis will be essential in the design phase.
In such problems, the boundary element method will result in “resonance” when there is a gap if adopting the Green function approach. This “resonance” was over predicted in numerical simulations and
thus considered as unphysical. Besides, the resonance resulted from the boundary element method can be divided into two categories: one is the resonance from the irregular frequencies and the other
perhaps from the limitation of the method. Sometimes, the resonance caused by the irregular frequencies will confuse the true resonance. Thus, it needs to be removed.
Irregular frequencies result from the ill condition of the linear system in boundary integral problems. In other words, the matrix to be solved is almost singular at some frequencies. The solver can
still provide some results because of limited accuracy of numerical techniques and computer round-off errors. The irregular frequency effect in wave–body interaction was first found by John [1]. The
harmful effects in hydrodynamic analysis were identified by Frank [2]. The characteristic “sharp spikes” resulted from the irregular frequency effect were observed in added mass and damping, which
affected the accuracy of the final results. The undesirable “spikes” will be confused with the physical resonance especially in multibody interaction problems. Therefore, this effect must be removed
in order to expand the applications of boundary integral methods in hydrodynamic analysis.
Researchers first sought for the feasible approaches to remove this effect. Afterward, several applicable methods for more frequencies and more general shapes were proposed. The modified-integral
method and the extended-boundary-condition one are the two approaches to resolve this problem.
In the study of the modified-integral method, Ursell [3] investigated this problem in an analytical way. He put wave source at the center of the circle and did not observe irregular frequencies for
shorter wavelengths. This analytical study indicates that such a technique can be adopted in numerical evaluation of such problems. Schenck [4] applied the combined integral equations at selected
internal points, leading to an overdetermined system. Burton and Miller [5] also adopted a modified Green function in acoustic wave scattering problem. Jones [6] added a source at the origin to
remove interior eigenmode effects in acoustics. Inspired by Ursell [3], Ogilvie and Shin [7] modified the Green function integral by adding a source or dipole at the center of the internal free
surface for the wave–body interaction problem. Sayer [8] also examined the suggestions from Ursell [3] on a symmetric body in finite depth. Later on, Ursell [9] demonstrated that a sequence of
singularities can remove all the irregular frequencies, resulting in a method applicable for the general case. Wu and Price [10] extended this method to a twin-hull problem. Lau and Hearn [11]
adopted the combined integral equation to study this problem. Lee and Sclavounos [12] further developed the modified integral method. Lee [13] pointed out that the final effect was determined by the
choices of linear combination coefficients. He converted this problem into an optimization one and used the condition number at the first irregular frequency as the objective function. Besides, Kress
[14] did the similar problem conversion in acoustic and electromagnetic scattering. Zhu [15] also discussed this method and proved its effectiveness.
On the other hand, Martin [16] applied null equations in setting up the equation system. Liapis [17] combined null equations to the original equation set, making it valid for all frequencies.
The extended-boundary-condition method was proposed by Paulling [18]. They enforced a fixed lid condition on the internal free surface. Ohmatsu [19] validated the method in the two-dimensional case.
Kleinman [20] revisited this method by strict mathematical derivation. He proved the uniqueness in the potential formulation. Rezayat et al. [21] improved the method from Schenck [4] and applied the
“lid” method in elastodynamics. Zhu [15] followed Kleinman [20], validating the effectiveness of his method. Lee et al. [22] further discussed this approach in a more general way, including the
second-order effect.
In numerical evaluation, special care is needed for the integral of the Green function across free surface panels. It is because there will exist a log singularity if the panel is located on z=0 [
23,24]. Newman and Sclavounos [25] proposed one method to evaluate the log singularity, however, the important information is missing about the final expression and assumptions. Based on the idea of
converting the integral across the pyramid bottom surface to that on the surrounding four surfaces, we have developed our own method for evaluation, ending up with different expressions but with
accuracy up to 10^−7 when comparing against Newman's results in the free surface panels only.
From the timeline of the development, the extended-boundary-condition method gradually showed its advantages. It is convenient to use, especially for users without abundant experience with such
problems. In this paper, we will briefly review the modified-integral method and extended-boundary-condition method. The incomplete parts in the previous literature are clarified. In the
implementation section, we will adopt the latter one. To achieve better accuracy, the integral of the wave Green function at the internal free surface will be discussed. Finally, the irregular
frequency removal effect will be evaluated for the single-body and two-body cases.
In this section, we will discuss the methodology of irregular frequency removal and the numerical evaluation of the wave Green function terms.
Formulation of Boundary Value Problem.
The Cartesian coordinate setting is indicated in Fig. 1: V[–] stands for the internal volume of the floater, bounded by the boundary surface S[b] and S[i]; S[f] stands for the outside free surface; V
is the fluid domain, bounded by free surface S[f], body surface S[b], bottom S[B], and infinite control surface S[c]. n is the unit normal vector pointing outside fluid domain V, inside the floater
Herein, we will implicitly consider
to be the definition domain. If eliminating the evaluation on
, the remaining part will naturally corresponds to the discussion of only
to be the definition domain. Besides, we adopt the deep water assumption. The Green function based on Ref. [
] is defined as
where, $f*=ω2L/g, ρ=[(x−ξ)2+(y−η)2](1/2)$, $r=[(ρ)2+(z−ζ)2](1/2), r′=[(ρ)2+(z+ζ)2](1/2), h=f*ρ, v=f*(z+ζ)$, J[0] is the Bessel function of the first kind of order 0, and R[0] is the function
defined in Telste and Noblesse [24].
The following case is the basic formulations for boundary integral problems when the field point is located on the body boundary. If the point of interest
approaches the body boundary from
, for the external potential
, we have
$2πϕ(x)=∬Sb∂ϕ(ξ)∂nξG(x;ξ)dSξ−∬Sbϕ(ξ)∂G(x;ξ)∂nξdSξ, x∈Sb$
For internal potential
$−2πϕi(x)=∬Sb∂ϕi(ξ)∂nξG(x;ξ)dSξ−∬Sbϕi(ξ)∂G(x;ξ)∂nξdSξ, x∈Sb$
The −2
is a result of consistent normal direction with that defined by the external domain. Taking the subtraction, then
$2π[ϕ(x)+ϕi(x)]=∬Sb[∂ϕ(ξ)∂nξ−∂ϕi(ξ)∂nξ]G(x;ξ)dSξ−∬Sb[ϕ(ξ)−ϕi(ξ)]∂G(x;ξ)∂nξdSξ, x∈Sb$
In this case of enforcing $ϕ(ξ)=ϕi(ξ)$ on S[b], we will get the source distribution equation; if $∂ϕ(ξ)/∂nξ=∂ϕi(ξ)/∂nξ$, then, we will get the doublet distribution equation.
Mathematical Background.
The final output is based on the solution of the potential or source strength. To get the potential value from body boundary conditions, we need to solve the following equation:
$2πϕ(x)+∬Sbϕ(ξ)∂G(x,ξ)∂nξdSξ=∬Sb∂ϕ(ξ)∂nξG(x;ξ)dSξ, x∈Sb$
If converting it into a matrix form, we get
From the theory of ordinary differential equations, we assume that the matrix equation has its homogeneous form. Thus, there will be a homogeneous solution and a particular one: $ϕ=ϕh+ϕp$. Based on
the property of matrices, the following conclusion can be drawn:
• (a)
If $det[A(ω)]=0, ϕh=0, ϕp$ has only one solution.
• (b)
If $det[A(ω)]=0, ϕh$ has infinite solutions, $ϕp$ has infinite solutions or none, depending on the value of [b].
Therefore, the second case needs to be avoided, and it is necessary to modify the structure of the matrix [A] to become [A*]. The objective is to make the $rank(A*)=rank(ϕ)=rank(A*,b)=n$ to ensure
the uniqueness of the solution. In other words, the homogeneous solution must be zero and only zero.
There are two general approaches to achieve this: One is to construct [A*](m×n), where m>n, rank(A*)=n, n is the number of unknowns. This method will result in an overdetermined matrix. In some
cases, [A*] has the same rank with [A], but a different structure.
The other is to construct a square matrix [A**](m×m), where rank(A**)=m, the number of unknowns equal to m. This method needs more unknowns and needs to extend the boundary conditions, ensuring
it is still a square matrix.
Methods to Remove Irregular Frequency
Method I: Modified Green Function Method.
The essence of this method is to find more equations which the unknowns could satisfy, leading to an overdetermined system but guaranteeing a unique solution. More exactly, it is to choose some
points inside the domain
that the unknowns could still satisfy some conditions. For the external potential
, we have
$2πϕ(x)=∬Sb∂ϕ(ξ)∂nξG(x;ξ)dSξ−∬Sbϕ(ξ)∂G(x;ξ)∂nξdSξ, x∈Sb$
$0=∬Sb∂ϕ(ξ)∂nξG(x;ξ)dSξ−∬Sbϕ(ξ)∂G(x;ξ)∂nξdSξ, x∈V−$
If the uniqueness of the solution can be proven, then, the irregular frequencies effects can be removed. The proof is similar to the procedure in Schenck [
]. Under the assumption that
satisfies Eqs.
. The key equation in the proof is
The conclusion is that $∬Sbϕz(∂G/∂nξ)$ is not necessarily zero for every $x∈V−$. Then A=0, ensuring that the solution is unique. So if the chosen points are not the node points of the homogeneous
solution for the Dirichlet internal potential problem, the solution will be unique. To ensure it, a sufficiently large number of interior points might be needed.
This method will result in an overdetermined problem, which can be solved using a least square approach. However, special treatment will be needed in selecting the interior points and setting up
parameters to remove the irregular frequency effect for an arbitrary shape. To make it convenient to users, the fixed lid method is discussed by Kleinman [20], Zhu [15], and Lee et al. [22].
Method II: Extended Boundary Condition Method.
The motivation of this method is to convert the overdetermined linear system into a square matrix. Then, a direct matrix solver can be utilized. Meanwhile, the irregular frequency corresponds to the
sloshing mode of interior space. Therefore, it would be natural to place a lid on the interior free surface to suppress it.
The final equations to be solved are listed here. For a detailed derivation, please refer to Liu and Falzarano [
$4π∂ϕ(x)∂nx=2πσ(x)+PV∬Sb σ(ξ)∂G(x;ξ)∂nxdSξ+∬Si σ′(ξ)∂G(x;ξ)∂nxdSξ, x∈Sb$
$4π∂ϕ−(x)∂nx=4πσ′(x)+∬Sb σ(ξ)∂G(x;ξ)∂nxdSξ+PV∬Si σ′(ξ)∂G(x;ξ)∂nxdSξ, x∈Si$
Evaluation of Green Function for Free Surface Panels
In both the potential formula and the source formula, the derivative of the Green function needs to be evaluated at the free surface. Based on Newman [23] and Telste and Noblesse [24], the
expressions for the Green function both contain a log term, which will result in a singular value when the field point is infinitesimally close to the source point and both are near the free surface.
$G(x;ξ)=1r+1r′−2kek(z+ζ)[log(r′+|z+ζ|)+(γ−log 2)+r′]+O(r′2 log r′)$
where $x$ is the position of field point, and $ξ$ is the source point. $r2=(x−ξ)2+(y−η)2+(z−ζ)2, r′2=(x−ξ)2+(y−η)2+(z+ζ)2, γ=0.577…$ is the Euler constant.
where $f*=ω2L/g, ρ=[(x−ξ)2+(y−η)2]1/2, r=[(ρ)2+(z−ζ)2]1/2, r′=[(ρ)2+(z+ζ)2]1/2, h=f*ρ, v=f*(z+ζ)$, J[0] is Bessel function of the first kind at order 0, and R[0] is the function defined in Telste
and Noblesse [24]. In the limiting case when $d→0, R0=−ln(d−v)+ln(2)−γ$ with error $O(d ln(d))$.
When the field point and the image source point become infinitesimally close, the log term inside both Green functions will become singular. However, the integral value of the log term across the
panel is still finite, depending on the panel size. Based on the numerical evaluation in Sec. 4, it is necessary to consider the log singular effect.
The Green function will have different behaviors when the field point is infinitesimally close to the source point and both are located at internal free surface. The relations are as below
$∂G∂z=kG, when z=0, ζ=0, x→ξ or x→ξ$
$∂G∂z=kG, when z→0−, ζ=0, x→ξ$
$∂G∂z=−2zr3+kG, when z→0−, ζ=0, x→ξ$
In the constant panel method, each panel has a uniform source distribution. All the internal lid panels are exactly on the z=0 surface, while the field points are inside the floater body. Thus,
Eqs. (13) and (14) are better descriptions of the practical model.
In calculating the panel effect on itself, the integral of –2z/r^3 will result in 4π. Also please note that the above three equations are derived based on Noblesse's Green function. Noblesse uses a
source point as the singular part, while Newman uses a sink point.
Evaluation of Log Singularity
When the source point is close to field point, we will have a natural log singularity in the Green function. This happens when studying the internal free surface panels. The log term can be written
$d=(x−ξ)2+(y−η)2+(z+ζ)2, v=z+ζ$
. To solve this log singular integral, the approach of Newman and Sclavounos [
] is adopted, which is to form a pyramid first and convert the bottom surface integral into that along the four triangular facets. The derivation herein is based on the panel coordinates, and the
final results are different from Newman's. The function
is harmonic in the pyramid region. Green's second identity can be used, and then, we can get
$∫S(ζ∇f−f∇ζ)·n dS+∫∑i=14Ti(ζ∇f−f∇ζ)·n dS=0L+∑i=14Li=0$
On this bottom panel,
, then
$L=−∫Sf dξdηLi=∫Ti(ζ∇f−f∇ζ)·n dS$
If the field point is at a finite distance from the panel,
is not zero. However, the log term will become a singularity only if the field point is infinitesimally close to the panel, which makes it necessary to evaluate the log integral in an alternative
way. Thus, in this case, we need to assume
→ 0 even on the triangular facets. Thus, we will have
Please note that on the triangular surface, the approximate normal vector n is pointing along negative z-axis, i.e., $(∂/∂n)=−(∂/∂z)$. From the above, we need to integrate f along the triangular
surface. It will be convenient to complete it in the polar coordinate on the triangular surface. Thus, the coordinate is described in Fig. 2.
We are more interested in the internal free surface panels. To have a better view of these angles, the panel is flipped over. That is the reason why z-axis is pointing downward.
is the Cartesian coordinate on the triangular surface. The
-axis is perpendicular to the surface BCE in Fig.
. We determine
such that
will point to the edge
, and the surface
is perpendicular to the edge
. This has ensured that the
-axis is perpendicular to edge
, and
-axis is, therefore, determined. The angle between the
-axis and negative
-axis is defined as
. For an arbitrary vector
on the surface BCE,
is the angle between the vector
, and the
-axis, and
is the angle between the vector
and the positive
-axis. Naturally, the
-axis becomes the reference axis when converting to the polar coordinate on the surface BCE. In the
coordinate system, we define
). The line equation can be
=1. If in polar coordinate and
$u=ρ cos α, v=ρ sin α$
, it is easy to get
$ρ(α)=1A0 cos α+B0 sin α=1C0 cos β$
$C0=A02+B02, β=α−δ, δ=arctan(B0/A0)$
. After solving for the following two equations:
Solve for
Based on the geometric relation, we have $cos θ=cos((π/2)−φ)cos α=sin φ cos α$.
Therefore, on the triangular surface, this equation can be transformed to
$f=ln[d(1−vd)]=ln[d(1+cos α sin φ)]$
In the discussion later, we distinguish the source point on the quadrilateral panel and those on the triangular facet. We will use
instead of
. Then, the integral equation will become
$∫α1α2dα∫0r(α)ln[r(1+cos α sin φ)]$
If enforcing the assumption that
→ 0, then
$φ→0,sin φ≈φ, cos φ≈1$
. The final result will be
$∫Tif dS=∫α1α2dα∫0r(α)ln[r(1+cos α sin φ)]rdr={− tan β2C2[lnC+ln(cos β)+32]−φ2C2[cos δln[tan(π4+β2)]− sin δ cos β]+β2C2}|β1β2$
Finally, we can get the integral of log singularity on the panel by this equation
$L=∫Sf dξdη=∑i=14∫Tif dS$
Results and Discussion
In this section, we will first verify the accuracy of our method to evaluate the log singularity, then, justify the method of subtracting the log singularity from the wavy Green function, and
finally, we evaluate the irregular removal effects.
Log Singularity.
As mentioned in Sec. 1, Newman and Sclavounos [25] did not provide complete details about the assumptions, and some other information for the final expression was also not included. To be
conservative, we choose some special cases in which Newman's final expression might be valid and applied a systematic trial-and-error approach to tune the parameters until we get similar results with
Maple. Afterward, based on the idea, we have developed our own method for evaluation, ending up with somewhat different expressions. When comparing against Newman's results, not surprisingly, they
are very close. The comparison validates our assumptions about Newman's expression and also proves our approach is accurate.
Table 1 contains the comparison results for the method in this paper and modified Newman's. As can be seen from the table, the difference is 10^−7. We can draw the conclusion that our method will be
accurate enough in evaluating the log singularity. Please note that Table 2 contains the node position vector in panel coordinate for different cases. The log analytical evaluation should be used for
panels on and near the free surface. Based on our numerical testing, when $|v|<0.05$, separate evaluation of the log singularity will be needed. The alternative approaches are discussed in Liu and
Falzarano [28].
Integral of R[0].
Since the log singularity was evaluated analytically, one may be curious about the function shape of R[0] after subtracting the log integral from it. This section will illustrate the shape and
justify a proper numerical method to evaluate R[0]. The numerical evaluation method for R[0] is from Telste and Noblesse [24].
Figures 3–6 illustrate the function value of $R0−ln(d−v)$ across a unit square panel.
As shown in Figs. 3–6, if the panel size is small, a four-node Gaussian quadrature method can still give accurate results. However, if the panel is a larger size, the wavy behavior will nullify the
four-node Gaussian quadrature. To balance accuracy and efficiency, it is preferable to construct smaller panels for the internal lid surface. Moreover, please note that in implementing the Gaussian
quadrature method, the input is dimensionless. The variable related to length is multiplied by f=ω^2L/g, where ω is the wave frequency, L is the wavelength, and g is the gravity constant. For a
floater with a large L, when wave frequency ω is relatively higher, the node position could be amplified. Therefore, the four-node quadrature method may not produce accurate enough results.
Nevertheless, the shorter wave may not lie in the range of interest for such floaters in sea-keeping analysis.
Irregular Frequency Removal Effect.
The accurate numerical evaluation of the log singularity is important to eliminate the irregular frequency effect. This section will demonstrate the irregular frequency removal effects for
single-body and multibody case. The results are generated by MDL-MultiDYN, an in-house program developed by Marine Dynamic Laboratory, Texas A&M University. This program is a redesigned program based
on MDL HydroD by Guha [29]. It is able to conduct multibody analysis with improved log singularity evaluation and an irregular frequency removal module [30]. It also has the capability to analyze the
forward speed effect of multiple floaters discussed in Liu and Falzarano [31,32] and the drift forces or added resistances addressed in Liu and Falzarano [33].
From John [1], the irregular frequency effect is more likely to happen in shorter waves. Thus, we adopted a miniboxbarge as the test case. The miniboxbarge has 500 panels. If we add a lid at the
internal free surface, the panel number will be 700. All the cases are evaluated under head sea condition. For two-body cases, the separation distance is 10m. Besides the miniboxbarge with the most
significant irregular frequency removal effects, we have also validated our approach using boxbarges side-by-side (10m apart), a cylindrical dock, the U.S. navy ships BOBO and Bob Hope. The
particulars are listed in Table 3 (unit: meter). Please note that all the cases are conducted in the head sea condition. We use 1 6 to denote the six degrees-of-freedom (DOFs: surge, sway, heave,
roll, pitch, and yaw) for the body 1, 7 12 for the body 2 in the multiple-body cases.
The irregular frequency effect is more significant in added mass and damping, phases of forces (Froude-Krylov and Diffraction (FKD)) and motion, less apparent in amplitude of forces and motion
response amplitude operator. Herein, we benchmark our results against WAMIT version 6. To follow the notation in WAMIT, we use “IRR” to denote whether the irregular frequency module is activated or
not. “IRR1” means that it is in effect, while “IRR0” means it is not.
Figures 7–14 are the results for the single miniboxbarge (Fig. 15). We may find the significant irregular frequency effect that appear in the higher frequency range. After implementing the lid
method, the irregular frequency effects are removed. However, we may observe a tiny discrepancy appearing in the results for added mass A15, B11, B55 near the irregular frequency. The relative error
is very small. The discrepancy may be caused by the numerical error in the discretization of the analytical equation. For example, it may be affected by the method we define the collocation point at
which the potential is calculated or the way we calculate the part of wavy Green function. The methods of our in-house program were discussed by Guha [29]. In the calculations of drift forces, we
observe an interesting phenomenon. In the drift force along the surge direction, we have obtained totally different results compared against WAMIT. The results from MDL multi-DYN are in dashed line,
while those from WAMIT are in dotted line. We find the dotted line has obvious discrepancy against the results when the irregular frequency module is not invoked. The dashed line is closer to the
results when not removing the irregular frequencies. All of the spikes almost disappear on the dashed line. We believe that our results are more reasonable. However, it is still not clear about the
causes of the discrepancy.
Figures 16–21 are the results for the two miniboxbarges (Fig. 22), side-by-side. From the results, we may find that the irregular frequencies are more likely to happen in the higher frequency range
and are successfully removed. From the results for the damping term B57, we may find that one spike is due to the irregular frequency, another spike nearby is due to the physical resonance. This has
validated our assumption that the irregular frequency confuses with the true resonance. Thus, in the cases of the box-shape floaters, we must invoke the irregular frequency module.
Figures 23–26 are the results for case of the larger boxbarge (Fig. 27). For the larger-size boxbarge, the irregular frequency effect happens in the relatively higher frequency range as well. It is
less significant compared with cases of the miniboxbarge. By comparison against the miniboxbarge, it also shows that the significance of the irregular frequency effect may be influenced by the sizes
of the floater. It is recommended to run the cases without invoking the module and with the module activated.
Figures 28–31 show the irregular frequency effect of the cylindrical dock (Fig. 32). It is also successfully removed. Similar to the large box barge, the irregular frequency effect is not
significant. The nondimensional irregular frequency in the figures is around 2. It shows that the irregular frequency is an intrinsic property of the shape of the floater. It should be removed to
obtain more accurate results.
When coming to the verification of the ships, we also output the results from WAMIT (Lee [34]) to demonstrate the effectiveness of our irregular frequency removal module. Figures 33–36 are the
results for the ship Bob Hope (Fig. 37). The irregular frequency effect is not significant. However, it is still removed. In the results of damping term B15, we find that the discrepancy between the
results with irregular frequency inactive and active is more obvious compared against the cases of the miniboxbarge. The reason is not clear yet. We will investigate it more in the future.
Figures 38–43 show the results for ship BOBO (Fig. 44). In the single body case, the irregular frequency effect is very significant in some components in the added mass matrix, for example, A13, A33,
etc. For this case, we also find that the irregular frequency effects exist in the drift forces of the floater. The results in the heave drift force and pitch drift force are also presented here to
demonstrate the effectiveness of our method.
Figures 45–50 are the results when ship BOBO is next to the ship Bob Hope (Fig. 51). The separation distance is 3m. Similar to the miniboxbarge case, the results prove that the irregular frequency
effect will confuse the resonance in the relatively higher frequency range. For this case, the irregular frequency effect may exist in a small range of the frequency. For example, in the results of
added mass terms A11 and A55, we observe that the discrepancy exists when nondimensional frequency varies from 6 to 7. In the results of the added mass term A33 and the damping term B11, the
irregular frequency effect results in a single spike. For the interactions of multiple bodies, it will be necessary to remove the irregular frequency effect.
This paper reviews the reason of the irregular frequency effects, evaluates the modified integral Green function approach and extended boundary condition approach for removing the irregular frequency
effects, thoroughly discusses the analytical method to evaluate the log singularity term inside wavy Green function, and finally, validates the irregular frequency removal effect of in-house program
“MDL MultiDYN.”
The comparison of the log singularity ensures the high accuracy of our method. Finally, the irregular frequency removal module is effective based on the figures in the results section. Therefore, the
incorporation of this module will enhance the capability of MDL MultiDYN on multibody problems. For the interactions of multiple bodies, the irregular frequency module must be invoked to ensure a
higher accuracy in capturing the true resonance. Moreover, the irregular frequency effects in the drift forces still require more investigation in the future.
The authors would like to thank Dr. Paul Hess for supporting this work. The authors also acknowledge the partial funding provided by Society of Naval Architects and Marine Engineers (SNAME) to
support this work. Finally, the authors would thank Dr. Francis Noblesse for his suggestions on Green function evaluations.
• Office of Naval Research (N00014-16-1-2281).
, “
On the Motion of Floating Bodies II
Commun. Pure Appl. Math.
), pp.
, “
Oscillation of Cylinders in or Below the Free Surface of Deep Fluids
,” Naval Ship Research and Development Center, Bethesda, MD, Technical Report No.
, “
Short Surface Waves Due to an Oscillating Immersed Body
Proc. R. Soc. A: Math., Phys. Eng. Sci.
), pp.
H. A.
, “
Improved Integral Formulation for Acoustic Radiation Problems
J. Acoust. Soc. Am.
), pp.
A. J.
, and
G. F.
, “
The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary-Value Problems
Proc. R. Soc. A: Math., Phys. Eng. Sci.
), pp.
, “
Integral Equations for the Exterior Acoustic Problem
Q. J. Mech. Appl. Math.
), pp.
T. F.
, and
Y. S.
, “
Integral Equation Solutions for Time Dependent Free Surface Problems
J. Soc. Nav. Arch. Jpn.
), pp.
, “
An Integral-Equation Method for Determining the Fluid Motion Due to a Cylinder Heaving on Water of Finite Depth
Proc. R. Soc. London, Ser. A
), pp.
, “
Irregular Frequencies and the Motion of Floating Bodies
J. Fluid Mech.
, pp.
, and
, “
A Multiple Green's Function Expression for the Hydrodynamic Analysis of Multi-Hull Structures
Appl. Ocean Res.
), pp.
S. M.
, and
G. E.
, “
Suppression of Irregular Frequency Effects in Fluid-Structure Interaction Problems Using a Combined Boundary Integral Equation Method
Int. J. Numer. Methods Fluids
), pp.
, and
P. D.
, “
Removing the Irregular Frequencies From Integral Equations in Wave-Body Interactions
J. Fluid Mech.
, pp.
, “
Numerical Methods for Boundary Integral Equations in Wave Body Interactions
,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
, “
Minimizing the Condition Number of Boundary Integral Operators in Acoustic and Electromagnetic Scattering
Q. J. Mech. Appl. Math.
), pp.
, “
On the Null-Field Equations for Water-Wave Radiation Problems
J. Fluid Mech
, pp.
, “
A Method for Suppressing the Irregular Frequencies From Integral Equations in Water Wave-Structure Interaction Problems
Comput. Mech.
), pp.
, “
On the Irregular Frequencies in the Theory of Oscillating Bodies in a Free Surface
,” Ship Research Institute, Tokyo, Japan, Paper No.
R. E.
, “
On the Mathematical Theory of the Motion of Floating Bodies: An Update
,” David W. Taylor Naval Ship Research and Development Center, Bethesda, MD, Report No.
, and
, “
On Time-Harmonic Elastic-Wave Analysis by the Boundary Element Method for Moderate to High Frequencies
Comput. Methods Appl. Mech. Eng.
), pp.
J. N.
, and
, “
An Extended Boundary Integral Equation Method for the Removal of Irregular Frequency Effects
Int. J. Numer. Methods Fluids
), pp.
J. N.
, “
Algorithms for the Free-Surface Green Function
J. Eng. Math.
), pp.
, and
, “
Numerical Evaluation of the Green Function of Water-Wave Radiation and Diffraction
J. Ship Res.
), pp.
, and
, “
The Computation of Wave Loads on Large Offshore Structures
International Conference on Behaviour of Offshore Structures
Trondheim, Norway
, pp.
, “
The Green Function in the Theory of Radiation and Diffraction of Regular Water Waves by a Body
J. Eng. Math.
), pp.
, and
J. M.
, “
Irregular Frequency Removal Methods: Theory and Applications in Hydrodynamics
J. Mar. Syst. Ocean Technol.
), pp.
, and
J. M.
, “
A Method to Remove Irregular Frequencies and Log Singularity Evaluation in Wave-Body Interaction Problems
J. Ocean Eng. Mar. Energy
), pp.
, and
J. M.
, “
Suppression of Irregular Frequency in Multi-Body Problem and Free-Surface Singularity Treatment
Paper No. OMAE2016-54957.
, and
J. M.
, “
Frequency Domain Analysis of the Interactions Between Multiple Ships With Nonzero Speed in Waves or Current-Wave Interactions
,” ASME Paper No. OMAE2017-62322.
, and
J. M.
, “
A Note on the Conclusion Based on the Generalized Stokes Theorem
J. Offshore Eng. Technol.
, in press.
, and
J. M.
, “
Improvement on the Accuracy of Mean Drift Force Calculation
Paper No. OMAE2017-62321.
, “
WAMIT Theory Manual
,” Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA. | {"url":"https://heattransfer.asmedigitalcollection.asme.org/offshoremechanics/article-split/139/5/051101/376604/Suppression-of-Irregular-Frequency-Effect-in","timestamp":"2024-11-04T10:56:00Z","content_type":"text/html","content_length":"634493","record_id":"<urn:uuid:b0b7e6af-cf90-471c-ae7e-b33d1d1d5baa>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00038.warc.gz"} |
HYPERSIM Documentation
This model is based on the ST4B excitation system in which is applied a low-pass filter to the terminal voltage transducer output, EC. This model is a variation of the Type ST3A model, with a
proportional plus integral (PI) regulator block replacing the lag-lead regulator characteristic that was in the ST3A model. Both potential-and compound source rectifier excitation systems are modeled
as shown in the figure below. The PI regulator blocks have nonwind up limits. The voltage regulator of this model is typically implemented digitally, so the model is identified with the suffix “B.”
IEEE, "IEEE Recommended Practice for Excitation System Models for Power System Stability Studies," in IEEE Std 421.5-2016
The other features of the regulator are a low value gate for the OEL limit function, and the UEL and V/Hz control are summed into the input to the regulator. This means that on a unit with PSS
control, the PSS will be active if the unit goes into UEL limit control, unlike some previous designs that had take-over type limiters. There is flexibility in the power component model to represent
bus-fed exciters (KI and XL both equal to zero), compound static systems (XL = 0), and potential- and compound-source systems where XL is not zero. The appropriate PSS model to use with the ST4B
excitation model is Type PSS2B.
Mask and Parameters
AVR Parameters
Expanding the "AVR diagram" displays the block diagram in the parameters window.
Tr Regulator input filter time constant s
Kpr Voltage regulator proportional gain -
Kir Voltage regulator integral gain -
Vrmin Minimum voltage regulator output pu
Vrmax Maximum voltage regulator output pu
Ta Voltage regulator time constant s
Kg Feedback gain constant of the inner loop field regulator -
Kpm Voltage regulator proportional gain -
Kim Voltage regulator integral gain -
Vmmin Minimal output factor of converter bridge corresponding to firing angle command to thyristors pu
Vmmax Maximal output factor of converter bridge corresponding to firing angle command to thyristors pu
Exciter Parameters
Expanding the "Exciter diagram" displays the block diagram in the parameters window.
Kp Potential circuit real part gain coefficient -
Kj Potential circuit imaginary part gain coefficient -
Xl Reactance associated with potential source pu
Kc Rectifier loading factor proportional to commutating reactance -
Vbmax Maximum available exciter voltage pu
Initial Values
Efd0 Initial exciter output voltage pu
Ifd0 Initial exciter output current pu
Inputs, Outputs and Signals Available for Monitoring
E[C] Output of terminal voltage transducer and load compensation elements pu
V[UEL] Underexcitation limiter output pu
V[S] Is defined as the output voltage of a Power System Stabilizer (PSS) [1]. pu
V[REF] Voltage regulator reference voltage pu
V[T] Synchronous machine terminal voltage pu
I[T] Synchronous machine terminal current pu
I[FD] Synchronous machine field current pu
EFD Exciter output voltage pu
[1] "IEEE Recommended Practice for Excitation System Models for Power System Stability Studies," in IEEE Std 421.5-2016 (Revision of IEEE Std 421.5-2005) , vol., no., pp.1-207, 26 Aug. 2016, doi: | {"url":"https://opal-rt.atlassian.net/wiki/spaces/PDOCHS/pages/939983629/ESST4B?atl_f=content-tree","timestamp":"2024-11-06T10:44:40Z","content_type":"text/html","content_length":"1050365","record_id":"<urn:uuid:b33c7610-2434-45f3-a656-f66b34898c7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00603.warc.gz"} |
Erlang: Functions (Part 4).
As we have seen in the previous articles, functions elevate some tiny little things to the new heights. I find it really cool you could create something new in terms of existing things just by
combining them in the right way (composition). Functions also allow you to manipulate complex objects if you guarantee you could break them to smaller pieces (recursion). Combined together these two
principles create a universe of beautiful solutions. In this article we're going to explore some new forms of composition and recursion.
Sometimes recursion could lead you to unexpected results. We once created a hyper operator by just utilizing the successor and predecessor functions. The predecessor function (just to remind you) is
a function that decrements its only parameter by one. If we start applying the predecessor function to any positive integer over and over we would eventually reach zero (which is an even number as
you might know). Thinking this way we could try and implement two functions that test positive integers for being even or odd numbers.
-export([is_even/1, is_odd/1]).
is_even(0) -> true;
is_even(N) -> is_odd(N - 1).
is_odd(0) -> false;
is_odd(N) -> is_even(N - 1).
Here we introduce a few new concepts. First of all, functions that return a boolean value are called predicates (yet this is not a complete definition and the term itself has different
interpretations, if you need more information please start here). So, is_even and is_odd are both unary predicates. It's also very common to utilize the is prefix for unary predicates when naming
your functions.
Secondly, an interesting fact here is that both predicates have mutual base cases. If we're calling is_even and the number is zero, then the answer is true, because zero is even. Otherwise, we're
decrementing the argument by 1 and giving the control to the is_odd function. If at this point the argument is zero then the result is false, otherwise the argument is decremented again and is_even
is called. And so on and so forth until it reaches zero and stops the recursion.
Functions with mutual base cases represent so called mutual recursion. While in everyday programming you will face such recursion rarely it might be common in such tasks as writing parsers and
producing some sequences for analytical purposes.
You might be also wondering why implement such simple thing that way while there are generic and performant alternatives. Let me illustrate them.
is_even2(N) -> N rem 2 =:= 0.
is_even3(N) -> 1 band N =:= 0.
I'm omitting is_odd here; those are just to illustrate the idea after all. You could see that the same function could be implemented in many different ways and it's your responsibility to utilize the
version that suits your needs best. However while working with negative integers both the above functions require you to have either rem (integer remainder) or band (bitwise and) operator to be
On the other hand our initial version of the is_even and is_odd functions requires nothing but the predecessor function, which could be easily replaced by an analogue operation over some other
non-integer structure (for instance, tuples). This might be confusing for now but this topic will be explained in detail as soon as we discuss functions and their relation to types.
Now, let's see how we could filter out a list of integers so that if you provide a lists of integers it would return you either a list of odd or even numbers depending on which version you're using.
filter_even([]) -> [];
filter_even([H|T]) ->
case is_even(H) of
true -> [H|filter_even(T)];
false -> filter_even(T)
filter_odd([]) -> [];
filter_odd([H|T]) ->
case is_odd(H) of
true -> [H|filter_odd(T)];
false -> filter_odd(T)
You might notice these functions are almost identical. The only difference is the exact function we call to test a single number. If only we had a mechanism to generalize such cases...
Erlang is a functional language. Among other properties Erlang treats functions as first-class citizens. It means we could assign functions to variables, test them for equality, consider functions as
arguments of other functions and also return them as a result of evaluation of other functions. In other words functions in Erlang are in fact like all other types we use regularly.
To refer to a function we use the keyword fun followed by either the function's name or a combination of module and function depending on where we refer it from and the function's arity. For
instance, if we refer to the is_even function from where it's defined (lesson4), we could just use the short form which is fun is_even/1.
The syntax could be generalized as fun M:F/A, where MFA stands for module, function and arity. You should also keep in mind that any component could be either constant or variable.
1> IsOdd = fun lesson4:is_odd/1.
fun lesson4:is_odd/1
2> IsOdd(42).
3> M = lesson4.
4> F = is_even.
5> A = 1.
6> IsEven = fun M:F/A.
fun lesson4:is_even/1
7> IsEven(3).
8> fun lesson4:is_even/1(0).
It's not always necessary to assign a reference to a variable to call it. Instead you could just call it right away by providing its arguments, as you could see in the very last example (step 8). You
could also wrap a reference in parentheses before calling it which is a common way to do it. Here is an example.
1> (fun lesson4:is_even/1)(6).
The fact we could associate any function with a variable and then call it means we could now implement a generic version of the filter function. Let's do this.
filter(_, []) -> [];
filter(P, [H|T]) ->
case P(H) of
true -> [H|filter(P, T)];
false -> filter(P, T)
Filter is a higher-order function. Higher-order functions (unlike first-order functions) take other functions as arguments or return other functions as a result, elevating our ability to abstract
things to a new level.
To call the filter function you should provide a reference to some unary predicate. Let's illustrate it by simplifying the filter_even and filter_odd functions.
filter_even_simple(L) -> filter(fun is_even/1, L).
filter_odd_simple(L) -> filter(fun is_odd/1, L).
One important thing I have to mention is that our generic filter function is already implemented as a part of the lists module. While it's fun to write our own versions of generic and fundamental
functions (not just fun but also really useful for educational purposes) it's always a good idea to utilize Erlang's built-in functions (BIFs) when possible due to performance reasons. It's also a
good practice to utilize the Standard Library even if you know some functions aren't BIFs because these modules and functions are well tested and documented.
Now let's get back to the predicates. It's sometimes useful to check whether at least one element in the list satisfies some condition or conditions. This task could be generalized as a higher-order
function which is a predicate itself called any. This function takes a predicate P (that's intended to test a single element in the list) and the list itself and returns true if at least one element
of the list satisfies the predicate. Otherwise it returns false.
any(_, []) -> false;
any(P, [H|T]) -> P(H) orelse any(P, T).
Oh boy, I do love this function! It's quite unique among all the previous functions we have seen so far because it has two base cases one of which is at the same clause that forms the recursion. I
mean, look at the second clause, you could see here is the orelse operator! It does work in that way so we call any (which is the second operand) if only the first operand is false. Hence, P(H) is a
base case! Let's do some tests first to get a better understanding.
1> false orelse 1.
2> true orelse 1.
3> true andalso 1.
4> false andalso 1.
5> 2 < 1 andalso io:fwrite("you won't see this~n").
The andalso and orelse operators represent so called Short-Circuit Evaluation, means their second operand won't be evaluated at all if not needed. In the very last example you could see the io:fwrite
function is not evaluated because 2 is more than 1 hence the equation is false already and there is no need to evaluate it further.
You could also notice that unlike in and and or the second operand could be evaluated to anything, not just some boolean value. And one last note is that you could use andalso and orelse in guards
(we will discuss it one day if there is a luck).
Any has its dual called all. The all function is almost the same but it does return true if all elements in the list satisfy the given predicate.
all(_, []) -> true;
all(P, [H|T]) -> P(H) andalso all(P, T).
As I said both functions are almost the same but you could notice the first clauses return different boolean values. That's because in the case of all if all elements satisfy the given predicate then
the recursion terminates when calling the all function against an empty list so that it must return true, otherwise it would produce incorrect results. In the case of any if all elements do not
satisfy the given predicate then the very last call would be made against an empty list, hence the result should be false (there are no elements that satisfy P).
Now let me introduce the zip function. This function takes 2 lists and combines them into a single list so that each element of each source list would be combined into a binary tuple. Let me show you
how it works.
1> lists:zip([1, 2, 3], [a, b, c]).
Cool, huh? Well, one thing though... I did mention you should always be using the Standard Library whenever possible. But for some weird reason the zip function that's provided by the lists module
does only work when there are 2 lists of same size. Giving it lists of different sizes would simply produce an error. So we're going to implement our own version!
% differs from lists:zip!
zip([HX|TX], [HY|TY]) -> [{HX, HY}|zip(TX, TY)];
zip(_, _) -> [].
Let's now try it in erl.
1> lesson4:zip([1, 2, 3], [a, b, c, d]).
Splendid! Now, let me show you something. Consider we have a list of integers. If we try to combine it with its own tail (the list itself minus the head element) we would get each element paired with
the element next to it. Let's see.
1> Numbers = [1, 2, 3, 4, 5].
2> lesson4:zip(Numbers, tl(Numbers)).
We could now use this property to define a predicate that checks whether a list is sorted or not.
gte({X, Y}) -> X =< Y.
lte({X, Y}) -> X >= Y.
is_sorted([]) -> true;
is_sorted([_|T] = L) ->
Pairs = zip(L, T),
all(fun gte/1, Pairs) orelse
all(fun lte/1, Pairs).
I suppose the gte and lte functions are simple and straightforward. The is_sorted predicate needs some little clarification though. If you take a look in the line number 7 we define a variable called
Pairs. That's because the result of calling the zip function is going to be used twice in the worst scenario (when the first all evaluates to false). So, involving a variable here is quite reasonable
as we're getting rid of unnecessary computations making the function more efficient. Though, involving variables in functions is an advanced topic (surprise!) as it requires you to understand
scoping. I do recommend you to avoid variables in functions during learning Erlang. This would force you to keep your functions short and simple and as a result teaches you good practices. Scoping
rules will be discussed in one of the upcoming articles.
Let's see how is_sorted works.
1> lesson4:is_sorted([1, 2, 3, 4, 5, 6]).
2> lesson4:is_sorted([1, 2, 3, 5, 4, 6]).
3> lesson4:is_sorted([6, 5, 4, 3, 2, 1]).
You could see that our predicate does work correctly for both ascending and descending orders, that's why we needed to involve 2 calls of the all function.
That's it for now. Play with the functions from this article, write your own ones and read more about higher-order functions using the links you could find here or over the Internet. In the next
article (articles?) I'm going to introduce you Funs and explain why these tiny little things are really a new level to abstract things. Stay tuned!
P.S.: Don't forget to inspect and download the sources.
August 08, 2020 | {"url":"https://kduman.com/post/20200808/erlang-functions-4/","timestamp":"2024-11-07T03:47:02Z","content_type":"text/html","content_length":"21495","record_id":"<urn:uuid:26d142f3-58f1-441c-8233-ca47d66fda58>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00381.warc.gz"} |
Dynamics of kink instability in a non-uniform magnetoplasma
Accounting for an external electron current gradient, a set of nonlinear fluid equations governing the dynamics of kink instability in an inhomogeneous magnetized plasma has been derived. In the
linear regime, the dispersion relation is analysed and the variation of the growth rate is graphically shown. In the nonlinear regime, it is shown that a quasi-stationary solution of the mode
coupling equations can be represented as a dipolar vortex. Conditions under which the latter arises are given.
Journal of Plasma Physics
Pub Date:
October 1987
□ Magnetohydrodynamic Stability;
□ Nonuniform Plasmas;
□ Plasma Frequencies;
□ Plasma Pinch;
□ Coupled Modes;
□ Electric Fields;
□ Plasma Currents;
□ Plasma Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1987JPlPh..38..309B/abstract","timestamp":"2024-11-10T21:50:10Z","content_type":"text/html","content_length":"35691","record_id":"<urn:uuid:94862b2b-fa0e-4449-b05e-bd99e1dfb345>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00403.warc.gz"} |
How do you find the T in T distribution?
The formula to calculate T distribution (which is also popularly known as Student’s T Distribution) is shown as Subtracting the population mean (mean of second sample) from the sample mean ( mean of
first sample) that is [ x̄ – μ ] which is then divided by the standard deviation of means which is initially Divided by …
How do you calculate t0?
To find the t value:
1. Subtract the null hypothesis mean from the sample mean value.
2. Divide the difference by the standard deviation of the sample.
3. Multiply the resultant with the square root of the sample size.
How do you find the t-value for a 95 confidence interval?
The t value for 95% confidence with df = 9 is t = 2.262.
How do you find t statistic on TI-84?
How to Perform a One Sample t-test on a TI-84 Calculator
1. Step 1: Select T-Test. Press Stat. Scroll over to TESTS.
2. Step 2: Fill in the necessary info. The calculator will ask for the following information:
3. Step 3: Interpret the results. Our calculator will automatically produce the results of the one-sample t-test:
How do you do t on a TI-84 Plus?
While your TI-84 is in Polar mode, press the [X,T,θ,n] key (just below the Mode key) to select and insert θ, together with any other characters you require for your expression.
How do you find T in stats?
Calculate the T-statistic Take the value you got from subtracting μ from x-bar and divide it by the value you got from dividing s by the square root of n: (x-bar – μ) ÷ (s ÷ √[n]).
What is the T value of 95 percentile?
Thus, the 95th percentile (aka 0.95 quantile) of the t(df=3) distribution is 2.353. (See the picture below.)
How do you use the t-distribution table of values?
In the t-distribution table, find the column which contains alpha = 0.05 for the two-tailed test. Then, find the row corresponding to 20 degrees of freedom. The truncated t-table below shows the
critical t-value. The t-table indicates that the critical values for our test are -2.086 and +2.086.
What is the t-value for 90 confidence interval?
The T-distribution
Confidence Level 80% 90%
Degrees of Freedom (df)
1 3.078 6.314
2 1.886 2.920
3 1.638 2.353
How do you find the t-value given the confidence level and sample size?
critical value: The critical t -value for a given confidence level c and sample size n is obtained by computing the quantity tα/2 t α / 2 for a t -distribution with n−1 degrees of freedom.
How do you find the T distribution on a TI 83 Plus?
Step 1: Press the “STAT” key, then press the left arrow key to arrive at the Tests menu. Step 2: Press “8” for TInterval. Step 3: Arrow right to choose STATS. Press ENTER….How to Find T Critical
Value on TI 83: Steps
1. Set the mean (Xbar) to 0.
2. Set Sx to the square root of your sample size.
3. Set n as your sample size.
When you do the t-test on a TI calculator you need to set?
Example: performing a t-test on the calculator
• Step 1: Write the null and alternative. hypotheses.
• Step 2: Calculate the p-value using your calculator and the correct test.
• Step 3: Compare the p-value to the significance level alpha and make your decision.
• Step 4: Interpret your decision in terms of the problem. | {"url":"https://pleasefireme.com/popular-questions/how-do-you-find-the-t-in-t-distribution/","timestamp":"2024-11-02T05:35:46Z","content_type":"text/html","content_length":"57227","record_id":"<urn:uuid:13d403d6-1908-4e1b-ad04-1ecd50c71d15>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00238.warc.gz"} |
Genuine N -partite entanglement without N -partite correlation functions
Title Genuine N -partite entanglement without N -partite correlation functions
Publication Journal Article
Year of 2017
Authors Tran, MC, Zuppardo, M, de Rosier, A, Knips, L, Laskowski, W, Paterek, T, Weinfurter, H
Journal Physical Review A
Volume 95
Issue 6
Pages 062331
Date 2017/06/26
A genuinely N-partite entangled state may display vanishing N-partite correlations measured for arbitrary local observables. In such states the genuine entanglement is noticeable solely
in correlations between subsets of particles. A straightforward way to obtain such states for odd N is to design an “antistate” in which all correlations between an odd number of
observers are exactly opposite. Evenly mixing a state with its antistate then produces a mixed state with no N-partite correlations, with many of them genuinely multiparty entangled.
Abstract Intriguingly, all known examples of “entanglement without correlations” involve an odd number of particles. Here we further develop the idea of antistates, thereby shedding light on the
different properties of even and odd particle systems. We conjecture that there is no antistate to any pure even-N-party entangled state making the simple construction scheme unfeasible.
However, as we prove by construction, higher-rank examples of entanglement without correlations for arbitrary even N indeed exist. These classes of states exhibit genuine entanglement and
even violate an N-partite Bell inequality, clearly demonstrating the nonclassical features of these states as well as showing their applicability for quantum information processing.
URL https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.062331
DOI 10.1103/PhysRevA.95.062331 | {"url":"https://quics.umd.edu/publications/genuine-n-partite-entanglement-without-n-partite-correlation-functions","timestamp":"2024-11-04T13:31:25Z","content_type":"text/html","content_length":"25029","record_id":"<urn:uuid:6a5dfb36-6817-444c-aa4e-eca76e4d28fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00174.warc.gz"} |
I Stopped Letting My Students Use Calculators in Class! Here's Why...
This article will push some peoples buttons I’m sure. However, I feel it is something that needs discussed. My school is one to one with iPads. Here’s my concern.
Why do we need calculators anymore?
What is the reasoning behind needing to learn to use the next new TI Calculator? Am I missing something or are these big bulky calculators completely obsolete and only being used because of testing?
When in the real world is a student ever going to whip out a big graphing calculator and use it? When would they ever choose that tool over an app on their iPad or phone so solve a problem?
My students NEVER use graphing calculators ever!
We use apps for everything. There is a fine line between teaching to tests, teaching students what they need to know about the content, and PREPARING THEM FOR THE REAL WORLD. I lean towards teaching
them to use current and updated technology. Most of our students will not go on to be math majors in college but they will have to know how to use the latest technologies.
Are we doing our students a disservice by using ancient technology?
My opinion is YES! I want my students to look back on their experience in my class and say “he taught me to think outside the box and that there is always a way to find the answer I need” rather than
giving up because they need a graphing calculator or some other form of outdated technology that they forget how to use or worse yet isn’t even for sale anymore.
I would like to know how many you are still using graphing calculators in your class room. Not ones on tablets, but actual graphing calculators like a TI-84+ or a TI-Inspire.
In the comments below we would love to hear your feedback and how you handle this in your classrooms!
Are You Looking for a New Algebra 2 Curriculum to Use in Class??
Why I Stopped Letting My Students Use Calculators in Class! Pint It! | {"url":"https://algebra2coach.com/i-stopped-letting-my-students-use-calculators-in-class/","timestamp":"2024-11-05T20:04:53Z","content_type":"text/html","content_length":"114164","record_id":"<urn:uuid:f8a97548-2ca6-44f9-b9f6-9a176ea0d5b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00609.warc.gz"} |
How to Determine Fluorescence Lifetimes | Fluorescence Fitting
A photoluminescent or fluorescent decay can be analysed to extract the lifetime(s). When fitting a decay, the sample’s underlying photophysical processes must be considered to help evaluate the
appropriateness of the fitting. Here, we will explore the analysis of single, multi, and non-exponential decays using discrete component fitting in Fluoracle®.
Key Points
• Measured decays may be defined by multiple emissive populations.
• Emissive populations with different lifetimes can be due to many sample-specific reasons.
• A system should always be fitted with the lowest number of components possible.
• The average lifetime of a system can be described by the amplitude or intensity weighted average lifetime.
• Evaluation of a fit can be optimised computationally, but suitability should be evaluated according to knowledge of the system.
Excited State Populations
Time-Correlated Single-Photon Counting examines the rate of decay for excited state populations, [M*]. The decay in the concentration of excited state molecules, [M*], at time, t, can be written as:
Here, [M*]0 is the concentration of molecules in the excited state at time = 0. Time = 0 is equivalent to the arrival time of the excitation pulse in TCSPC fluorescence measurements. There are
multiple competing deactivation processes, split into radiative and non-radiative pathways. The rate constants associated with these parameters are k[r] and k[nr], the rate constants for the
radiative and sum of the non-radiative processes respectively. The term k may be introduced to signify the sum of all rate constants.
However, we can’t directly measure the concentration of excited state molecules. Instead, a measurable parameter is the fluorescence intensity, I. The relationship between the fluorescence intensity
at a time, t, and the decay in the fluorescence intensity is shown in Equation 2.
where τ is the fluorescence lifetime. τ is defined in relation to the rate constants according to:
How to Fit Single Exponentials Decays
The simplest fluorescence decays exhibit single exponential behaviour:
where t is time, τ is the fluorescence lifetime, and B is the pre-exponential factor. The fluorescence lifetime is defined as the time it takes the intensity to drop to 1/e (=0.368) of its initial
value (Figure 1a). Fluorescence decays are most often shown on a logarithmic scale which gives a linear response for a single exponential decay (Figure 1b).
Figure 1. Representation of a fluorescence decay following a short excitation pulse on a (a) linear and (b) logarithmic scale.^1
Fitting Methods
To extract components from a decay, it must be fitted to an appropriate function. There are two common methods of fitting TCSPC data, tail fitting and reconvolution fitting. Over multiple iterations,
the parameters of lifetime and B-factor are varied to optimise the fit to the collected decay.
When a sample has long fluorescence lifetimes, tail fitting can be used. This involves fitting from the peak of the decay without convolution with the Instrument Response Function (IRF).
When studying short-lived excited state populations, iterative reconvolution is used. The theoretical decay function is convoluted with the measured IRF.
Fitting Evaluation
The fit can be evaluated by calculation of the χ^2. This function calculates the difference between the raw data points and the fitting points to quantify how well the data has been replicated.
The difference between the measured fluorescence decay function, N(t[k]), and the calculated decay function, N[C](t[k]), are evaluated across the number of data points, n.
A χ^2 value of 1 indicates that the system is replicated by the fit. A value above 1.2 can indicate that the system is not well described by the fit. The bounds for an acceptable χ^2 will vary on the
experimental system, specifically with the noise associated with it. Good practice should include measurement of a well-established single component fluorophore prior to sample measurement to
establish the achievable sensitivity of the system.
The fluorescent dye 9-aminoacridine (9AA) in solution is a classic example of single exponential behaviour. The fluorescence decay of 9AA dye measured using TCSPC is shown in Figure 2.
Figure 2. (a) Decay of 9AA measured using an FS5 Spectrofluorometer and (b) single exponential fit analysis in Fluoracle.
Example Single Component Result
Here, τ is determined by tail fitting a single exponential component to the measured decay in Fluoracle. Fluoracle adjusts the value of τ and B until the least squares fit with the lowest residuals
is achieved, which is 16.2ns for 9AA (Figure 2b).
Fluoracle can also calculate the lifetime using the time it takes the intensity to reduce to 1/e which has the advantage of not requiring least squares fitting. The 1/e lifetime for the 9AA decay was
16.4 ns, close to the fitted value of 16.2 ns. For single exponential decays, the 1/e calculation provides a quick estimate of the lifetime, but this method cannot be applied to more complex
Untangling Multi-Exponential Decays
When a sample contains multiple fluorophores, or a single molecule has multiple fluorescent populations (e.g. conformers or tautomers), a multi-exponential model must be used:
Where I(t) is the fluorescence intensity as a function of time, t, normalised to the intensity at t = 0, τ[i] is the fluorescence lifetime of the ith decay component and B[i] is the component
In theory, there is no limit to the number of exponential components that can be included in the model but there is a practical limit before the model loses physical meaning. A ‘better’ fit can
always be achieved by increasing the number of components but that does not mean it is sensible to do so. For the lifetime components to be meaningful they must represent distinct photophysical
processes occurring in the sample and the number of components should therefore be chosen based on knowledge of the expected photophysics.
An example of a multiexponential decay is from a thermally activated delayed fluorescence (TADF) dye. The characteristic biexponential behaviour of a TADF dye can be seen in Figure 3a. The decay can
be accurately modelled with two exponential decay components, τ[1] = 65 ns and τ[2] =1061 ns which correspond to prompt fluorescence and delayed (after reverse intersystem crossing from the T[1])
fluorescence from the S[1] excited state of the dye.
Figure 3. (a) A biexponential decay of TADF measured using the FS5 Spectrofluorometer and (b) fitting result in Fluoracle.
Non-Exponential Decays
There are many sample types which do not follow exponential emission behaviour. This is often the case with photoluminescence decays from inorganic materials including semiconductors and ensemble
emitters such as quantum dots. An example of a non-exponential decay is photoluminescence from indium phosphide quantum dots coated with a layer of zinc sulphide (InP/ZnS) shown in Figure 4.
Figure 4. (a) Photoluminescence decay of InP/ZnS quantum dots measured using the FS5 Spectrofluorometer and (b) the corresponding fit analysis in Fluoracle with the amplitude weighted average
lifetime (blue box) and the intensity weighted average lifetime (orange box) highlighted.
A popular approach for this type of decay is to fit the decay with multiple exponential components and then calculate the average lifetime of the decay from the fit. This approach is shown in Figure
4a and 4b where Fluoracle has been used to fit the decay with four exponential components. The four individual lifetime components have no physical relation to specific states or interactions but are
simply a means to fit the decay curve accurately. From the four lifetime components Fluoracle then calculates two average lifetime values, the amplitude weighted average lifetime and the intensity
weighted average lifetime which can be used as a figure of merit to describe the photoluminescence lifetime properties of the sample.
Average Lifetime
Two average lifetimes are shown in Fluoracle as there is more than one definition of the average lifetime. The two average lifetimes most often reported in the literature are the amplitude weighted
average lifetime and the intensity weighted average lifetime. Prior knowledge of the intrinsic mechanisms that lead to the depopulation of the material’s excited states is required to choose which
average lifetime should be used.
Amplitude Average Lifetime
The amplitude average lifetime, <τ>[amp], weights each lifetime component (τ[i]) by its amplitude (B[i]):^4
The amplitude average lifetime is characteristic of the fluorophore in the steady state and may be mathematically related to the rate constants.^3 The amplitude average lifetime is commonly used in
biological systems where energy transfer between fluorophores occurs, and hence, their lifetime decay is multiexponential due to their heterogeneity and their interaction with their ambient
It is worth noting that if the B[i] component amplitudes are normalised (add up to 1) then the denominator in Eq. 7 becomes 1 and it therefore common to see Eq. 7 written as the numerator only.
Fluoracle does not normalise the B[i ]values and the full form of Eq. 7 is therefore used.
Intensity Average Lifetime
The intensity average lifetime weights each lifetime component (τ[i]) by the intensity of that component (B[i] τ[i]):^4
The intensity average places greater emphasis on the longer lifetimes, reducing the visibility of changes in fractional amplitude and shorter lifetimes. This makes the average appear more stable to
changes in the number of components during fitting.^2 The intensity average lifetime is applicable in, e.g., an ensemble of emitters, such as quantum dots embedded in photonic crystals, and
semiconductor nanocrystals whose fluorescence is nanocrystal size dependent.^5 Nanocrystals of the same material but of different sizes will emit at different wavelengths and therefore, the average
lifetime for the whole excited population of the nanocrystals should be considered.
Knowing Your Samples
Fits are best evaluated by careful evaluation of the photophysics of the chemical system, and the mathematically calculated ‘best’ solution is not always the correct answer. Fitting algorithms are
blind to the system they are analysing. A single component rare-earth sample is treated in the same way as a non-exponential QD system. This means that the experimentalist should always make the
final decision using sample-specific knowledge. Fitting is heavily shaped by the values input by the user (fitting range, background, suggested lifetimes). The fitting result is greatly affected by
even small changes in these values, and these changes will have a non-calculable influence on the final fit.
Analysing Multiple Data Sets
This blog explores the basics of fitting for a single sample decay profile. To improve the understanding of a system multiple decay functions may be analysed together. A single sample may be measured
with varied excitation or emission wavelength, or a series of samples with varying composition, will all have shared lifetimes. This can be linked using global fitting where multiple decay curves are
fitted in parallel with some fitting parameters in common.
1. D. M. Jameson, Introduction to Fluorescence, 2014.
2. E. Fišerová and M. Kubala, Mean fluorescence lifetime and its error, J Lumin, 2012, 132, 2059–2064.
3. A. Sillen and Y. Engelborghs, The Correct Use of ‘Average’ Fluorescence Parameters, Photochem Photobiol, 1998, 67, 475–486.
4. B. Valeur and M. N. Berberan-Santos, Molecular Fluorescence, Wiley-VCH, 2nd edn., 2012.
5. G. Zatryb and M. M. Klak, On the choice of proper average lifetime formula for an ensemble of emitters showing non-single exponential photoluminescence decay, Journal of Physics Condensed Matter,
What is Fluorescence Lifetime? | {"url":"https://www.edinst.com/blog/determining-fluorescence-lifetimes/","timestamp":"2024-11-04T02:49:14Z","content_type":"text/html","content_length":"199233","record_id":"<urn:uuid:3ac03583-6aa8-4a66-bc36-b0489f293cc0>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00026.warc.gz"} |
Multiple Merit Function Optimization - Dyoptr
Multiple Merit Function Optimization
Learn how to optimize multiple merit functions and sequences simultaneously.
In this video, I will show you how to set up multiple merit functions.
What is the advantage of setting up multiple merit functions?
So first, it is possible to set up a unique merit function for individual assemblies or elements.
This is of great importance, for example, for larger optical systems, which consists of several assemblies or of several elements and multiple ray paths as, for example, here this FISO
For example, here in this FISO example, you want to optimize for the wave front error of the whole interferometer, and you also want to optimize for the imaging quality of the imaging optics, which
you can see here, for example.
Another big advantage is in the case of comparison of two different system configurations to check which configuration performs the best. And a third very important point is that it is possible to
define multiple merit functions for tolerancing. For example, one merit function only for the evaluation of the optical performance and one merit function for the compensators and so on.
And to show you how to define multiple method functions, I’ve set up here this simple system with those two assemblies, the first assembly containing the stoplight lens and the second assembly
containing here the singlet lens, and I’ve defined two sequences. One sequence which is going here until the first surface of the second lens highlighted here in red and the second sequence which is
going through the whole system.
And I will define two merit functions, and the first merit function will be optimized using the first sequence and the second merit function will use here the second sequence. To add the merit
functions, we go here to the system setup and click here with the right mouse button on optimization, and there we click on add merit function.
In my case, I will add two merit functions.
So the first merit function here for the first assembly using the first sequence and the second merit function using the second assembly with the singlet in the second sequence.
To define the merit function, I will open the merit function here. There, I will define the ray trace here, the sequence one.
The second merit function, the same, but I will, select the sequence two.
The next step would be to define the ray tracing goals.
So in my first merit function, I would like to collimate the rays, and in the second merit function, I would like to focus the race onto a single point. So in the next step, I will define the merit
functions with the targets and the constraints.
In the next step, we will define the variables for our optimization, and for that, we go to the optical design editor, select here down from the drop down menu the merit function for which we would
like to, define the variables.
So first, I would like to define the variables for merit function one.
Then we open the assembly, which would should like to optimize with the merit function one, and then we just select here the check boxes in order to set those parameters as a variable.
In the second step, I will set up the variables for the merit function two. So again, I go here to the drop down menu, select merit function two, then I go here to the singlet lens, which will focus
the rays on a single point, Open here the singlet lens, and then I will select here the variables.
So now we have to find the merit function and set up the variables, and now we can optimize our system. And for that, we go to the optimization tab and open, for example, here this local
optimization. And here in the local optimization wizard, we select the merit function which we would like to optimize. In my case, I would like to use first the merit function one to optimize here
this doublet lens in order to collimate the rays here with the first sequence colored in red.
And as you can see now, the red sequence here and the the lens has been optimized and the rays are collimated.
And now again here in the optimization wizard, I will select the merit function two in order to optimize here the second assembly with the singlet in order to optimize the sequence that the rays will
focus on a single point.
Yeah. And as you can see, we have optimized our system with two unique merit functions, one merit function for each assembly and using two different sequences.
Yeah. This was my short tutorial about using multiple merit functions for the optimization and using multiple sequences for the optimization.
Thanks for watching. | {"url":"https://dyoptr.com/multiple-merit-function-optimization/","timestamp":"2024-11-14T16:36:35Z","content_type":"text/html","content_length":"129211","record_id":"<urn:uuid:e88834ae-7505-425d-84e1-2ee1c0e30bb8>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00874.warc.gz"} |
FRIEDMAN procedure • Genstat v21
Performs Friedman’s nonparametric analysis of variance (S. Langton).
PRINT = string tokens Output required (test, ranks); default test
TREATMENTS = factor Treatment factor
BLOCKS = factor Block factor
DATA = variates Identifier of the variate holding the data values
RANKS = variates Saves the ranks
STATISTIC = scalars Saves the test statistic
DF = scalars Saves the degrees of freedom for the chi-square approximation
PROBABILITY = scalars Saves the probability value for the chi-square statistic
Friedman’s test is a nonparametric test for analysing a randomized complete block design. That is, the data consists of observations on k treatments assessed under n different conditions (blocks).
The variate of observations is specified using the DATA parameter, whilst options TREATMENTS and BLOCKS supply the treatment and blocking factors. FRIEDMAN calculates the test statistic together with
a probability value based on a chi-square approximation. If sample sizes are small, stored tabulated values are printed in addition.
The PRINT option controls printed output, with settings test to print the various test statistics, and ranks to print the ranks (together with the BLOCKS, TREATMENTS and DATA).
Parameters RANKS, STATISTIC, DF and PROBABILITY can be used to save the ranks, the test statistic (adjusted for ties), the degrees of freedom for the chi-square approximation, and the probability
value for the chi-square approximation.
Options: PRINT, TREATMENTS, BLOCKS.
Parameters: DATA, RANKS, STATISTIC, DF, PROBABILITY.
The Friedman test is a test for treatment differences in a randomized complete block design: i.e. a test of the null hypothesis that the samples arise from distributions with the same mean versus the
alternative that the distribution means differ. Each block is checked in turn to ensure that it consists of exactly one replicate of each treatment, after excluding any units which are restricted out
or which have missing values for DATA, TREATMENTS or BLOCKS. Any block not meeting this condition is excluded from analysis and a warning is printed. The treatments are ranked within each block and
the sum of the ranks is calculated for each treatment group over all valid blocks. The sum of the squared values of these rank sums is calculated, as is the sum of the cubed sizes of all groups of
ranks (i.e. 1 for an untied observation, 2^3=8 for pairs of ties, etc.).
The test statistic is formed using the equation (Siegel & Castellan 1988):
Fr = 12 × R / (n × k × (k + 1)) – 3 × n × (k + 1)
where R is the sum of the squared rank sums,
k is the number of treatments, and
n is the number of blocks.
A version adjusted for ties is also formed, and this version is used for calculating the significance level:
Fr = {12 × R – 3 × n^2 × k × (k + 1)^2} / {n × k × (k + 1) + (n × k – T) / (k – 1)}
where T is the sum of the cubed sizes of rank groups.
Any units that are restricted for DATA, BLOCKS or TREATMENTS (or which have missing values) are excluded from the analysis. Any block which no longer has one replicate of each treatment as a result
of such restrictions is excluded in its entirety.
Siegel, S. & Castellan, N.J. (1988). Nonparametric Statistics for the Behavioural Sciences (second edition). McGraw-Hill, New York.
See also
Procedures: APERMTEST, A2WAY, KRUSKAL.
Commands for: Basic and nonparametric statistics, Analysis of variance.
CAPTION 'FRIEDMAN example','1) Siegel & Castellan, 1988, p. 176';\
VARIATE Score
READ Score
9 1 2 6 :
FACTOR [LEVELS=3; VALUES=4(1...3)] Group
& [LEVELS=4; VALUES=(1...4)3] Condition
FRIEDMAN [TREATMENTS=Condition; BLOCKS=Group] Score
CAPTION '2) Siegal & Castellan, 1988, p. 179'
VARIATE Rank
READ Rank
3 2 1 2 3 1 2.5 2.5 1 3 2 1 3 2 1 2 3 1 :
FACTOR [LEVELS=18; VALUES=3(1...18)] Group
& [LEVELS=3; LABELS=!t(RR,RU,UR); VALUES=(1...3)18] Type
FRIEDMAN [TREATMENTS=Type; BLOCKS=Group] Rank | {"url":"https://genstat21.kb.vsni.co.uk/knowledge-base/friedman/","timestamp":"2024-11-14T11:14:51Z","content_type":"text/html","content_length":"41849","record_id":"<urn:uuid:be54c048-b0e5-4e64-b548-dce94c874863>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00827.warc.gz"} |
What is Quantization Of ChargeWhat is Quantization Of Charge
What is quantization of charge
Hi guise let's explore what is quantization of charge ,Quantization of charge is a fundamental concept in physics that refers to the discrete nature of electric charge. In simple terms, it means that
electric charge cannot exist in arbitrary amounts but rather in multiples of a fundamental unit. This concept was introduced by the French physicist Albert Einstein in the early 20th century.
• - The electric charge of any object is quantized, meaning it exists only in discrete values.
• - Now if we talk about elementary charge, denoted as 'e,' is the fundamental unit of charge.
• - The quantization of charge is a consequence of the discrete nature of matter at the atomic and subatomic levels.
after getting information of what is quantization of charge let's know why is it important?
some things i want to say if you want to learn more about charge and want to dive into information of electrostatics please visit this page also thanks.
Why is charge quantized?
Charge quantization stems from the fundamental properties of matter at the atomic and subatomic scales. It is closely linked to the discrete nature of elementary particles, such as electrons and
protons, which constitute matter.
• - The quantization of charge is a result of the discrete distribution of electrons in atoms.
• - Elementary particles, like electrons, carry a specific charge in multiples of the elementary charge 'e.'
• - The quantization of charge ensures stability and predictability in electrical interactions at microscopic levels.
How was the quantization of charge discovered?
The discovery of the quantization of charge is attributed to the renowned physicist Robert A. Millikan, who conducted the famous oil-drop experiment in 1909. This groundbreaking experiment provided
direct evidence for the quantization of electric charge.
• - Robert A. Millikan's oil-drop experiment demonstrated the discrete nature of electric charge.
• - The experiment involved measuring the charge on tiny oil droplets suspended in an electric field.
• - Millikan's work confirmed the existence of the elementary charge and its multiples.
What is the elementary charge?
The elementary charge, often denoted as 'e,' is the basic unit of electric charge in the quantization of charge theory. It plays a crucial role in understanding the discrete nature of electrical
phenomena at the atomic and subatomic levels.
• - If we talk about elementary charge is approximately equal to 1.602 x 10^-19 coulombs.
• - All charges observed in nature are integral multiples of the elementary charge.
• - Electrons, one of the fundamental particles, carry a negative elementary charge, while protons carry a positive charge of the same magnitude.
How does quantization of charge impact electronic devices?
The quantization of charge has profound implications for the functioning of electronic devices, shaping the principles underlying modern electronics and technology.
• - The discrete nature of charge ensures the stability and reliability of electronic components.
• - Digital information processing relies on the quantization of charge to represent binary states (0s and 1s).
• - Semiconductor devices, such as transistors, operate based on the controlled flow of quantized charge carriers.
In summary, quantization of charge is a foundational concept in physics, describing the discrete nature of electric charge at the atomic and subatomic levels. This concept, discovered through
experiments like Millikan's oil-drop experiment, is vital for understanding the behavior of matter and has significant implications for electronic devices and technology.
Post a Comment | {"url":"https://www.electronicsystemsengineering.co.in/2024/01/What-is-Quantization-Of-Charge.html","timestamp":"2024-11-14T14:18:21Z","content_type":"application/xhtml+xml","content_length":"276599","record_id":"<urn:uuid:a3dad0a8-3b1b-4e62-b46a-27c033d2b97c>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00510.warc.gz"} |
What are SQL Window Functions? | Complete Beginner's Guide | SQL Queries
What are SQL Window Functions? | Complete Beginner’s Guide
In this article, we will see what are the SQL Window Functions and common issues that we face with practical examples.
SQL Window Functions
We have 3 different categories of window functions, which are given below.
• Aggregate Functions – These are AVG, SUM, MIN, MAX, COUNT e.t.c
• Ranking Functions – These are RANK, DENSE RANK, ROW_NUMBER e.t.c
• Analytic Functions – These are LEAD, LAG, FIRST_VALUE, LAST_VALUE e.tc
With all these analytical functions we can use the over by clause.
For our better understanding we will write some queries and see the behaviour of these functions.
Interview point of view these are very important.
Let us consider the below table.
Now let us write a query to get the average on bonus column
SELECT AVG(Bonus)
From Employee;
Result for this query will be
Now we will write another query to capture all other records from the employee table along with average.
Employee ID, Name, Work, Bonus,
FROM Employee;
Results for the above query will be
What happened here?. If you can obeserve, I got a different average salary for each employee. The last employee I got the correct value which is 550.
Let us see the below diagram to understand how this average is being computed.
SQL Window Functions Average Bonus Calculation
Here we have not specified the explicit value for the rows or arrange clause. So that is the reason all average values for the bonus column is being computed differently except the last value.
Since we have not specified explicit value it is going to take the default value.
It is going to take the Unbounded Preceding and Current Row.
Let us understand what this value means.
Say for example this average function is being applied to the second row. In that context what is this range between unbounded preceding and current row?
Current row is the one which average being computed and unbounded preceding is the window for this average function starts at the first row within the result set.
Now the order by clause is emposed on the rows for the employee table that is our result set.
Within the orderset results unbounded preceding means the window starts for this average function at the first row.
When it is in the second row the range of average is for the first row and current row.
Similarly when we are on the third row the range is first row and the current row which is third row. So average of these three salary would be captured for the third row.
At the moment this is how it is being calculated. Now look at the last row it is calculated correctly. Why because it is going to calculate the average for all records.
We will have the same problem whenever we use the any of window functions that we have aggregate functions, Ranking Function and Analytical Functions because of the default value for that clause.
Now in the query we will also take Count and SUM
For that our query will be
SELECT Name, Work, Bonus,
OVER( ORDER BY BONUS) AS [Average]
OVER(ORDER BY BONUS) AS [Count]
OVER(ORDER BY BONUS) AS [SUM]
Results for this query will be
The window for these functions is, It has to starts with the first row and ends with the last row.
But what is the default value here? Which is the range between unbounded preceding and current row.
Now i am going to change that range between Unbounded Preceding And Unbounded Following. Means window ends with the last row within the result set.
Now let us writte a query for this
SELECT Name, Work, Bonus,
AVG(Bonus) OVER
( ORDER BY BONUS ROWS BETWEEN
UNBOUNDED FOLLOWING ) AS [Average],
UNBOUNDED FOLLOWING) AS [Count],
UNBOUNDED FOLLOWING) AS [SUM]
Results for this query will be
What we achieved here. Total count, Total average and Total sum for all rows. This is what we expected here.
At the moment there is no partition. If the partition is involved this is going to slightly change.
Now let us take the partition by work. Here window starts with partition first row and ends with the partition last row.
Let us write a query to achieve this
SELECT Name, Work, Bonus,
AVG(Bonus) OVER
( PARTITION BY Work
UNBOUNDED FOLLOWING ) AS [Average],
OVER( PARTITION BY Work
UNBOUNDED FOLLOWING) AS [Count],
OVER(PARTITION BY Work
AND UNBOUNDED FOLLOWING) AS [SUM]
Results for this query will be
Now there is one more thing here I want to calculate the one row preceding and one row following how i can do that?
Let us see.
If you can take an average. Consider I am in third row and I want to calculate average between second row and fourth row.
Let us look at the first row we dont have any row before that and one row after that it calculate average, sum and count respectively .
Let us write a query for this
SELECT Name, Work, Bonus,
OVER( PARTITION BY Work
ROWS BETWEEN 1 PRECEDING AND 1 FOLLOWING )
AS [Average],
OVER( PARTITION BY Work
ROWS BETWEEN 1
FOLLOWING ) AS [Count],
OVER(PARTITION BY Work
ROWS BETWEEN 1
FOLLOWING) AS [SUM]
Results for this query will be
Here you need to calculate the result set very carefully because partition is applied on the work column. There are two sets here one is Home and other one is Office. When new partition starts you
need to calculate it separately.
I hope this article is useful to you. Please leave your comment in the comment section below.
You can read related article SQL Date Functions here. | {"url":"https://sqlqueries.in/sql-window-functions/","timestamp":"2024-11-03T06:57:38Z","content_type":"text/html","content_length":"114366","record_id":"<urn:uuid:d3233ab8-0a8e-4df7-8f93-2ef1fbef277b>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00138.warc.gz"} |
Are We Being Simulated by a Clock?
Are we living in a simulation?
I don’t mean the brain-in-a-jar (or body-in-a-pod) kind of simulation shown in The Matrix. I mean, is our whole universe, including our brains and minds, being simulated by some machine in some
“larger” universe? Are you being simulated? At least, what kind of machine is capable of doing this?
Are We Being Discretely Simulated?
Let us first consider machines that treat time discretely rather than continuously. That is, in the simulated universe, time is a sequence of moments, there’s a first moment, and each moment has a
following moment. For each moment the machine stores the state of the universe, and then calculates the next state.
More mathematically, the machine runs a discrete simulator, which is a structure (S, i, T) where S is the set of possible states, i ∈ S is the state at the first moment, and T ∈ S → S is the
“transformation” function, that maps the state at a given moment to the state at the next moment. Note that S does not have to be finite or even countable: it can be any set.
For example, we might consider moments as a very small gradation of time in our universe (perhaps the Planck time). S might be the phase space of the universe: the set of all possible states of all
particles & fields in the universe. T is a function that tells the machine how to obtain the next state of the universe from the current state of the universe.
Let us say that a simulator is feasible if it is possible that our present universe is being simulated by a machine running it. If a given simulator is feasible, what other simulators are feasible?
It seems to me that it doesn’t really matter how a given simulator (S, i, T) represents the state of our universe in S, provided T works correctly. For example, if such a simulator is feasible, then
these other simulators are feasible:
• (S, R(i), R · T · R), where R maps the universe to a universe reflected in some co-ordinate,
• (S‘, R(i), R · T · R^-1), where R ∈ S → S‘ performs some kind of lossless data compression of the state, and R^-1 ∈ S‘ → S decompresses it,
• (S‘, R(i), R · T · R^-1), where R ∈ S → S‘ is any function with a retraction R^-1,
• (S × S‘, (i, i‘), (s, s‘) ↦ (T s, T‘ s‘)), where (S‘, i‘, T‘) is another simulator (feasible or not).
In fact, all that matters for one simulator to be as feasible as another is that the state of the latter can be obtained from the state of the former. Consider two simulators A = (S[A], i[A], T[A])
and B = (S[B], i[B], T[B]). Then a morphism from A to B is a function f ∈ S[A] → S[B] such that
1. f(i[A]) = i[B]
2. f(T[A](s)) = T[B](f(s))
My assumption is that if such a morphism exists and B is feasible, then A is feasible. This is because A is capable of representing B, since for any state in A, there’s a function that obtains the
corresponding state in B. So if it’s possible that a machine running B is simulating our universe, it’s possible that it’s running A instead. This seems reasonable to me.
Now consider a very trivial simulator K = (N, 0, s ↦ s + 1), which I call the counting simulator. Its state set is simply N, the natural numbers starting from zero. Its initial state is zero. Its
transformation function just adds one. A machine running K is just a machine that counts.
As it turns out, for any simulator A = (S, i, T), we can construct a morphism from K to A like this:
f = s ↦ T^s(i)
To spell it out recursively:
f(0) = i
f(s + 1) = T(f(s))
For f to be a morphism, we need to satisfy the two conditions:
1. f(0) = i
2. f(s + 1) = T(f(s))
But these are exactly the definition of f, so it is indeed a morphism. (In fact it’s a unique morphism. Category theorists will recognise K as an “initial object”.)
So this means if any simulator is feasible, then the counting simulator is feasible. A simple “machine that counts” is just as capable of simulating the universe as any other.
In fact, if you want to discretise time by the Planck time, then the machine you’re reading this on, if only it could last long enough, is capable of simulating the universe from Big Bang to present
day. The age of the universe is about 8×10^60 Planck times. To represent a number this large, you’ll need 203 bits, or 26 bytes. Each number counted out by your desktop counting machine fully
represents the state of a universe just like ours: it’s just encoded in a rather unusual way.
Are We Being Continuously Simulated?
Perhaps instead the machine that simulates our universe treats time continuously. Assuming the simulation started at some point (time zero), we’ll define Time as the set of non-negative real numbers.
The machine runs a continuous simulator (a dynamical system), defined as a structure (S, i, T) where S is a set (of possible states), i ∈ S is the state at time zero, and T ∈ Time × S → S is the
function that determines how the state changes over time, with these conditions:
1. T(0, s) = s [no time means no change]
2. T(t[1] + t[2], s) = T(t[2], T(t[1], s)) [the state alone determines how the state changes]
Let us make the equivalent assumption from the discrete case. If A encodes the state of B, and B is feasible, then A is feasible. Specifically, given two continuous simulators A = (S[A], i[A], T[A])
and B = (S[B], i[B], T[B]), a morphism f from A to B is a function S[A] → S[B] such that
1. f(i[A]) = i[B]
2. f(T[A](t, s)) = T[B](t, f(s))
By this assumption, if A and B are continuous simulators, and there exists a morphism from A to B, and B is feasible, then A is feasible.
Now consider the “clock” continuous simulator K = (Time, 0, (t, s) ↦ s + t). Its state set is time itself. Its initial state is zero. Its transformation function just adds time. This obviously
matches the two conditions necessary to be a continuous simulator.
The “clock” continuous simulator plays the same role as the “counting” discrete simulator. For any continuous simulator A = (S, i, T), we can construct a morphism from K to A like this:
f = t ↦ T(t, i)
Let’s check the conditions:
1. f(0) = i
1. f(0) = T(0, i) [definition of f]
2. T(0, i) = i [first condition of continuous simulators]
2. f(s + t) = T(t, f(s))
1. f(s + t) = T(s + t, i) [definition of f]
2. T(s + t, i) = T(t, T(s, i)) [second condition of continuous simulators]
3. T(t, T(s, i)) = T(t, f(s)) [definition of f]
So it is indeed a morphism.
So this means if any continuous simulator is feasible, then the clock simulator is feasible. A simple clock is at least as capable of simulating the universe as any other machine, if only its motion
is perfectly continuous.
— Ashley Yakeley | {"url":"https://immanence.org/post/are-we-being-simulated-by-a-clock/","timestamp":"2024-11-13T04:10:43Z","content_type":"text/html","content_length":"49248","record_id":"<urn:uuid:7cce51aa-a519-460d-9f14-d055cc068fbd>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00025.warc.gz"} |
Understanding the Odds in Sports Betting | Ncord Church
The Basics of Sports Betting
Sports betting is a popular form of gambling that involves predicting the outcome of sports events and placing wagers on the predicted outcome. It is a thrilling activity that adds an extra level of
excitement and engagement to watching sports. However, to be successful in sports betting, it is crucial to have a good understanding of odds and how they work.
Odds are a way to quantify the probability of a specific outcome in a sports event. They determine the potential payout of a bet and help bettors make informed decisions. If you’re new to sports
betting, understanding how odds are calculated and what they mean is essential.
The Three Types of Odds
There are three commonly used types of odds in sports betting: American odds, Decimal odds, and Fractional odds. Each type represents the same probability but is presented differently. Let’s take a
closer look at each type:
American Odds
American odds, also known as moneyline odds, are widely used in the United States. They are represented by a plus (+) or minus (-) sign followed by a number. Positive odds indicate the potential
profit if a $100 bet is successful, while negative odds indicate the amount you need to bet to win $100.
For example, if the odds are +200, you would win $200 for every $100 wagered. On the other hand, if the odds are -150, you would need to bet $150 to win $100.
Decimal Odds
Decimal odds are commonly used in Europe and are a popular format for online sportsbooks. They represent the total potential return, including the initial stake, in decimal format. The odds are a
simple reflection of the probability of an outcome.
For example, if the odds are 2.50, a $100 bet would result in a total return of $250 ($100 initial stake + $150 profit).
Fractional Odds
Fractional odds, also called British odds, are most commonly used in the UK. They are represented as fractions and show the potential profit relative to the stake. The first number in the fraction
represents the profit, while the second number represents the stake.
For example, if the odds are 2/1, a $100 bet would result in a profit of $200 ($100 profit + $100 stake).
Calculating Probabilities from Odds
Understanding how to convert odds into probabilities is crucial for making informed betting decisions. To calculate the implied probability from odds, you can use the following formulas:
American Odds
To calculate the implied probability of positive American odds, use the formula:
Implied Probability = 100 / (Odds + 100)
For example, if the odds are +200, the implied probability would be 100 / (200 + 100) = 100 / 300 = 0.33 (33%).
To calculate the implied probability of negative American odds, use the formula:
Implied Probability = Odds / (Odds – 100)
For example, if the odds are -150, the implied probability would be -150 / (-150 – 100) = -150 / -250 = 0.6 (60%). Note that the result is negative, indicating that the bookmaker has an advantage in
this bet.
Decimal Odds
To calculate the implied probability of decimal odds, use the formula:
Implied Probability = 1 / Odds
For example, if the odds are 2.50, the implied probability would be 1 / 2.50 = 0.4 (40%).
Fractional Odds
To calculate the implied probability of fractional odds, use the following formula:
Implied Probability = Denominator / (Denominator + Numerator)
For example, if the odds are 2/1, the implied probability would be 1 / (1 + 2) = 1 / 3 = 0.33 (33%).
Understanding Betting Lines and Expected Value
In addition to odds, betting lines are an important factor to consider when placing sports bets. Betting lines are created by bookmakers to level the playing field between two teams or competitors by
assigning a point spread.
For example, in a football game, one team may be favored to win by 7 points. This means they would need to win by more than 7 points for a bet on them to be successful. On the other hand, the
underdog would need to lose by less than 7 points or win the game for a bet on them to be successful.
Expected value (EV) is another critical concept in sports betting. It refers to the average amount a bettor can expect to win or lose on a specific bet over the long term. A positive EV indicates a
profitable bet, while a negative EV suggests a losing bet.
Managing Bankroll and Applying Strategies
Bankroll management is a vital aspect of successful sports betting. It involves setting aside a specific budget for betting and sticking to it. A general rule of thumb is to never bet more than you
can afford to lose.
Applying strategies can also improve your chances of winning in sports betting. Some popular strategies include: Find more relevant information on the subject by visiting this carefully selected
external resource. 1win, extra information available.
The Martingale System: This strategy involves doubling your bet after a loss to recover previous losses. It can be risky, so it’s important to set limits.
The Kelly Criterion: This strategy suggests calculating the optimal bet size based on the odds and your perceived edge in the bet.
The Fade the Public Strategy: This strategy involves betting against popular opinion or the public’s perception, as they are often influenced by biases.
Understanding the odds in sports betting is crucial for making informed decisions and increasing your chances of success. By familiarizing yourself with the different types of odds, calculating
probabilities, and considering betting lines and expected value, you can enhance your overall betting experience. Remember to manage your bankroll effectively and consider using proven strategies to
maximize your winnings. Happy betting!
Check out the related links for additional information on the subject:
Learn more from this helpful source | {"url":"https://ncordchurch.com/understanding-the-odds-in-sports-betting/","timestamp":"2024-11-03T00:45:13Z","content_type":"text/html","content_length":"174493","record_id":"<urn:uuid:da72eb7c-e913-47a0-a34f-e9839e93ac38>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00413.warc.gz"} |
Emergent Behaviour, Population-based Search and Low-pass Filtering
Emergent Behaviour, Population-based Search and Low-pass Filtering
Created by W.Langdon from gp-bibliography.bib Revision:1.8010
□ author = "Riccardo Poli and Alden H. Wright and Nicholas F. McPhee and William B. Langdon",
□ title = "Emergent Behaviour, Population-based Search and Low-pass Filtering",
□ institution = "Department of Computer Science, University of Essex",
□ year = "2006",
□ type = "Technical Report",
□ number = "CSM-446",
□ month = feb,
□ ISSN = "1744-8050",
□ keywords = "genetic algorithms, genetic programming",
□ abstract = "In recent work we have formulated a model of emergent coordinated behaviour for a population of interacting entities. The model is a modified spring mass model where the masses
can perceive the environment and generate external forces. As a result of the interactions the population behaves like a single organism moving under the effect the vector sum of the external
forces generated by each entity. When such forces are proportional to the gradient of a resource distribution f(x), the resultant force controlling the single emergent organism is
proportional to the gradient of a modified food distribution. This is the result of applying a filtering kernel to f(x). The kernel is typically a low-pass filter.
This model can be applied to genetic algorithms (GAs) and other population-based search algorithms. For example, in previous research, we have found kernels (via genetic programming) that
allow the single organism model to track the motion of the centre of mass of GAs and particle swarm optimisers accurately for many generations.
we corroborate this model in several ways. Firstly, we provide a mathematical proof that on any problem and for any crossover operator, the effect of crossover is that of reducing the
amplitude of the derivatives (slopes) of the population distribution. This implies that a GA perceives an effective fitness landscape which is a smoothed, low-pass filtered version of the
original. Then, taking inspiration from this result and our active mass-spring model, we propose a class of fitness functions, OneMix, where there is an area of the landscape with high
frequency variations. This area contains the global optimum but a genetic algorithm with high crossover probability should not be able ``see'' it due to its low-pass behaviour. So, a GA with
strong crossover should be deceived and attracted towards a local optimum, while with low crossover probability this should not happen. This is, indeed, what happens as we demonstrate with a
variety of empirical runs and with infinite-population model simulations. Finally, following our earlier approach, we also evolved kernels for OneMix, obtaining again a good fit between the
behaviour of the ``single-organism'' hill-climber and the GA.",
□ notes = "See also \cite{poli:2006:cec}",
□ size = "23 pages",
Genetic Programming entries for Riccardo Poli Alden H Wright Nicholas Freitag McPhee William B Langdon | {"url":"https://gpbib.cs.ucl.ac.uk/gp-html/poli_2006_CSM446.html","timestamp":"2024-11-07T20:35:25Z","content_type":"text/html","content_length":"5396","record_id":"<urn:uuid:a1da4b7f-d90f-41e4-8c52-5b1faee6ea0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00657.warc.gz"} |
Linear Motion - Mathematics Form 2 Notes - EasyElimu: Learning Simplified
• Distance between the two points is the length of the path joining them while displacement is the distance in a specified direction
Average speed = distance covered
time taken
A man walks for 40 minutes at 60 km/hour, then travels for two hours in a minibus at 80 km/hour. Finally, he travels by bus for one hour at 60 km/h. Find his speed for the whole journey.
Average speed = distance covered
time taken
Total distance =(^40/[60] x 60)km + (2 x 80)km + (1 x 60)km = 260 km
Total time = ^4/[6] + 2 + 1 = 3^2/[3] hrs
Average speed = 260
=260 x 3 = 70.9 km/h
• For motion under constant acceleration;
Average velocity = initial velocity + final velocity
A car moving in a given direction under constant acceleration. If its velocity at a certain time is 75 km/h and 1 0 seconds later its 90 km/hr.
Accelaration = change in velocity
time taken
=(90 − 75)km/h
= (90 − 75) x 1000 m/s²
10 x 60 x 60
= ^5/[12] m/s²
A car moving with a velocity of 50 km/h then the brakes are applied so that it stops after 20 seconds. In this case the final velocity is 0 km/h and initial velocity is 50 km/h.
Acceleration = (0−50) x 1000 m/s²
20 x 60 x 60
= −^25/[36] m/s²
Negative acceleration is always referred to as deceleration or retardation
• When distance is plotted against time, a distance time graph is obtained.
• When describing the motion of an object try to be as detailed as possible. For instance...
□ During 'Part A' of the journey the object travels +8m in 4s. It is travelling at a constant velocity of +2ms^-1
□ During 'Part B' of the journey the object travels 0m in 3s. It is stationary for 3 seconds
□ During 'Part C' of the journey the object travels -8m in 3s. It is travelling at a 'constant velocity' of '-2.7ms^-1' back to its starting point, our reference point 0.
• When velocity is plotted against time, a velocity time graph is obtained.
• The distance travelled is the area under the graph
• The acceleration and deceleration can be found by finding the gradient of the lines.
• Consider two bodies moving in the same direction at different speeds. Their relative speed is the difference between the individual speeds.
A van left Nairobi for kakamega at an average speed of 80 km/h. After half an hour, a car left Nairobi for Kakamega at a speed of 100 km/h.
1. Find the relative speed of the two vehicles.
2. How far from Nairobi did the car over take the van
Relative speed = difference between the speeds
= 100 – 80
= 20 km/h
Distance covered by the van in 30 minutes
Distance = ^30/[60] x 80 = 40 km
Time taken for car to overtake matatu = ^40/[20]
= 2 hours
Distance from Nairobi = 2 x 100 =200 km
A truck left Nyeri at 7.00 am for Nairobi at an average speed of 60 km/h. At 8.00 am a bus left Nairobi for Nyeri at speed of 1 20 km/h .How far from nyeri did the vehicles meet if Nyeri is 1 60 km
from Nairobi?
Distance covered by the lorry in 1 hour = 1 x 60
= 60 km
Distance between the two vehicle at 8.00 am = 160 – 1 00
= 100km
Relative speed = 60 km/h + 120 km/h
Time taken for the vehicle to meet = ^100/[180]
=^5/[9] hours
Distance from Nyeri = 60 x ^5/[9] x 60
= 60 + 33.3
= 93.3 km
1. A bus takes 195 minutes to travel a distance of (2x + 30) km at an average speed of (x - 20) km/h Calculate the actual distance traveled. Give your answers in kilometers.
2. The table shows the height metres of an object thrown vertically upwards varies with the time t seconds. The relationship between s and t is represented by the equations s = at^2+bt +10 where b
are constants.
│t│0│1 │2│3│4│5│6│7│8│9│10 │
│s│ │45.1 │ │ │ │ │ │ │ │ │ │
1. Using the information in the table, determine the values of a and b ( 2 marks)
2. Complete the table (1 mark)
1. Draw a graph to represent the relationship between s and t (3 marks)
2. Using the graph determine the velocity of the object when t = 5 seconds (2 marks)
3. Two Lorries A and B ferry goods between two towns which are 31 20 km apart. Lorry A traveled at km/h faster than lorry B and B takes 4 hours more than lorry A to cover the distance.Calculate the
speed of lorry B
4. A matatus left town A at 7 a.m. and travelled towards a town B at an average speed of 60 km/h. A second matatus left town B at 8 a.m. and travelled towards town A at 60 km/h. If the distance
between the two towns is 400 km, find;
1. The time at which the two matatus met
2. The distance of the meeting point from town A
5. The figure below is a velocity time graph for a car.
1. Find the total distance traveled by the car. (2 marks)
2. Calculate the deceleration of the car. (2 marks)
6. A bus started from rest and accelerated to a speed of 60 km/h as it passed a billboard. A car moving in the same direction at a speed of 100 km/h passed the billboard 45 minutes later. How far
from the billboard did the car catch up with the bus? (3mks)
7. Nairobi and Eldoret are each 250km from Nakuru. At 8.1 5am a lorry leaves Nakuru for Nairobi. At 9.30am a car leaves Eldoret for Nairobi along the same route at 1 00km/h. Both vehicles arrive at
Nairobi at the same time.
1. Calculate their time of arrival in Nairobi (2 marks)
2. Find the cars speed relative to that of the lorry. (4 marks)
3. How far apart are the vehicles at 1 2.45pm. (4 marks)
8. Two towns P and Q are 400 km apart. A bus left P for Q. It stopped at Q for one hour and then started the return journey to P. One hour after the departure of the bus from P, a trailer also
heading for Q left P. The trailer met the returning bus ¾ of the way from P to Q. They met t hours after the departure of the bus from P.
1. Express the average speed of the trailer in terms of t
2. Find the ration of the speed of the bus so that of the trailer.
9. The athletes in an 800 metres race take 104 seconds and 108 seconds respectively to complete the race. Assuming each athlete is running at a constant speed. Calculate the distance between them
when the faster athlete is at the finishing line.
10. A and B are towns 360 km apart. An express bus departs form A at 8 am and maintains an average speed of 90 km/h between A and B. Another bus starts from B also at 8 am and moves towards A making
four stops at four equally spaced points between B and A. Each stop is of duration 5 minutes and the average speed between any two spots is 60 km/h. Calculate distance between the two buses at 10
11. Two towns A and B are 220 km apart. A bus left town A at 11 . 00 am and traveled towards B at 60 km/h. At the same time, a matatu left town B for town A and traveled at 80 km/h. The matatu
stopped for a total of 45 minutes on the way before meeting the bus. Calculate the distance covered by the bus before meeting the matatu.
12. A bus travels from Nairobi to Kakamega and back. The average speed from Nairobi to Kakamega is 80 km/hr while that from Kakamega to Nairobi is 50 km/hr, the fuel consumption is 0.35 litres per
kilometer and at 80 km/h, the consumption is 0.3 litres per kilometer .Find
1. Total fuel consumption for the round trip
2. Average fuel consumption per hour for the round trip.
13. The distance between towns M and N is 280 km. A car and a lorry travel from M to N. The average speed of the lorry is 20 km/h less than that of the car. The lorry takes 1h 10 min more than the
car to travel from M and N.
1. If the speed of the lorry is x km/h, find x (5 marks)
2. The lorry left town M at 8: 15 a.m. The car left town M and overtook the lorry at 12.15 p.m.
Calculate the time the car left town M.
14. A bus left Mombasa and traveled towards Nairobi at an average speed of 60 km/hr. after 2^1/[2] hours; a car left Mombasa and traveled along the same road at an average speed of 100 km/ hr. If the
distance between Mombasa and Nairobi is 500 km, Determine
1. The distance of the bus from Nairobi when the car took off (2mks)
2. The distance the car traveled to catch up with the bus
2. Immediately the car caught up with the bus
3. The car stopped for 25 minutes. Find the new average speed at which the car traveled in order to reach Nairobi at the same time as the bus.
15. A rally car traveled for 2 hours 40 minutes at an average speed of 120 km/h. The car consumes an average of 1 litre of fuel for every 4 kilometers. A litre of the fuel costs Kshs 59. Calculate
the amount of money spent on fuel
16. A passenger notices that she had forgotten her bag in a bus 1 2 minutes after the bus had left. To catch up with the bus she immediately took a taxi which traveled at 95 km/hr. The bus maintained
an average speed of 75 km/hr. Determine
1. The distance covered by the bus in 12 minutes
2. The distance covered by the taxi to catch up with the bus
17. The athletes in an 800 metre race take 1 04 seconds and 1 08 seconds respectively to complete the race. Assuming each athlete is running at a constant speed. Calculate the distance between them
when the faster athlete is at the finishing line.
18. Mwangi and Otieno live 40 km apart. Mwangi starts from his home at 7.30 am and cycles towards Otieno’s house at 1 6 km/ h Otieno starts from his home at 8.00 and cycles at 8 km/h towards Mwangi
at what time do they meet?
19. A train moving at an average speed of 72 km/h takes 1 5 seconds to completely cross a bridge that is 80m long.
1. Express 72 km/h in metres per second
2. Find the length of the train in metres | {"url":"https://www.easyelimu.com/high-school-notes/maths/form-2/item/1946-linear-motion","timestamp":"2024-11-13T15:04:22Z","content_type":"text/html","content_length":"171725","record_id":"<urn:uuid:b59e4b4e-7254-44af-9d4b-8fb657fb5dfa>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00708.warc.gz"} |
Current Refinance Rates
Property Value
To see what mortgage and refinance rates you could qualify for, enter your current property value. You likely won't have done an appraisal, so make this a best-guess effort. Take a look at some other
homes in your area that have recently sold and find ones that are comparable to yours. Once you look at a few of those homes, you should start to get a broad sense of what your home is worth. Put
what you think your home is worth in this field.
Loan Balance
Enter your current loan balance. You'll usually be able to find this information if you request a mortgage payoff statement from your lender. This statement will say something like payoff amount of
$250,000 good through December 30. This statement means that if you pay that amount of money before December 30, the bank will consider the loan closed. Since a refinance is merely taking out a new
mortgage and paying off the old loan with the money, that payoff amount is the exact value you'll want for these calculations!
Cash Out
If you'd like to cash out some of your home's equity, enter the amount you'd like in this text box. When looking at refinance rates on this calculator, you'll likely see that they are much less
expensive than other forms of borrowing, such as personal loans or credit cards. When refinancing your loan, you can typically borrow up to 80% of your home's value. Let's suppose you have a $200,000
balance on your loan and a home worth $500,000. Therefore, you can refinance for any amount between $200,000 and $400,000. The bank will give you any amount above $200,000 as cash. Many people use
this money to pay for trips, pay off debt, make big purchases, or invest. If you'd like to get some money out as cash, enter it here.
Loan Types
Select the type of loan for which you would like to see the refinance rates. There are three possible loan types from which you can choose.
The first is a conventional loan. This loan type is the most common and the most straightforward. When applying for a conventional refinance mortgage, you can refinance up to 80% of your home's
value. The bank has a set of criteria that they'll use to determine if you qualify or not, which typically means a credit score of at least 620 (although 680+ is preferable) and a debt-to-income
ratio of no more than 45% when factoring in the refinanced loan.
The second loan type is an FHA loan. With this replacement mortgage, you'll have a loan backed by the Federal Housing Administration. These loans have looser requirements. You'll need a minimum
credit score of 580 and a debt-to-income ratio of 43% maximum. You can refinance up to 80% of your home's value. The downside is that you must pay for private mortgage insurance - even with 20%
equity - which tends to make this option more expensive.
Finally, the last (and best option, if you qualify) is a VA refinance. This option has significant benefits, allowing refinancing for up to 100% of your home's value (including cash out!), as well as
no mortgage insurance costs. The minimum credit score is 580, and, while there is no specific debt-to-income maximum, anything above 41% will encounter additional scrutiny.
Loan Products
Choose the type of loan that you want. Broadly, there are two types of loans, and each one has a drastic impact on your refinance rates.
Fixed loans are the first loan category. They have a fixed interest rate and monthly payment for the duration of the loan. Therefore, if your term is 30 years with a monthly cost of $1,500 per month
at 3.5% APR, that payment, term length, and interest rate will never change. The only way you can adjust it is to refinance. These loans typically have 30, 20, 15, or 10-year durations. The shorter
the term, the lower the interest rate!
The second loan category is an ARM or adjustable-rate mortgage. With this loan, you will have a fixed rate for a certain length of time (say, five years). After that, your interest rate will adjust
based on an index. They'll take that index (say, the one-year Treasury bills) and add a percentage to it. As an example, let's say you have a 5/1 ARM over a 30-year term. The interest rate is 2% for
the first five years, and then the one-year T-bill rate plus 2% each year after. Your monthly payment will be constant for the first five years, calculated at a 2% interest rate to pay off the loan
in 30 years. After year five, let's suppose the T-bill rate is 4%. Now, your interest rate jumps from 2% to 6% (4% T-bill + 2% markup)! This jump will boost your monthly payment significantly. Then,
in year seven, the T-bill rate goes back down to 1%. Now, your monthly payment accounts for a 3% interest rate (1% T-bill + 2% markup).
Due to the consistent monthly payments, usually fixed loans are the better choice for those looking to stay in their homes for the long-term. | {"url":"https://www.amortization-calc.com/refinance-rates/","timestamp":"2024-11-03T07:08:13Z","content_type":"text/html","content_length":"29090","record_id":"<urn:uuid:3d5a2f3f-f4db-41fa-95bc-82b5a5b7685d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00082.warc.gz"} |
Given that the remainder when f(x) = x3 − 2x2 − ax + b is divided by x − 1is 96 and that x − 4 is a factor, determine the values of a and b.
Given that the remainder when f(x) = x3 − 2x2 − ax + b is divided by x − 1is 96 and that x − 4 is a factor, determine the values of a and b.
Solution 1
To solve this problem, we will use the Remainder Theorem and the Factor Theorem.
Step 1: Use the Remainder Theorem The Remainder Theorem states that if a polynomial f(x) is divided by x - k, the remainder is f(k). In this case, we know that the remainder when f(x) is divided by x
- 1 is 96. Therefo Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions. | {"url":"https://knowee.ai/questions/38615817-given-that-the-remainder-when-fx-x-x-ax-b-is-divided-by-x-is-and-that","timestamp":"2024-11-08T19:15:09Z","content_type":"text/html","content_length":"364556","record_id":"<urn:uuid:e2705b42-2ad6-45c8-9e0e-6aeae620752d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00799.warc.gz"} |
What is def2 TZVP?
def2-TZVPP – Valence triple-zeta with two sets of polarization functions. def2-TZVPPD – Valence triple-zeta with two sets of polarization functions and a set of diffuse functions.
What is 6 31G basis set?
6-31G means each inner shell (1s orbital) STO is a linear combination of 6 primitives and each valence shell STO is split into an inner and outer part (double zeta) using 3 and 1 primitive Gaussians,
respectively (see Table 11.2. 1 for other examples). Basis set.
What is TZVP basis set?
TZVP (triple zeta for valence electrons plus polarzation function) and DZVP should be basis sets not density functionals. TZVP is a larger basis set then DZVP. Since TZVP is a larger basis set set it
should give more accurate results then DZVP but may take a long time depending on the size of the system.
What is LANL2DZ basis set?
LANL2DZ (Los Alamos National Laboratory 2 Double-Zeta), which is a widely used ECP type basis set was used to model the metal atoms19. This mixed basis set was created through the use of the GEN
keyword in Gaussian 03.
How do you select basis sets?
Here, you can choose three different situations.
1. Use different basis set for each atom in order to reduce the calculation time.
2. Use the same basis set for all atom.
3. Use Complete basis set extrapolation (CBS). In my opinion this is one of the best option for high accuracy calculations.
What is effective core potential?
Effective core potentials (ECPs) are used to replace the inner (core) electrons of atomic and molecular systems by an effective potential and treat only the valence electrons explicitly in quantum
mechanical calculations.
What is a basis set in quantum mechanics?
A basis set is a set of functions combined linearly to model molecular orbitals. Basis functions can be considered as representing the atomic orbitals of the atoms and are introduced in quantum
chemical calculations because the equations defining the molecular orbitals are otherwise very difficult to solve. | {"url":"https://www.sweatlodgeradio.com/what-is-def2-tzvp/","timestamp":"2024-11-06T06:05:35Z","content_type":"text/html","content_length":"128137","record_id":"<urn:uuid:700bc286-2c74-4bb3-ba52-dea53b40cc70>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00600.warc.gz"} |
Introduction to Stacks
Stacks are one of the fundamental data structures in computer science, essential for a wide range of algorithms and applications. A stack is a linear data structure that follows a particular order in
which operations are performed. The order may be LIFO (Last In, First Out), which means that the last element added to the stack will be the first one to be removed. This characteristic makes stacks
particularly useful in scenarios where you need to reverse the order of items or handle nested structures, such as in recursive algorithms.
Basic Operations
A stack typically supports the following core operations:
1. Push: Add an item to the top of the stack.
2. Pop: Remove the item from the top of the stack.
3. Peek (or Top): Get the value of the top item without removing it.
4. IsEmpty: Check whether the stack is empty.
5. IsFull (optional, for fixed-size stacks): Check whether the stack is full.
These operations are designed to have an O(1) time complexity, meaning they can be performed in constant time.
Real-World Analogy
A common analogy to understand how a stack works is to think of a stack of plates. You add plates to the stack one by one, and when you need a plate, you take the one on top. You can't take a plate
from the middle or the bottom without first removing the ones on top. This is exactly how a stack operates.
Representation of Stacks
Stacks can be implemented in various ways, but the two most common methods are:
1. Array-Based Implementation: This approach uses an array to store the stack elements. It's simple and efficient but requires a predefined size. If the stack exceeds this size, it either needs to
be resized (which can be costly) or throws an overflow error.
2. Linked List-Based Implementation: This method uses a linked list where each node points to the next node in the sequence. This implementation is more flexible in terms of size but may require
more memory due to the storage of pointers.
Applications of Stacks
Stacks are used in numerous algorithms and real-world applications:
• Expression Evaluation: Stacks are used to evaluate expressions in postfix (Reverse Polish notation) and convert infix expressions (common arithmetic notation) to postfix or prefix.
• Backtracking: Many algorithms, like solving mazes or puzzles, use stacks to remember paths and backtrack when a dead-end is reached.
• Function Calls: In most programming languages, the call stack is used to manage function calls, local variables, and return addresses.
• Undo Mechanism: Many software applications use stacks to implement the undo feature, where the last operation can be undone first.
Implementing Stacks Using Arrays
Implementing a stack using an array is one of the simplest and most efficient ways to represent a stack. In this guide, we'll walk through the process of creating a stack using an array in the C
programming language. We'll cover the core stack operations: push, pop, peek, isEmpty, and isFull.
Step 1: Define the Stack Structure
First, let's define the structure of the stack. We need to store the stack's elements in an array, keep track of the top element, and define the maximum size of the stack.
#include <stdio.h>
#include <stdlib.h>
#define MAX 100 // Define the maximum size of the stack
typedef struct Stack {
int items[MAX]; // Array to store the stack elements
int top; // Index of the top element in the stack
} Stack;
Step 2: Initialize the Stack
Next, we'll create a function to initialize the stack. Initially, the stack is empty, so we set top to -1.
void initStack(Stack* stack) {
stack->top = -1;
Step 3: Check if the Stack is Full
Before pushing an element onto the stack, we need to ensure there's enough space. The stack is full when top equals MAX - 1.
int isFull(Stack* stack) {
return stack->top == MAX - 1;
Step 4: Check if the Stack is Empty
Similarly, before popping an element, we should check if the stack is empty. The stack is empty when top equals -1.
int isEmpty(Stack* stack) {
return stack->top == -1;
Step 5: Push an Element onto the Stack
To push an element onto the stack, we first check if the stack is full. If not, we increment top and add the new element at that position.
void push(Stack* stack, int item) {
if (isFull(stack)) {
printf("Stack Overflow\n");
stack->items[++(stack->top)] = item;
printf("%d pushed to stack\n", item);
Step 6: Pop an Element from the Stack
To pop an element, we first check if the stack is empty. If it's not, we return the element at the top and decrement the top index.
int pop(Stack* stack) {
if (isEmpty(stack)) {
printf("Stack Underflow\n");
return -1; // Return an invalid value to indicate underflow
return stack->items[(stack->top)--];
Step 7: Peek at the Top Element of the Stack
The peek operation returns the element at the top of the stack without removing it.
int peek(Stack* stack) {
if (isEmpty(stack)) {
printf("Stack is empty\n");
return -1; // Return an invalid value to indicate the stack is empty
return stack->items[stack->top];
Step 8: Demonstrate the Stack Operations
Finally, let's create a main function to demonstrate the stack operations.
int main() {
Stack stack;
push(&stack, 10);
push(&stack, 20);
push(&stack, 30);
printf("%d popped from stack\n", pop(&stack));
printf("Top element is %d\n", peek(&stack));
return 0;
Step 9: Compile and Run the Program
To compile and run the program, you can use the following commands in your terminal or command prompt:
gcc -o stack stack.c
Expected Output
When you run the program, you should see the following output:
10 pushed to stack
20 pushed to stack
30 pushed to stack
30 popped from stack
Top element is 20
1. The push operation adds elements 10, 20, and 30 to the stack.
2. The pop operation removes the top element (30) from the stack.
3. The peek operation returns the current top element (20) without removing it.
Implementing Stacks Using Linked Lists
A stack can also be implemented using a linked list, which offers greater flexibility than an array-based implementation. Unlike an array, a linked list does not require a predetermined size, making
it a dynamic structure that can grow or shrink as needed. In this guide, we'll walk through implementing a stack using a linked list in C, followed by a comparison of the pros and cons relative to an
array-based stack.
Step 1: Define the Node Structure
In a linked list, each element (node) contains the data and a pointer to the next node. For a stack, we need to define a node structure that holds an integer value and a pointer to the next node.
#include <stdio.h>
#include <stdlib.h>
// Define the structure of a node in the linked list
typedef struct Node {
int data;
struct Node* next;
} Node;
Step 2: Define the Stack Structure
For the stack, we only need a pointer to the top node of the linked list.
typedef struct Stack {
Node* top; // Pointer to the top node
} Stack;
Step 3: Initialize the Stack
We'll create a function to initialize the stack. Initially, the stack is empty, so the top pointer is set to NULL.
void initStack(Stack* stack) {
stack->top = NULL;
Step 4: Check if the Stack is Empty
Before performing operations like pop or peek, we should check if the stack is empty. The stack is empty when top is NULL.
int isEmpty(Stack* stack) {
return stack->top == NULL;
Step 5: Push an Element onto the Stack
To push an element onto the stack, we need to create a new node, set its data field, and point its next field to the current top node. Then, we update the top pointer to point to this new node.
void push(Stack* stack, int item) {
Node* newNode = (Node*)malloc(sizeof(Node));
if (newNode == NULL) {
printf("Stack Overflow\n");
newNode->data = item;
newNode->next = stack->top;
stack->top = newNode;
printf("%d pushed to stack\n", item);
Step 6: Pop an Element from the Stack
To pop an element from the stack, we first check if the stack is empty. If it's not, we store the current top node's data, update the top pointer to point to the next node, free the old top node, and
return the stored data.
int pop(Stack* stack) {
if (isEmpty(stack)) {
printf("Stack Underflow\n");
return -1;
Node* temp = stack->top;
int poppedData = temp->data;
stack->top = stack->top->next;
return poppedData;
Step 7: Peek at the Top Element of the Stack
The peek operation returns the element at the top of the stack without removing it.
int peek(Stack* stack) {
if (isEmpty(stack)) {
printf("Stack is empty\n");
return -1;
return stack->top->data;
Step 8: Demonstrate the Stack Operations
Let's create a main function to demonstrate the stack operations.
int main() {
Stack stack;
push(&stack, 10);
push(&stack, 20);
push(&stack, 30);
printf("%d popped from stack\n", pop(&stack));
printf("Top element is %d\n", peek(&stack));
return 0;
Step 9: Compile and Run the Program
To compile and run the program, you can use the following commands in your terminal or command prompt:
gcc -o stack stack.c
Expected Output
When you run the program, you should see the following output:
10 pushed to stack
20 pushed to stack
30 pushed to stack
30 popped from stack
Top element is 20
Pros and Cons of Linked List-Based Stacks
1. Dynamic Size: Unlike array-based stacks, linked list-based stacks can grow or shrink dynamically, so there's no need to worry about stack overflow (as long as there is memory available).
2. No Wasted Space: Memory is allocated as needed, so there’s no need to allocate a fixed-size block of memory upfront.
3. No Resizing Needed: There’s no need to resize the stack, as would be necessary with an array when the number of elements exceeds its capacity.
1. Memory Overhead: Each node requires additional memory for the pointer to the next node, which can lead to higher memory usage compared to array-based stacks, especially for small data elements.
2. Slower Access: Linked list elements are not stored in contiguous memory locations, leading to potentially slower access times compared to arrays due to cache misses.
3. Complexity: Linked list-based implementations are generally more complex to implement and manage due to the need for dynamic memory allocation and deallocation.
Stack ADT: Interface and Descriptions
A Stack Abstract Data Type (ADT) defines a stack as a collection of elements with specific operations, regardless of the underlying implementation (array-based, linked list-based, etc.). The Stack
ADT provides a clear and consistent way to interact with the stack, ensuring that users of the stack can perform the necessary operations without needing to know the details of how the stack is
Following is the typical interface of a Stack ADT and what each function provides:
1. push(item)
□ Description: Adds an item to the top of the stack.
□ Purpose: This operation increases the size of the stack by one. The newly added element becomes the top element of the stack.
□ Preconditions: The stack must not be full (if there's a maximum capacity).
□ Postconditions: The stack contains the new item at the top.
2. pop()
□ Description: Removes and returns the item at the top of the stack.
□ Purpose: This operation decreases the size of the stack by one. It provides access to the most recently added item and removes it from the stack.
□ Preconditions: The stack must not be empty.
□ Postconditions: The stack no longer contains the item that was at the top, and the new top is the element that was below the removed item.
3. peek() (or top())
□ Description: Returns the item at the top of the stack without removing it.
□ Purpose: Allows the user to view the top element of the stack without altering the stack’s state.
□ Preconditions: The stack must not be empty.
□ Postconditions: The stack remains unchanged.
4. isEmpty()
□ Description: Returns true if the stack is empty, false otherwise.
□ Purpose: This operation checks if the stack contains any elements.
□ Preconditions: None.
□ Postconditions: None, the stack remains unchanged.
5. isFull() (optional, for fixed-size stacks):
□ Description: Returns true if the stack is full, false otherwise.
□ Purpose: This operation checks if the stack has reached its maximum capacity, preventing further push operations if true.
□ Preconditions: The stack has a maximum size.
□ Postconditions: None, the stack remains unchanged.
6. size() (optional):
□ Description: Returns the number of elements currently in the stack.
□ Purpose: This operation allows the user to query how many items are currently stored in the stack.
□ Preconditions: None.
□ Postconditions: None, the stack remains unchanged.
Arithmetic Expressions: Infix, Prefix, and Postfix Notations
Arithmetic expressions can be represented in three different notations: Infix, Prefix, and Postfix. Understanding how to convert between these notations is essential for working with expression
evaluation algorithms, especially in the context of compilers and interpreters.
1. Infix Notation
Infix notation is the most common way of writing expressions, where operators are placed between operands.
2. Prefix Notation (Polish Notation)
In prefix notation, also known as Polish notation, the operator precedes its operands. This notation eliminates the need for parentheses to indicate operator precedence, as the order of operations is
dictated by the position of the operators.
• Infix: A + B
• Prefix: + A B
3. Postfix Notation (Reverse Polish Notation)
In postfix notation, also known as Reverse Polish Notation (RPN), the operator follows its operands. Like prefix notation, postfix notation does not require parentheses to enforce operator
• Infix: A + B
• Postfix: A B +
Example 1: Infix to Prefix and Postfix
Infix Expression:
(A + B) * (C - D)
Step-by-Step Conversion:
1. Infix to Prefix:
□ Start by identifying the operator with the lowest precedence that is outside parentheses. In this case, the multiplication operator * is the main operator.
□ The expression can be split into two sub-expressions: (A + B) and (C - D).
□ Convert these sub-expressions:
☆ (A + B) becomes + A B
☆ (C - D) becomes - C D
□ Finally, place the * operator before these sub-expressions.
□ Prefix: * + A B - C D
2. Infix to Postfix:
□ Start from the innermost expressions and move outward.
□ Convert (A + B) to A B +
□ Convert (C - D) to C D -
□ Finally, place the multiplication operator after these results.
□ Postfix: A B + C D - *
Summary for Example 1:
• Infix: (A + B) * (C - D)
• Prefix: * + A B - C D
• Postfix: A B + C D - *
Example 2: Infix to Prefix and Postfix
Infix Expression:
A * (B + C) / D - E
Step-by-Step Conversion:
1. Infix to Prefix:
□ Identify the main operator with the lowest precedence. Here, the subtraction - is the main operator.
□ The expression is split into two parts: A * (B + C) / D and E.
□ Convert A * (B + C) / D:
☆ * takes precedence over /, so it becomes * A + B C
☆ This then combines with / D to become / * A + B C D
□ Place the - operator before these results.
□ Prefix: - / * A + B C D E
2. Infix to Postfix:
□ Begin with the innermost expressions and convert them.
□ Convert (B + C) to B C +
□ Combine with A * (B + C) to get A B C + *
□ Combine with / D to get A B C + * D /
□ Finally, subtract E:
□ Postfix: A B C + * D / E -
Summary for Example 2:
• Infix: A * (B + C) / D - E
• Prefix: - / * A + B C D E
• Postfix: A B C + * D / E -
Evaluating Arithmetic Expressions Using Stacks
Evaluating arithmetic expressions is one of the classic applications of stacks in computer science. Let's focus on evaluating postfix expressions using stacks. This approach is straightforward
because postfix expressions eliminate the need for parentheses, and the order of operations is implicitly handled by the position of the operators.
Steps to Evaluate a Postfix Expression
1. Initialize an empty stack.
2. Scan the postfix expression from left to right.
□ If the token is an operand (number), push it onto the stack.
□ If the token is an operator, pop the top two elements from the stack, apply the operator, and push the result back onto the stack.
3. At the end of the expression, the stack will contain the final result.
For the postfix expression 2 3 1 * + 9 -:
• Push 2 (stack: 2)
• Push 3 (stack: 2, 3)
• Push 1 (stack: 2, 3, 1)
• Encounter *: Pop 1 and 3, compute 3 * 1 = 3, push 3 (stack: 2, 3)
• Encounter +: Pop 3 and 2, compute 2 + 3 = 5, push 5 (stack: 5)
• Push 9 (stack: 5, 9)
• Encounter -: Pop 9 and 5, compute 5 - 9 = -4, push -4 (stack: -4)
Final result is -4.
Below is a C program that evaluates a postfix expression:
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
// Define a stack structure
typedef struct Stack {
int* arr;
int top;
int capacity;
} Stack;
// Function to create a stack
Stack* createStack(int capacity) {
Stack* stack = (Stack*)malloc(sizeof(Stack));
stack->capacity = capacity;
stack->top = -1;
stack->arr = (int*)malloc(stack->capacity * sizeof(int));
return stack;
// Function to push an element onto the stack
void push(Stack* stack, int value) {
stack->arr[++stack->top] = value;
// Function to pop an element from the stack
int pop(Stack* stack) {
return stack->arr[stack->top--];
// Function to evaluate a postfix expression
int evaluatePostfix(char* exp) {
Stack* stack = createStack(100); // Assuming expression won't be longer than 100 characters
int i;
for (i = 0; exp[i]; i++) {
// If the character is a digit, push it onto the stack
if (isdigit(exp[i])) {
push(stack, exp[i] - '0');
// If the character is an operator, pop two elements from the stack, apply the operator, and push the result back
else {
int val1 = pop(stack);
int val2 = pop(stack);
switch (exp[i]) {
case '+': push(stack, val2 + val1); break;
case '-': push(stack, val2 - val1); break;
case '*': push(stack, val2 * val1); break;
case '/': push(stack, val2 / val1); break;
// The result of the expression is the last remaining element in the stack
return pop(stack);
// Main function to test the evaluation
int main() {
char exp[] = "231*+9-";
printf("Postfix Expression: %s\n", exp);
printf("Result: %d\n", evaluatePostfix(exp));
return 0;
Explanation of the Code
1. Stack Creation: The createStack function initializes a stack with a specified capacity.
2. Pushing and Popping: The push and pop functions handle pushing and popping elements from the stack.
3. Evaluating Postfix Expression: The evaluatePostfix function iterates through each character of the postfix expression. If it encounters a digit, it pushes it onto the stack. If it encounters an
operator, it pops the top two elements, applies the operator, and pushes the result back onto the stack.
4. Result: After processing the entire expression, the final result will be at the top of the stack.
1. Expression: "231*+9-"
□ Postfix Evaluation Steps:
☆ Stack after pushing 2: 2
☆ Stack after pushing 3: 2, 3
☆ Stack after pushing 1: 2, 3, 1
☆ After *: 2, 3
☆ After +: 5
☆ Stack after pushing 9: 5, 9
☆ After -: -4
□ Result: -4
2. Expression: "123+*8-"
□ Postfix Evaluation Steps:
☆ Stack after pushing 1: 1
☆ Stack after pushing 2: 1, 2
☆ Stack after pushing 3: 1, 2, 3
☆ After +: 1, 5
☆ After *: 5
☆ Stack after pushing 8: 5, 8
☆ After -: -3
□ Result: -3
• Check for Balanced Parentheses: Write a C program using a stack to check whether an input string of parentheses (including {}, [], and ()) is balanced. The program should return true if the
parentheses are balanced, and false otherwise.
• Reverse a String Using a Stack: Implement a C function that uses a stack to reverse a given string. The program should take the string as input and print the reversed string as output.
• Evaluate a Postfix Expression: Write a C program to evaluate a given postfix expression using a stack. The program should support basic arithmetic operators (+, -, *, /).
• Convert Infix Expression to Postfix: Develop a C program to convert an infix expression to its postfix form using a stack. The program should handle operators with different precedence and
• Find the Next Greater Element Using a Stack: Write a C program to find the next greater element for each element in an array using a stack. The program should output the next greater element or
-1 if there is none.
• Design a Special Stack that Supports getMin() in O(1) Time: Develop a C program to design a stack that, in addition to push and pop, supports a function getMin() that returns the minimum element
in the stack in O(1) time. The program should not use any auxiliary data structures.
1. Check for Balanced Parentheses
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// Structure for stack
struct Stack {
int top;
unsigned capacity;
char* array;
// Function to create a stack of given capacity
struct Stack* createStack(unsigned capacity) {
struct Stack* stack = (struct Stack*)malloc(sizeof(struct Stack));
stack->capacity = capacity;
stack->top = -1;
stack->array = (char*)malloc(stack->capacity * sizeof(char));
return stack;
// Stack utility functions
int isFull(struct Stack* stack) { return stack->top == stack->capacity - 1; }
int isEmpty(struct Stack* stack) { return stack->top == -1; }
void push(struct Stack* stack, char item) { if (!isFull(stack)) stack->array[++stack->top] = item; }
char pop(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top--]; return '\0'; }
char peek(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top]; return '\0'; }
// Function to check if the input character is a matching pair
int isMatchingPair(char char1, char char2) {
return ((char1 == '(' && char2 == ')') ||
(char1 == '{' && char2 == '}') ||
(char1 == '[' && char2 == ']'));
// Function to check if the parentheses are balanced
int areParenthesesBalanced(char* expr) {
int i = 0;
struct Stack* stack = createStack(strlen(expr));
while (expr[i]) {
// Push opening brackets to stack
if (expr[i] == '(' || expr[i] == '{' || expr[i] == '[')
push(stack, expr[i]);
// For closing brackets, check if top of stack matches
if (expr[i] == ')' || expr[i] == '}' || expr[i] == ']') {
if (isEmpty(stack) || !isMatchingPair(pop(stack), expr[i]))
return 0;
return isEmpty(stack);
int main() {
char expr[] = "{()}[]";
if (areParenthesesBalanced(expr))
printf("Not Balanced\n");
return 0;
2. Reverse a String Using a Stack
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// Structure for stack
struct Stack {
int top;
unsigned capacity;
char* array;
// Function to create a stack of given capacity
struct Stack* createStack(unsigned capacity) {
struct Stack* stack = (struct Stack*)malloc(sizeof(struct Stack));
stack->capacity = capacity;
stack->top = -1;
stack->array = (char*)malloc(stack->capacity * sizeof(char));
return stack;
// Stack utility functions
int isFull(struct Stack* stack) { return stack->top == stack->capacity - 1; }
int isEmpty(struct Stack* stack) { return stack->top == -1; }
void push(struct Stack* stack, char item) { if (!isFull(stack)) stack->array[++stack->top] = item; }
char pop(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top--]; return '\0'; }
// Function to reverse a string using a stack
void reverseString(char* str) {
int n = strlen(str);
struct Stack* stack = createStack(n);
// Push all characters of the string to the stack
for (int i = 0; i < n; i++) push(stack, str[i]);
// Pop characters from stack and put them back into the string
for (int i = 0; i < n; i++) str[i] = pop(stack);
int main() {
char str[] = "HelloWorld";
printf("Original String: %s\n", str);
printf("Reversed String: %s\n", str);
return 0;
3. Evaluate a Postfix Expression
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
// Structure for stack
struct Stack {
int top;
unsigned capacity;
int* array;
// Function to create a stack of given capacity
struct Stack* createStack(unsigned capacity) {
struct Stack* stack = (struct Stack*)malloc(sizeof(struct Stack));
stack->capacity = capacity;
stack->top = -1;
stack->array = (int*)malloc(stack->capacity * sizeof(int));
return stack;
// Stack utility functions
int isFull(struct Stack* stack) { return stack->top == stack->capacity - 1; }
int isEmpty(struct Stack* stack) { return stack->top == -1; }
void push(struct Stack* stack, int item) { if (!isFull(stack)) stack->array[++stack->top] = item; }
int pop(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top--]; return -1; }
// Function to evaluate a postfix expression
int evaluatePostfix(char* expr) {
struct Stack* stack = createStack(strlen(expr));
for (int i = 0; expr[i]; i++) {
if (isdigit(expr[i])) {
push(stack, expr[i] - '0'); // Push operand to stack
} else {
int val1 = pop(stack);
int val2 = pop(stack);
switch (expr[i]) {
case '+': push(stack, val2 + val1); break;
case '-': push(stack, val2 - val1); break;
case '*': push(stack, val2 * val1); break;
case '/': push(stack, val2 / val1); break;
return pop(stack); // The final result is on top of the stack
int main() {
char expr[] = "231*+9-";
printf("Postfix Expression: %s\n", expr);
printf("Evaluated Result: %d\n", evaluatePostfix(expr));
return 0;
4. Convert Infix Expression to Postfix
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// Structure for stack
struct Stack {
int top;
unsigned capacity;
int* array;
// Function to create a stack of given capacity
struct Stack* createStack(unsigned capacity) {
struct Stack* stack = (struct Stack*)malloc(sizeof(struct Stack));
stack->capacity = capacity;
stack->top = -1;
stack->array = (int*)malloc(stack->capacity * sizeof(int));
return stack;
// Stack utility functions
int isFull(struct Stack* stack) { return stack->top == stack->capacity - 1; }
int isEmpty(struct Stack* stack) { return stack->top == -1; }
void push(struct Stack* stack, char op) { if (!isFull(stack)) stack->array[++stack->top] = op; }
char pop(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top--]; return -1; }
char peek(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top]; return -1; }
// Function to check if character is an operand
int isOperand(char ch) { return (ch >= 'A' && ch <= 'Z') || (ch >= 'a' && ch <= 'z'); }
// Function to get the precedence of operators
int precedence(char ch) {
switch (ch) {
case '+':
case '-':
return 1;
case '*':
case '/':
return 2;
case '^':
return 3;
return -1;
// Function to convert infix expression to postfix
int infixToPostfix(char* exp) {
int i, k;
struct Stack* stack = createStack(strlen(exp));
if (!stack) return -1;
for (i = 0, k = -1; exp[i]; ++i) {
// If the character is an operand, add it to output
if (isOperand(exp[i])) exp[++k] = exp[i];
// If the character is '(', push it to stack
else if (exp[i] == '(') push(stack, exp[i]);
// If the character is ')', pop and output from the stack until '(' is found
else if (exp[i] == ')') {
while (!isEmpty(stack) && peek(stack) != '(')
exp[++k] = pop(stack);
pop(stack); // Discard '('
// If an operator is encountered
else {
while (!isEmpty(stack) && precedence(exp[i]) <= precedence(peek(stack)))
exp[++k] = pop(stack);
push(stack, exp[i]);
// Pop all the operators from the stack
while (!isEmpty(stack)) exp[++k] = pop(stack);
exp[++k] = '\0'; // Null-terminate the postfix expression
return k;
int main() {
char exp[] = "A+B*(C^D-E)";
printf("Infix Expression: %s\n", exp);
fix Expression: %s\n", exp);
return 0;
5. Find the Next Greater Element Using a Stack
#include <stdio.h>
#include <stdlib.h>
// Structure for stack
struct Stack {
int top;
unsigned capacity;
int* array;
// Function to create a stack of given capacity
struct Stack* createStack(unsigned capacity) {
struct Stack* stack = (struct Stack*)malloc(sizeof(struct Stack));
stack->capacity = capacity;
stack->top = -1;
stack->array = (int*)malloc(stack->capacity * sizeof(int));
return stack;
// Stack utility functions
int isFull(struct Stack* stack) { return stack->top == stack->capacity - 1; }
int isEmpty(struct Stack* stack) { return stack->top == -1; }
void push(struct Stack* stack, int item) { if (!isFull(stack)) stack->array[++stack->top] = item; }
int pop(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top--]; return -1; }
int peek(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top]; return -1; }
// Function to print the next greater element for each element in the array
void printNGE(int arr[], int n) {
int i = 0;
struct Stack* stack = createStack(n);
push(stack, arr[0]);
for (i = 1; i < n; i++) {
if (isEmpty(stack)) {
push(stack, arr[i]);
while (!isEmpty(stack) && peek(stack) < arr[i]) {
printf("%d -> %d\n", pop(stack), arr[i]);
push(stack, arr[i]);
while (!isEmpty(stack)) {
printf("%d -> %d\n", pop(stack), -1);
int main() {
int arr[] = {11, 13, 21, 3};
int n = sizeof(arr)/sizeof(arr[0]);
printNGE(arr, n);
return 0;
6. Design a Special Stack that Supports getMin() in O(1) Time
#include <stdio.h>
#include <stdlib.h>
#include <limits.h>
// Structure for the special stack
struct Stack {
int top;
unsigned capacity;
int* array;
int* minArray; // Additional array to keep track of minimum elements
// Function to create a stack of given capacity
struct Stack* createStack(unsigned capacity) {
struct Stack* stack = (struct Stack*)malloc(sizeof(struct Stack));
stack->capacity = capacity;
stack->top = -1;
stack->array = (int*)malloc(stack->capacity * sizeof(int));
stack->minArray = (int*)malloc(stack->capacity * sizeof(int));
return stack;
// Stack utility functions
int isFull(struct Stack* stack) { return stack->top == stack->capacity - 1; }
int isEmpty(struct Stack* stack) { return stack->top == -1; }
void push(struct Stack* stack, int item) {
if (!isFull(stack)) {
stack->array[++stack->top] = item;
if (stack->top == 0)
stack->minArray[stack->top] = item;
stack->minArray[stack->top] = (item < stack->minArray[stack->top - 1]) ? item : stack->minArray[stack->top - 1];
int pop(struct Stack* stack) { if (!isEmpty(stack)) return stack->array[stack->top--]; return INT_MAX; }
int getMin(struct Stack* stack) { if (!isEmpty(stack)) return stack->minArray[stack->top]; return INT_MAX; }
// Function to return the minimum element in O(1) time
int getMinimum(struct Stack* stack) {
return getMin(stack);
int main() {
struct Stack* stack = createStack(100);
push(stack, 10);
push(stack, 20);
printf("Minimum element is %d\n", getMinimum(stack));
push(stack, 5);
printf("Minimum element is %d\n", getMinimum(stack));
printf("Minimum element is %d\n", getMinimum(stack));
return 0; | {"url":"https://blog.jyotiprakash.org/introduction-to-stacks","timestamp":"2024-11-11T00:01:14Z","content_type":"text/html","content_length":"859918","record_id":"<urn:uuid:37c8946c-d294-4d8e-a023-f22300fd06c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00230.warc.gz"} |
AMA3007 Assignment 1 solution
Exercises 1.2.10, 1.3.9, 1.4.3, 1.5.6, 2.2.2 (b), 2.3.1 (b), 2.3.3, 2.4.3 (b),
2.5.5, 2.6.5, 2.7.2 (c), (e).
Exercise 1.2.10.
Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not:
(a) Two real numbers satisfy a <b if and only if a <b+e for every € > 0. (b) Two real numbers satisfy a <b if a < be for every € > 0.
(c) Two real numbers satisfy a <b if and only if a <b+€ for every € > 0.
Exercise 1.3.9.
(a) If sup A < sup B, show that there exists an element be B that is an upper bound for A.
(b) Give an example to show that this is not always the case if we only assume sup A≤ sup B.
Exercise 1.4.3.
Prove that 1(0,1/n) = 0. Notice that this demonstrates that the intervals in the Nested Interval Property must be closed for the con- clusion of the theorem to hold.
Exercise 1.5.6.
(a) Give an example of a countable collection of disjoint open intervals.
(b) Give an example of an uncountable collection of disjoint open intervals, or argue that no such collection exists.
Exercise 2.2.2.
Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. (a) lim 2n+1
(b) lim
2n2 n3+3
= 0.
(c) lim sin()
Exercise 2.3.1.
Let n > 0 for all n e N.
(a) If (n) 0, show that (√) → 0.
(b) If (n), show that (√) →√x.
Exercise 2.3.3 (Squeeze Theorem).
Show that if nyn zn for all nЄN, and if liman lim z = 1, then lim yn as well.
Exercise 2.4.3.
(a) Show that
√2, √2+ √2,√2+ √2+ √2,…
converges and find the limit.
(b) Does the sequence
converge? If so, find the limit.
Exercise 2.5.5.
Assume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a € R. Show that (an) must converge to a.
Exercise 2.6.5.
Consider the following (invented) definition: A sequence (sn) is pseudo-Cauchy if, for all > 0, there exists an N such that if n > N, then Sn+1
Sn❘ < €.
Decide which one of the following two propositions is actually true. Supply a proof for the valid statement and a counterexample for the other.
(i) Pseudo-Cauchy sequences are bounded.
(ii) If (2) and (v) are pseudo-Cauchy, then (a,+ n) is pseudo-Cauchy as well.
Exercise 2.7.2.
Decide whether each of the following series converges or diverges:
(2) ΣΕΞ1
n=1 2″+n
(1) ΣΕΞ1
sin(n) n2
(c) 1 – 1+砉-音+湯一西 +. 12
(d) 1+ + + + + +
(e) 1-+-+-+-+.. | {"url":"https://jarviscodinghub.com/assignment/ama3007-assignment-1-solution/","timestamp":"2024-11-03T19:51:39Z","content_type":"text/html","content_length":"102250","record_id":"<urn:uuid:3a783d16-e183-4f7d-8bde-9cbe0fba1ba7>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00010.warc.gz"} |
back start
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [ 30 ]
III 73 < B1S6»
10 Baa let
h;i$6 iso iti i67 wa
) 891 Q9j*94ki6« 97 93» eefliou
;?j 7 4 m JT« 7 » i» « ta?\jai m » / 9oi eti
56 688 6 66 67 7 6Tf 6T3 674 67$1s
9(U 9M 005
90C 10»
Bin WII mi WI3 tgu ! 1011 iwaioa 1 \fti
THE Ganns Pyramid, Square of Nine Essentials By Daniel T. Ferrera
The following titles relate to the contents of the Square of Nine Essentials Course. These titles present background theories from science and metaphycs which relate speciScatty lo the Square of
Nine, its origins and uses as a calculator. Most of these books are available through the Satred Science Institute:
<• The CoUecUd Writings Of W. D. Gaim Voluma /- Wlliam Ddbert G«ib 1909 -1955. Anyone who desires to undemand the ideas of W. D. Gann should not be without his complete writings, particularly the
stock and commodity courses, wherein all of the basis his techniques and theories is presented with the greatest elaboration. This set, published by the Sacred Science Institute, is the most complete
and best organized collection of Ganns writings from which much of the work in this course was derived, and is recommended as the most usehil collection of Gann*s writings.
❖ Magic Squares Cubes. W. S. Andrews. 1917. The most complete collection of writings oo every aspect of Mac Squares and Cubes by over a Dozen Excellent Authors. Fundamental for understanding the
Logic behind Magical Numerical and Geometric Representational Calculatcrs and Ganns Squares.
<t The Temple Of Man: Apet Of The South ofLuxar. R. A. Schwaller de Lubicz. Truncated From Tbe French By Robert & Deborah Lawlor. 1999. This 51 ! , besides containing the most developed presentation
of esoteric mathematics and geometry usehil for market study, includes a section on the Square of Nine as derived from India and connected with the Temple of Luxor,
An Important Question tn Metrology, Charles Totteo. 1884. A very important work on metrology (the sacred science of measure) with a particularly iinportant appendix on Ihe 5x5 square (the basis of
Ganns square of nine) as the besis of the grest pyramid & as a template squaring the circle.
•> The Itttriusie Harmony Of Number, Parts I-IV, Part 1 The Squares (rfBeBJamin Franklin. Fart The Magk Squares Of The Fifth & Seventh Orders. Part Magic Squares of The Orders Three-Six-Nine &
Twelve. Part IV The Asxiliary Square. Clarence Marker. 1940-41.
A very original and detailed work on number squares and their mathematical significance.
❖ The Cosrnoiogicai Freemasonry of Frank Higgiti5.X)xt < 1 haVwtiiiazoilb Greetett Of All Ancieat Mysteries. A.U.M.e Loat Wcn)".The BegbiningOf Masoary, Forty Papers On the Hklden Mysteries Of
Ancient Freemasonry. Franli C. Higghis. 1912-16. These writings are amongst the best geometrical, maAematical, & arithmetical unveilings ofthe secret Freemasonic cosmological science including the
use of mathematical squares as calculators.
❖ Science Based Upon Symmetry. Chakravorty. 1974. An interesting wotK on how the 5 x 5 square worics as a mathematic calculator to solve many forms of mathematical and scientific problems.
The 1 1 Ttmplef 2 Volumes. Stella Kramrisch. 1976. An excellent work on the symbolism of Hindu Temple architecture, giving us one ofthe oldest examplcsof the Square of Nine with its symbolism and
meaning. An essential WOTk!
❖ The Booh Of Magic Squares, 3 Volumes. Jain. 2001. An cxcaJlent presentation of magic squares, their properties, meanings, how to crente them and use them as number calculators.
The Cosmological Freemasonry of Frank Higgim, 1912-16. This important work contains many excellent keys to geometric and numerical calculators used in Gann v/ork.
❖ lo Unveiled. Bozcna Brydlova. 1922. Very important work cn number theory, not to be missed!
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [ 30 ] | {"url":"http://eng.yax.su/finlab/ir018/30/index.shtml","timestamp":"2024-11-06T02:46:24Z","content_type":"text/html","content_length":"11952","record_id":"<urn:uuid:50f85c0c-68f6-417d-a69b-e4b2416587de>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00540.warc.gz"} |
What is Probability?
When a coin is tossed, the mathematician says there is 50% chance it will be head and 50% chance it will be tail. What does it all mean? The outcome is determined to be either a head or a tail, we do
not know the outcome prior to the event. If we do not know the outcome, what meaning is there to speak of the unknown by assigning equally meaningless numbers to the unknown outcome.
My understanding is that our description of probability is dependant on our feeling of certainty. In instances when we are very certain of an outcome we say its probability is 100%. when we are more
than ordinarily certain, we say its probability is 140%. when we are not sure, we say probability is 50%. however, the probability we assigned has no mathematical basis rather, it is a description of
our will to certainty.
My rule is that if we do not know about it meaning it is a unknown, it is meaningless to speak about it, so we ought not to speak about it.
Let’s look at it this way, if there is 50% chance for heads or tails, which one would you pick? Does the ‘50% chance’ say anything, apart from an admission of uncertainty?
(pertaining to the bolded part): I don’t think you can say this in terms of predictive probability. You can say something like… ‘There was a 130% rise in earnings this year’ but I don’t think you can
use anything above 100 in the sense you’re talking about… then again I could be wrong.
You’re right that induction is basically a guess… but I mean, in terms of science, it’s worked fairly good so far I’d say. | {"url":"https://www.ilovephilosophy.com/t/what-is-probability/7163","timestamp":"2024-11-15T00:48:53Z","content_type":"text/html","content_length":"17809","record_id":"<urn:uuid:0703486d-a750-4aa3-9771-8fc6356822c0>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00569.warc.gz"} |
LIGON – Link Discovery with Noisy Oracles (OM-2020 - Long technical paper)
← Go back
LIGON – Link Discovery with Noisy Oracles (OM-2020 - Long technical paper)
4 years ago by Dr. rer. nat. Mohamed Ahmed Sherif
The provision of links between knowledge graphs in RDF3 is of central importance for numerous tasks on the Semantic Web, including federated queries, question answering and data fusion. While links
can be created manually for small knowledge bases, the sheer size and number of knowledge bases commonly used in modern applications (e.g., DBpedia with more than 3 × 106 resources) demands the use
of automated link discovery mechanisms.
In this work, we focus on active learning for link discovery. State-of-the-art approaches that rely on active learning assume that the oracle they rely upon is perfect. Formally, this means that
given an oracle ω, the probability of the oracle returning a wrong result (i.e., returning false when an example is to be classified as true) is exactly 0. While these approaches show pertinent
results in evaluation scenarios, (in which the need for a perfect oracle can be fulfilled) this need is difficult, if not impossible to uphold in real-world settings (e.g., when crowdsourcing
training data). No previous work has addressed link discovery based on oracles that are not perfect.
We address this research gap by presenting a novel approach for learning link specifications (LS) from noisy oracles, i.e., oracles that are not guaranteed to return correct classifications. This
approach is motivated by the problem of learning LS using crowdsourcing. Previous works have shown that agents in real crowdsourcing scenarios are often not fully reliable. We model these agents as
noisy oracles, which provides erroneous answers to questions with a fixed probability. We address the problem of learning from such oracles by using a probabilistic model, which approximates the odds
of the answer of a set of oracles being correct. Our approach, dubbed Ligon, assumes that the underlying oracles are independent, i.e., that the probability distributions underlying oracles are
pairwise independent. Moreover, we assume that the oracles have a static behavior, i.e., that the probability of them generating correct/incorrect answers is constant over time.
The contributions of this paper are as follows:
1. We present a formalization of the problem of learning LS from noisy oracles. We derive a probabilistic model for learning from such oracles.
2. We develop the first learning algorithm dedicated to learning LS from noisy data. The approach combines iterative operators for LS with an entropy-based approach for selecting most informative
training examples. In addition, it uses cumulative evidence to approximate the probability distribution underlying the noisy oracles that provide it with training data.
3. We present a thorough evaluation of Ligon and show that it is robust against noise, scales well and converges with 10 learning iterations to more than 95% of the average F-measure achieved by
Wombat—a state-of-the-art approach for learning LS—provided with a perfect oracle.
Authors: Mohamed Ahmed Sherif, Kevin Dreßler, and Axel-Cyrille Ngonga Ngomo
Paper: https://papers.dice-research.org/2020/OM_LIGON/public.pdf
Github repository: https://github.com/dice-group/LIMES
Cite as:
author = {Sherif, Mohamed Ahmed and {Kevin Dreßler} and {Ngonga Ngomo}, Axel-Cyrille},
booktitle = {Proceedings of Ontology Matching Workshop 2020},
title = {{LIGON – Link Discovery with Noisy Oracles}},
url = {https://papers.dice-research.org/2020/OM_LIGON/public.pdf},
year = 2020 | {"url":"https://dice-research.org/news/2020-09-29-LIGON/","timestamp":"2024-11-15T04:21:51Z","content_type":"text/html","content_length":"116450","record_id":"<urn:uuid:28ec1194-3091-4c03-9456-d30cb354485f>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00769.warc.gz"} |
Our users:
Its been a long time since I needed to understand algebra and when it came time to helping my son, I couldnt do it. Now, with your algebra software, we are both learning together.
Linda Taylor, KY
I think the program is tremendous. We have only been using it a week, but it has already paid for itself. We are currently using it to "check" homework assignment on a child struggling in Algebra 2
in High School. I set it to save "99" steps and we can see every step of the solution. The explanations at each step are invaluable, since it has been many years since my Algebra days. We haven't had
a problem yet it couldn't solve. It is pretty user friendly, and, as long as you enter the problem correctly, there are no problems. Next year we have another child starting High School and Algebra
1. I will be looking forward to the next release of the Algebrator. It is great!
Charles B.,WI
I can't tell you how happy I am to finally find a program that actually teaches me something!!!
R.B., New Mexico
Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among
Search phrases used on 2010-01-13:
• math sats sheets to practis for children in year 2
• glencoe algebra answers
• slope using y intercept and slop
• free subtracting polynomials worksheets
• tricky aptitude questions
• texas instruments + solving equations + TI-84
• subtracting a negative fraction
• free help with college algebra midpoint
• maths rotations quiz
• math websites with 6th grade worksheets
• algebra substitution test ks3
• scale word problems
• square root of 108 in radical form
• What are the four fundamental math concepts used in evaluating an expression?
• free clep algebra papers
• free math homework cheating answers
• 8th grade algebra worksheet answers
• nc pre algebra eoc
• apptitude questions & answer
• parabolas algebra 11th grade
• examples of mathematical KS3 projects
• "Algebra With Pizzazz!
• help to graph the equation
• trigonometry basics for idiots
• stretch hyperbola
• percent formulas
• test + ks2 + doc + test + math
• polynomial cubed formula
• elementary algebra worksheets
• Evaluate rational expressions
• free school worksheets for year 8 to print of
• slope and y-intercept activities and TI-83
• saxon math algebra 2 answers
• terms,expressions and equations worksheet
• when do you need highest common factor
• Prentice Hall Mathematics: Algebra 1
• vertical stretches and comparisons solver
• grade 11 university math exam review
• 7th grade division sheet
• example of polynomial division in everyday life
• pre algebra software
• solving linear equations with two variables
• printable games involving linear equations and graphing
• solving non linear 2nd order differential
• how to put systems of equations in a TI-83
• mathmatical dictionary
• free online fraction calculator
• online holt algebra 1 books
• complex equations on ti89
• equation of a curved line
• fluid mechanics for dummies
• Real life examples of Combinations(Algebra 2)
• maths for dummies
• Advanced Algebra Essentials Holt TEahcers Book
• examples of parabolas in basketball
• square root (x-y)
• year 10 math algebra
• cube root on TI83
• intermedia algebra Whisker chart
• ti-84 summation
• factors of third order trinomials
• solving linear equations problems calculator
• Glencoe physics solution teacher edition download
• integrated 2 book mcdougal online
• graphing linear equations worksheets
• pre-algebra formulas
• Algebra 1 Holt notes
• interval notation converter
• linear Algebra with Applications free download
• free trig problem solver
• maths logarithm games seniors
• free downloadable textbooks for 9th grade
• vertex form
• free 9th grade algebra 1 printable worksheets
• factoring cubed polynomial
• download yr 9 maths sats paper
• least common denominator calculator
• esl handout worksheet
• how to divide polynomial fractions on the TI-89
• algebra powers addition
• fraction books 7th grade
• Square Root Formula
• Probability test - GRADE 2
• simplify equations
• free accounting worksheets | {"url":"https://algbera.com/school-algebra-math/explain-the-difference-between.html","timestamp":"2024-11-10T02:21:46Z","content_type":"text/html","content_length":"85369","record_id":"<urn:uuid:403d6e41-0d97-4d85-8170-0f8fd06a770c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00884.warc.gz"} |
On Dec 31, 2007, at 2:26 AM, BASHA ONE wrote:
I was forwarded your site from a person who bought one of my NOTHING graffiti art canvases. I like your site and I am especially glad to see nothing. I love nothing and have been painting nothing for
a while now. Check me at ebay under "BASHA ART"
To which I replied:
Thank you for your e-mail regarding nothing interesting. I must mirror your sentiments about nothing and add that I enjoy nothing more than you do. I know this may come as a shock to someone who has
painted nothing for so many years, but I have literally only made 3 paintings in the last 8 years. Absolutely all of the remaining time was dedicated to painting nothing. Once you develop that type
of will power and strength of vision, you too will understand what it is to paint nothing interesting, nothing unique, nothing original or even nothing important every time you don't paint. With just
a little more (or less) effort maybe even while you are painting you'll draw a blank, then perhaps a vortex, then maybe a black hole...
If you thought I was going somewhere with all this you still have a lot to learn about nothing. I looked at your work, and well I must say it seemed a bit agnomenistic (yes, it's a made up word, but
you know what I mean) I could tell you really cared about nothing, and by nothing I mean not just the word nothing, but the real meaning of that word as well.
Perhaps we will collaborate one day by not doing anything to all the dead space in all the galleries in the world. It will be the largest installation ever constructed, what will be even more amazing
is that it will require no effort. Are you with me? I'm on board, If you say yes then we've already done it. Of course, even if you don't say yes, we've pretty much already done it.
Here's to nothing collaborative!
--Xymyl (KON)
On Dec 13, 2007, at 3:13 AM, Nishant John wrote:
Change is constant, but nothing will be the same.
To which I replied:
Dear nothing unthusiast,
I appreciate your comments, however, I am expected to refute direct claims made regarding nothing. In this recent note you have given me (at most) very little to work with.
I must point out however, that it could also be said that nothing is constant because there will always be change. I personally don't like to use nothing in this thingesque way but it does have its
occasional advantages.
Actually, we appreciated your wording, because you used a single sentence tying the "change" and "nothing" together, then used "but" to show that the one and nothing weren't necessarily two things
that were related but could very likely be one thing and nothing with no further relationship beyond the sentence containing them. Very nice indeed!
May we uncourage you to continue to develop your appreciation for nothing specific.
--Xymyl (KON)
On Nov 19, 2007, at 9:44 AM, Per-Niklas Longberg wrote:
Where do you manufacture nothing?
To which I replied:
When we received your question we thought it was a very good question that required detailed lab work to determine the best answer. After more than 20 days in our lab, we were certain that the
research team would have a great answer for us. We were somewhat disappointed at the results. The research clearly shows that your question was not a very good question after all.
Nothing is a naturally occurring limited infinite resource. What I mean by this is that we don't need to manufacture nothing because "it" "is" in ready "supply" due to the fact that "it" "is" limited
to an infinite state of "being" due mostly to "it's" lack of properties, and partly due to the fact that nothing exists independently of anything. And we mean that in both ways.
So to get straight to the point, we manufacture nothing everywhere, because we don't. I hope this clears things (as they may or may not relate to nothing, which they don't) up.
--Xymyl (KON)
On Oct 12, 2007, at 8:23 AM, Andrew Sullivan wrote:
Hello, XYMYL. I am completely shocked to have discovered a website related, in every way possible, to a theory that my friend and I developed in our Physics class out of sheer boredom, and have
developing for the past month. I would like to hear back from you regarding nothing, and have an (if at all possible) discussion about nothing.
To which I replied:
Hi Andrew,
A conversation about nothing? There is nothing I would enjoy more. I have been very busy lately though, as busy as nothing you have ever seen. But I still make time for nothing, just as I always
I'm assuming that this theory of yours is actually the absence of a theory, or the theory that such a theory wouldn't exist. Am I close on that one? I would have a conversation regarding this
impossibility or others. Of course, the less said, the better.
Thanks for your interest in nothing.
--Xymyl (KON)
To which On Oct 17, 2007, at 8:14 AM, Andrew Sullivan wrote:
Well, as far as our theory, it's not so much the object of laziness, but an object itself. Nothing is an element. Atomic number 0. It is nowhere on the periodic table of elements, of course, because
nothing is not tangible. But nothing is. Nothing is everywhere. Everything is nowhere. If you think about it, nothing makes sense.
To which I replied:
I'm certainly happy to see that you've proved one of my theories false thru your response. My theory was, of course, that you can't learn nothing in school.
I have to say though, that assigning an atomic number to nothing (even if that number is zero, if that is what you propose) could generate even greater confusion. I'll give an example, not only is
nothing not an atom, "it" "is" also not light, gravity, or a thing made up of atoms such as a refrigerator. Should we be expected to make a new listing within every category just to point out that
there is nothing missing? Indeed, should every refrigerator manufacturer be required to create a ZERO model that doesn't exist? And what about Sub Zero? People might think the products they sell are
inverted refrigerators. They may have to re-brand and could (in the process) lose their proverbial shirts, thus having zero shirts.
I think that the fact that nothing isn't in the periodic table or the "chart" as we like to call it, is one of the greatest scientific achievements of all time. Imagine a world where science was so
accepting of nothing as a thing. Sucks right? In case you are not convinced, look at our modern dark age of communication. Nothing being categorized as a pronoun or even as a noun is commonplace, and
yet, "it" "is" neither. This is the linguistic equivalent of what you propose with nothing being added to the chart.
Imagine a world where science ebbed and flowed with the whims of the masses, as does language. Now imagine nothing. One totally different thing and no thing at all, right?
True science will often use the testable to "scratch the surface" of the intangible, but it will never test the intangible. For example, a vacuum such as the void of space can be interacted with
because there are properties to it. Although it may contain nothing in a general sense, it isn't nothing. True nothingness, would never be testable due to the total lack of properties, and the fact
that a testing device would introduce something into what once wasn't. Of course, here at nothing.net, we embrace nothing in all "its" forms, but especially in the lack of form.
To conclude... Nothing as a placeholder is. Nothing infinite isn't. Nothing as a joke is. Nothing serious isn't. Nothing real "is" because "it" isn't.
Thank you for your continued interest in nothing scientific.
--Xymyl (KON)
To Which On Oct 24, 2007, at 8:13 AM, Andrew Sullivan wrote:
If a refrigerator company manufactured a refrigerator brand 'zero', it'd have to exist. But by creating a brand 'zero', the company has created a meaning for 'zero'. As false meaning. 'Zero' does not
mean 'refrigerator'. 'Zero' means 'Nothing'. The absence of something. Nothing can define Zero. Nothing is indefinite. But, I suppose that outside the reaches of the result of the big bang, is
Nothing in its purist form. The element Nothing I'm referring to. It is void of all that is and isn't tangible. The universal absence.
To which I replied:
Hello again Andrew,
Obviously, the brand 'zero' refrigerator would have to exist if it was manufactured. However, a zero model could exist without the corresponding refrigerator having been manufactured. A zero model
that does not exist would be analogous to what you proposed regarding the element nothing. I said it wasn't such a hot idea. You seemed to think it was a great idea.
I appreciate that you point out that "zero" does not mean "refrigerator". I would like to add my corresponding claim that zero does not always equal nothing. I must also point out that nothing does
not always mean the absence of something.
Zero is functional, zero is an integral part of a modern ten based numerical system. Zero is also highly useful as an intermediate point between positive and negative numbers. Zero can point to a
lack or diminishment. And yes, as you rightly point out, zero can even mean nothing.
Although nothing CAN be used in functional in ways similar to zero, nothing is most widely used as no thing. This is not simply an absence of something, but nothing at all, making "it" independent of
everything. Since you make a "big bang" reference, I'm assuming that you appreciate the idea of a time when everything was not in existence. I'd like to think that such a time existed, but sadly, a
universe without anything would not be possible. And since nothing in the purest "form" would be featureless, "it" could never act or even react. This means that nothing would ever exist. So, clearly
that dream could never be reality, and if it could, no one would be around to know.
So, although we may use the word nothing for all of its purposes, we always hold in highest regard, true nothing. Unbounded, unfettered, unfiltered, un-acted upon by anything, and unable to interact
with anything. Incapable of propagating a universe or a grain of sand, or even a subatomic particle. I hope this helps you to see even more clearly why nothing amazes me.
Just as zero is the universal placeholder, nothing fills the void in our universe and beyond.
--Xymyl (KON)
On Sat, 24 Apr 1999, Brienne DeJong wrote:
Stupid question, but my kinda adopted little sis wants to know... If I send you $5 and my friends adress, will you send her an empty box, or nothing at all? If it's an empty box, can I send a
personal message with it?
Thanks, Brienne
To which I replied:
We send nothing, then, several weeks later we send a plain envelope with a small instruction book enclosed.
It explains nothing.
On Fri, 5 Mar 1999 BEIL99 wrote:
I'm really pissed. I order nothing and I got nothing in return. This is the last time I will order nothing from you!! I may even take this nothing order up with the consumer affairs divsion of my
local internet provider,who has nothing to say about nothing. From now on I will look forward to getting nothing from you and you will get nothing from me!! , Thanks for nothing
Dear Cunsumer,
To order nothing may mean not to order, or it may mean to pay money and get nothing in return. Since we've never received an order from you I'd have to assume it's the former.
Your interest in our lack of product brings us great pleasure. We are glad that nothing is such an important part of your life as to motivate you write such a spirited letter to us. I do hope,
however, you send *actual* money (in the form of a check or money order) towards the purchase of nothing (remember, you not only get nothing but you'll get our instruction book free with every
Before you order, let me remind you. You get what you pay for.
Thank you.
Xymyl (KON)
On Tue, 23 Feb 1999 Ellimist45 wrote:
> can you send me back an e-mail with nothing in it? please, and thanks
To which I responded:
On Tue, 2 Feb 1999, ho goht wrote:
hi, i'm not a prankster who doesn't have anything better to do than do anything, i'm just Mr nothing on a everything world ...by queer coincidence, i happened to click on your web link from
webcrawler search cos i searched for nothing....so why am i saying all these..could it be for something,,,i guess not cos i just have this to say to you.....YOU MUST BE A GOOD PEOPLE 'S GAME PLAYER
.......i said something but now it's ( ).
To which I responded:
I'm not sure what "A GOOD PEOPLE 'S GAME PLAYER" is but it seems too extreme for someone like me. I like to just sit back and do nothing.
Xymyl (KON)
On Thu, 21 Jan 1999, Don wrote:
Hi. I was impressed with nothing at your site. You have, by far, exceeded my expectations with respect to nothing and I felt compelled to comment on nothing at all.
I have been using nothing for years, and until recently, nothing was acceptable. But, I have a question; Is nothing Y2K compliant? Can I get a certified letter stating that nothing is Y2K? On Jan 1,
2000, will nothing function? I depend on nothing for almost everything. My wife thinks about nothing all day. My kids play with nothing, and I make my living getting better and better at nothing each
day. (No I don't work for the government - I'm not that good at doing nothing).
I need more. Where can I get next to nothing?
To which I replied:
I'm glad you were impressed by nothing. Some may say that one who is impressed by nothing is easily impressed. If you know nothing like I do, you understand how difficult it can be to get all worked
up over "it".
So, someone who is impressed by nothing is hardly impressed. (That is as opposed to easily (however not opposed as in opposite, just different.)
I'm glad to hear you and your family have been enjoying our products. (I use the term products in the sense far beyond the loosest possible sense, and by such statement mean lack thereof.)
Nonetheless I enjoy hearing positive feedback about nothing (it's really difficult to come by these days).
Now on to your Y2K problem.
If you mean nothing at all...
I can not guarantee that nothing will work on (or after) January 1, 2000. In fact, it is my painful obligation to inform you that some things will be working that day. I could go into exhaustive
detail... but... actually I don't have the time (so I can't).
If you mean nothing in general...
Yes, you have my guarantee that nothing will function on into eternity. Nothing lasts forever. Nothing will function perfectly, forever. Nothing will have nothing to do with the Y2K problems that
will pop up shortly. In fact, on effected systems mission critical data may be lost, systems may fail, programs may halt or have fatal errors.
What does this mean? Nothing. No more errors, blue screens, lockups, shutdowns, disconnects, routeflaps, collisions, fragments, truncations... I could go on and on... but... actually...
Yes, January 1, 2000 is install nothing day! It's not a Y2K problem it's NO/OS extravaganza! A black hole blowout! I could go on and on...
If I only had the time.
Anyway, no, you can't get a certified letter nothing will work because it could be taken to mean that everything will stop.
All I'm saying is something will go wrong ... but never nothing.
Xymyl (KON)
P.S. I take exception to your statement about government workers being good at doing nothing. Yes, there are some government workers who seem to be doing nothing, but how did they get there? That's
right, they had to walk to the bus.
On Mon, 18 Jan 1999, Killboy Powerhead wrote:
> why are your tshirts so fu**ing expensive?
To which I replied:
Because we waste so much time answering e-mail.
On Tue, 5 Jan 1999, Jonathan Davis wrote:
i hate to be the one to tell you, but you can't do nothing without dying. if you truly want to do nothing don't breath,suffucate, eat, starve, sleep, stay awake, die, or live.
dumb ass.
To which I replied:
I hate to be the one to tell you this but...
You state, "you can't do nothing without dying" which is a double negative. That could mean either you can do anything without dying, you can do anything while alive, you must do something without
dying, you must do something instead of dying, or you can do nothing with dying.
Let's examine these briefly:
1) you can do anything without dying
This statement is incorrect. I could jump off a tall building, tie dynamite to my head and ignite it, get hit by a bus etc...
2) you can do anything while alive
This is also a false statement. I can't go to the sun, or be completely eaten by maggots while alive (I could go on and on).
3) you must do something without dying
This is a lie. Whoever told you this should be shot while skiing. I've heard of people who died while flying or falling. In fact, it seems that you must be doing something if you are going to die.
4) you must do something instead of dying
This also is a total falsehood. I could choose to die, to live and die later on, or even to die while drinking my favorite wine.
5) you can do nothing with dying
This also is not entirely correct. Although you can do nothing while dying, you generally end up doing more than ever before. Many a mans last days are far busier than his earlier days. With trips to
the hospital, calling 911, often speaking to the police, getting limbs amputated, I could go on and on... I think you get the idea.
I'm sure there are many more meanings for this sentence. You can be sure whatever they are, they will all prove you wrong (and probably stupid).
Xymyl (KON)
On Wed, 5 Aug 1998 blueie@webtv.net wrote:
> YOU People are veary stupd
To which I quipped:
That may be but we can spell.
On Wed, 15 Jul 1998, RON & WILMA OLSON wrote:
> Are you really for REAL?
To which I replied:
No, not really. Wait I take that back... it only costs five dollars U.S.
to find out.
On Sat, 11 Jul 1998, kathi ricci wrote:
To which I replied:
Good question.
The answer is as follows:
On Thu, 30 Apr 1998, robert spaulding wrote:
Dear Xymyl,
I wrote a check for nothing and felt less than nothing, and by this nothingly, and non-receiving 32 pages of nothing, I increase my nothingness beyond by expectations! noughly vacuumi, robert
PS are there any openings at your university? something in the sports program?
To which I seem to have somewhat incorrectly responded:
If you write a check out for nothing, you will indeed recieve nothing but
also something (that is our 32 page instruction booklet which is
certainly not nothing). It is of course about nothing, that is to say on
the subject of nothing. Of course all this (and nothing) can be yours
only if you actually send us the check you wrote out for nothing.
Often people make the mistake of thinking that a booklet or tee-shirt is
nothing, obviously tee-shirts and instructions are something and not just
anything at all, but (and here's the point) they are tee-shirts and
instructions. There is no way to make 32 pages of nothing... it doesn't
even make sense. Once you have a page you have something.
Nothing is greater than all these things, because nothing exists only in
concept, and to people who think only in concept nothing is real!
"It's" beautiful.
On Tue, 28 Apr 1998, robert spaulding wrote:
i'm glad to see the universe has a redundancy of ideas as i was talking about a web site like this myself about 6 months ago, great to see that you have nothing up and runing. are you going to issue
stock? deep space sub-division of nothing? franchise nothing? passports? cds? cassettes? empty promises? in nothingness, robert spaulding
To which I replied:
The web site is nothing in comparison to the nothing I have nurtured over
the years from not even a speck to even less. If you'd like to learn more
about less and less, or less and less about more (progressively) just buy
our 32 page booklet. If you don't get it then you don't know nothing.
P.S. nothing.net has been alive since 1995.
On Thu, 9 Apr 1998, Ethereal wrote:
hope nothing happens to ya!
To which I responded:
You're too late somebody just stole my car stereo and cd player.
Of course now I keep nothing in that spot.
On Mon, 23 Mar 1998, Peggy & Bob wrote:
I have absolutely nothing to say to you.
Peggy S Carlan RN
To which I mused:
I'm not surprised.
On Thu, 19 Feb 1998, Ted Brasky wrote:
hilarious...but what is it?
what exactly are you selling? -- please don't say "nothing" because that's why I'm confused.... whatever it is -- be it a joke or an actual business --- it's hilarious!!!
To which I replied:
Thank you for your compliments,
"It" isn't.
I won't say "nothing" because what we're selling is actually nothing. No
cheap imitations, no gimmicks (well, maybe some gimmicks, but
we're not selling any of our gimmicks, we give them away free with each
order of nothing). And we have tee-shirts that say nothing on them which
is something. So, to answer your question, we sell something AND nothing.
Our blatant honesty in advertising seems to confuse some people but we're
sure that it'll catch on (or we don't know nothing, and I think we do).
We hope we'll make money off "it". The great thing is nothing's flexible
and we're sure nothing's going to work for us!
Thanks for your question and always remember, there's something for anyone but nothing's for everyone.
--Xymyl (the king of nothing)
On Thu, 12 Feb 1998, B. Shively wrote:
> Really, if I order nothing, what do I get? And what's in the
> instruction manual.
To which I apparently replied:
You get nothing. and detailed instructions on how "it" can be used to
enrich your (otherwise full and fulfilling) life.
Yes, nothing can be yours, but only if you act now!
Thanks for writing with your question!
Xymyl (the king of nothing)
On Nov 20, 2007, at 5:02 PM, Travis Kirkwood wrote:
I don't quite get "it". If you truly "believe" in the concept of nothing, or understand "it", then it should be entirely impossible to discuss "it". There is no need to discuss something that is
naught, in fact quite ridiculous since I'm sure we all know much about "it" for as you tell us it is basically everywhere at all times. Even in the case where someone is low enough to discuss
nothing, (being truly loyal to the concept) all that there is that they are capable of saying to prove so ofcouse is nothing. Incase you are wondering I am not contradicting myself by discussing
nothing, I am just pointing out the absurdity in doing so. Nevertheless, when talking about nothing the subject is naught which leaves plenty of time and space to discuss absolutely everything since
it fits all-so nicely into the vasteness of nothing. Well... looks like everybody's wrong.
By the way, neglecting what i have said, you have truly made something out of nothing and gotten people involved. That's pretty sweet. And don't say that you have not made something. Otherwise to all
this talk of nothing our minds would be void. Thanks for being interesting.
As you know, I can't reply to everybody who sends an e-mail, but sometimes I like to post a few without replies, just to let everyone know that I care... about nothing. Unjoy!
Wayne said: i have believed in NOTHING since forever its just SOMETHING very dear to me
Adam said: Do you guys have anything, or nothing, to do with this? Check out the link, in actuality, it's a link to nothing, and that's what you'll see for sale there.
Thanks, Adam
Jeff said: Really don’t have nothing to say.
Freddy said: i dont get it :(
Then later he said: i get it hahahahah its freeking hilarious
Still later he said: i love this site seriously and you deserve a hug
gracias ☻
Dave said: nothing,nothing,nothing,nothing,nothing,nothing,nothing
Maria said: hello.
On Sep 18, 2007, at 12:44 PM, Gina Turano wrote:
you are hilarious and might even consider putting you on my top 10 hero's list.
my stomach hurts just reading your site. too much..hilarious!!!!
Sincerely, Nothing
To which I responded:
Hi Gina,
I think you are somehow misunderstanding us. We take nothing very seriously!
Following much discussion, we have come to the conclusion that you are reading something into what we are barely saying. Clearly, we are saying something, but this is only a means to no end, to get
to the point, which isn't. We know that at times there can be confusion as to what we are trying to say when the real fact of the matter is that we are just trying to say nothing, but we choose to
take the long way around. We firmly believe that if the journey is its own reward, then the journey that ends in nothing must be at least twice that. Well, we don't FIRMLY believe that, however we do
something forcefully vague about, towards or in reference to it. But we do believe in nothing, and that's something. That is to say belief is something, obviously you already know that nothing isn't
something. I could go on and on, but I think you get the point.
I, personally, am very excited about the possibility of being included in your heros list, mostly due to the fact that I have never heard of it. Being included in something so vague and unreferenced
as the "Gina Turano Heros List" could be one of my greatest achievements, provided you were willing to make a couple of changes. First off, the number "one" kinda makes me itch. Any chance that you
could make a top zero list and put me in a footnote as a former honorary runner up for possible membership consideration? That's more my style (or lack thereof). Also, would you mind not typing so
loud? Modern keyboards almost type by themselves, and if there is one thing that we are not about here at nothing.net, it's something, and typing loudly is a thing.
Thank you for your kind words, but thank you even more for your interest in nothing. Without people such as yourself - who truly appreciate nothing - everything would seem a little more interesting,
and nobody wants that.
P.S. Sorry about the stomach pain, although we take no responsibility for it, we are deeply (possibly to the point of inversion) disappointed that anything at all happened to you while reading our
On Sep 17, 2007, at 12:50 PM, Larry wrote:
So what you’r saying is if I send you nothing I will still get nothing. Kewl!
I responded thusly:
Hi Larry,
I was shocked to receive your letter. We are not used to being spoken to in this way, even by our (nearly) dear nothing unthusiast core group. But what you say is true, in a sense. So I must address
this topic in a way that is not only truthful, heartfelt and sincere, but in a way that is at the same time vague, meaningless and (most importantly) still makes it seem reasonable for people buy
nothing from us.
First of all, we never said that you can get nothing without spending money. If you want nothing, you will be expected to pay for "it". If you find that you already have nothing lying around and
would like to use "it", then you are probably just noticing our "overflow stock" or "underflow stock" or rather " ", as we prefer to call it. Since the entire universe is within our (redundant
overlapping) warehouses, and nothing "is" almost everywhere (although not always in the purest form) your home may be used as one of our storage areas. Be careful, it is commonly found that when
nothing is stored in this manner, "it" is highly polluted with anything from dust bunnies to aromatic essences. Although this "nothing" may have nothing in it, there's certainly something more to it.
We pride ourselves on not trying to get you the finest quality nothing available, mostly because nothing isn't available in the same sense that any thing would be. Certainly nothing is available from
us, but not in the same way a tee-shirt might be. For example, we sell tee-shirts, but we haven't been making tee-shirts lately. So there is a demand, but not really much of a supply. In the case of
nothing, there is both an infinite supply and a modest demand, on the other hand, there is no supply at all and a modest demand. The laws of supply and demand indicate that on the one count nothing
should be very inexpensive because "it" doesn't do anything, and there is very little demand for "it". On the other count, there is a modest demand and no supply at all, so it should be nearly
infinitely expensive. The problem with a product that is infinitely expensive is that it is very difficult to close the sale on it. Now imagine there wasn't even a product. That's the dilemma that
confronted our marketeers when they sat down with us at the very large, yet seemingly infinitely small table that we all sit around to talk about nothing. As a side note, you should really see this
table, it is like something (or almost nothing) that would be dreamt up by the right halves M.C. Escher's and Salvador Dali's brains if they could somehow have been functionally fused together.
Anyway, you are probably wondering what the outcome of that discussion was. The two prong answer was simple:
#1 "Do Nothing". Live the product. Be the product. Promote the product in everything you do. Since all of these ideas were impossible, we chose to use the concept as a general jumping off point. We
really did nothing. Well, not really, but in the sense of not promoting nothing and doing as little as possible, we sorta did nothing.
#2 Nothing costs $5. It was that simple. What is a high enough price that people who think there is an infinite supply of nothing are almost guaranteed not to buy "it"? What is a low enough price
that the people who think nothing is anything will believe the price is too good to be true? Well, as you already know, the answer is "Five Bucks".
It turned out to be a huge success, and an almost insignificant failure! Countless millions of viewers later, we still haven't been promoting ourselves or our products (or lack thereof). Sure, we
waste our valuable time. But sales are scarce to the point that many think our sole purpose is to spread good will, which makes little sense to me, but I can't argue with the market. Well, actually I
can and will argue with the market...
What was the point I was trying to make? Oh, that's right, nothing. Please, buy nothing. If you don't buy "it" you may have it, but you'll never get "it". Or do you need some kind of multi-million
dollar ad campaign to convince you that you need to pay for nothing?
Thank you for your interest in nothing. Peace off!
On Aug 11, 2007, at 8:09 AM, jon dieks wrote:
oh nothing . net how do i thank you it has been many ,many month's since our last e-mail. i wanted to thank you for your encourgement in my quest for nothingness, and as in our last corispodance you
told me my journey to nothing has begun. i wondered what you meant about taking nothing seriously and ponderd enormously about it , and then all these signs started pointing my in directions, leading
me nowhere, and then i found it! my golden apple, and am now a proud poee of discordia! and now all things make sense and nothing makes sense. in a way nothing did before. so i just wanted to thank
you (actually the goddess eris commanded it of me) and as a acting poee of discordianism a declare you
!! (and remember "to eris human"
To which I replied:
Hi Jon,
Thank you for your further comments on a topic that means nothing more to me than it does to you. As for my words of uncouragement, I need clarify my statement before I go any further. I am pretty
sure that what I said was, "Your infinite journey has begun." I admire your eagerness, but I have to express that in my arrogant opinion, a few months is a very short turnaround on an infinite
journey. In fact, it seems that your journey has ended (give or take) about an eternity too soon.
I already know what you're thinking, so I'll just say it, "Infinity loops back on itself, ergo any exit point on my journey could be considered as the end of an infinite cycle." First, I must applaud
you for thinking of such a brilliant wrong argument. If time were circular, that argument would have some merit. With so much talk about space-time we may sometimes forget that time has no spacial
properties. Time is something that is measured by its arrival and/or passing. Or to look at it another way, the function a clock performs is keeping track of time. If a clock stops, we do not
conclude that time has stopped, or that our work on this planet is done. We simply replace the clock.
I submit that you have been following broken signs leading you to a big bad apple. I think that you simply started in the wrong figurative direction, by pondering enormously about it. (By your use of
the word "it" I'm assuming you mean the infinite journey.) You may have done better by thinking about it "at length". Although at first glance, "at length" may seem to be attributing spacial
qualities to time, it is merely a way of expressing that one has taken his course through time dealing with or pondering on the matter at hand. Even if we were to view time as a chart we would see a
linear progression in this case rather than a circle or a lazy 8.
You also call yourself a POEE of the goddess Eris. By POEE, I can only assume that you mean Powertrain Operations & Engine Engineer. What I'm wondering is what sort of operation is Eris running over
there? I knew that the American Auto industry was in the proverbial poo bucket, but now this is all beginning to make sense. Personally, I wouldn't put someone with her track record at the helm of a
sinking ship.
I have a tip for you, don't listen to commands from an apple tossin' banshee. It tends to cause trouble, and as we all know, trouble isn't nothing.
I wish you well on your CONTINUING journey. As for your e-mail, great use of negative space and a wickedly bad pun, really, really bad. Freud has been referenced many times as saying that the pun is
the lowest form of wit. But Freud would have said anything to get a peek inside the pants of your mind. If indeed Freud did say this, he could have at least delivered it as a pun. Low is almost
Thank you for your continued interest in nothing.
On Aug 10, 2007, at 3:14 PM, BF wrote:
Hello Xymyl:
I have been enjoying very little about your blog. I hesitate to say "nothing" about it, but there is something about your nothingness you may have overlooked. That "is" this: Nothingness has "it's"
flaws. "It's" flaws are an unnecessary apostrophe at the least. "Its" is a word which is possessive without adding an apostrophe. So the unneeded punctuation adds something to an otherwise quite
clean and featureless nothingness. Please ignore any typos of my own, as they must be construed as nothing of importance. Have a day!
Your whatever,
B. French
To which I responded:
Hi BF,
Glad to see that you are unjoying the blog. Thank you too for your con-structive comments. However, I take exception to the idea that there is anything to my nothingness.
Actually, I can't remember the original reason that I started doing that (by "that", I mean "it's"). Although it probably had something to do with the fact that pure nothing, the REAL nothing, the
"best" nothing, isn't. Thus, when referring to nothing "it" isn't a pronoun (which is also why it's in quotes). I believe that the original thought was along the lines that "it" was going to be
separated from its "s" by an apostrophe as a way of further defining that the possessive form was merely a literary tool and not in any way an indication that "it" was anything.
Since I have noticed over the years that this device has not necessarily cleared up the issue, I have often thought of changing my "it's"'s. But a coping device I can use is to pretend that having
the point obscured means that nothing (nothing being the general comprehension of nothing as a concept) makes sense, thus everything (everything being the literal devices required to explain nothing)
makes sense. Any intelligent person can see that this is a feeble way of reasoning, but it seems to be working anyway. Besides, I can't go back to 1995 and edit those old e-mails up to the present
day. So, I have mostly chosen to stick with it. It, of course, being my use of "it's"'s.
None of us should take this here mutt language of English too seriously. We should embrace its flaws and capitalize on them, of course. But we should also feel free to make up new words as needed,
and change spellings to fit road signs rather than changing the sign width or font size. Put some florp in your snazzle, because your lingo won't have any flavor if it ain't got juice. People may
come here for the grammar, but they stay for no good reason.
Thanks for your general disinterest in nothing. Please keep up the good work.
--Xymyl (KON)
Okay, I don't usually post my replies to blank e-mails. So by posting this, I am not suggesting that I will start posting responses to all the blank e-mails I receive. This particular e-mail had a
subtle confidence to it that unspired me. Obviously, anyone else who sends similar blank e-mails will be acknowledged as well. So without further ado, the letter and my response...
On Aug 5, 2007, at 7:12 AM, allan godshall wrote:
To which I replied:
Thanks for the uncouragement. Your e-mail said "it" all. Just the boost I needed to keep doing nothing all the live-long day. Sometimes nothing seems a vain pursuit, or even a poorly thought out
business, but then someone like you comes along and put's everything into perspective. Or do I mean out of perspective? Doesn't really matter. The fact is that you've given me a renewed sense of lack
of purpose. I'm going to get back off that horse and try and find a blank space to lie in and just... dwell on the past.
Thanks. I can tell that you really know nothing, and that is a gift to treasure (treasure meaning to put aside or lock away).
--Xymyl (KON)
On Aug 4, 2007, at 5:51 PM, Caroline and Annalee wrote:
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing
nothing nothing nothing!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!?
To which I responded:
Thanks for the great work!
And you know what 324 times nothing is right? That's right, it's nothing. You almost made something out of it with the triumphant addition of 39 exclamation marks. But you packaged it up neatly with
a question mark to save the day. You are to be commended for your poignant use of repetition and punctuation to emphasize nothing. Of course I had to take the time to count them all. It's the least I
could do for someone who was willing to do all this for nothing. You score extra points for for your excellent time sponging skills. Keep "it" up.
On Jul 22, 2007, at 6:56 PM, No one wrote:
To whom it may not concern,
None of this isn't making sense. Should I be concerned ?
Thanks for "Nothing" , No one.
To which I replied:
Dear No one,
You present a very good and bad question. The answer is clearly, yes and no. Yes, you should be concerned, because nothing should be clear. Apparently nothing isn't clear to you because (to reword
your statement), "every little bit of this is making sense". The operative word here is "this". "This" is not nothing. "This" is something. Notwithstanding, "this" is still nothing to be concerned
about. So, no, you shouldn't be concerned. Unless you were referring to nothing when you said "this", in which case you should be very concerned. If, however, you were simply referring to nothing as
a general concept and thus indicating the existence of that concept when you said "this", don't worry. "This" (in that context) is barely anything. And certainly nothing to worry about. Of course, I
don't expect that you will be so pedantic as to worry about "that" (my most recent use of the word nothing) just because of that last sentence.
Thank you for your interest in nothing and your desire to learn whether or not you should be concerned about nothing. I hope this helps.
On Jul 5, 2007, at 2:50 PM, Kurt G. wrote:
I know how tiring it is to collect all that Nothing which is why I may be of assistance. I am a Nothing aficionado and an avid collector. I get my Nothing from many different settings to observe if
the some-thing that was tainting it at the source has an influence on the Nothing after it is removed. My favorite collecting grounds are deep in caves and desolate mountain peaks. It seems to me
that the greater lengths (or depths) you go to locate a specific specimen makes it more pure and noteworthy. Holes located underneath rocks is one of my favorite wellsprings for Nothing. Distilling
Nothing from a something is of course more difficult and involved but the rewards are much sweeter and vapid. So if you need help in lifting the burden of obtaining all that Nothing, I'm your man.
With over 10 years in the pursuit of Nothing, adept at successful capturing and cataloguing of Nothing as well as the distillation of its essence, I can guarantee that the Nothing you provide to your
customers will be of the purest grade possible. You of all people realize that the saying "Nothing is free" is a lie and will understand my need for direct compensation. One fifth of your retail
price per shipment could possibly cover my expenses. Good news for you is that my methods of Nothing containment allow for expansion when you repackage (in a sterile environment I hope!) so that one
shipment from me could supply several of your retail packages. Intrigued and available,
Kurt Gindling
To which I replied:
We appreciate your offer to collect nothing for us, but we regret to inform you that you have actually been relocating catalogued nothing within our warehouses. We must insist that you return nothing
to where "it" was (or was not) found. We have had to delay shipment to many nothing unthusiasts who have had nothing on "lay away" in or near caves, under seashells and in other hard to reach
locations such as holes beneath rocks and the collective consciousness of the public. You have made it very difficult, and sometimes impossible to locate nothing to send to our patrons. I know, we
make nothing look easy, but sometimes nothing isn't as easy as "it" seems. Especially when well-meaning folks such as yourself are just making nothing more difficult for us.
We appreciate that you love nothing enough to travel to the ends of the earth in search of the most vacuous nothing imaginable, but we have nothing catalogued by inversion, pun, hyperbole, general
figure of speech, compound pun, thought experiment, and many other concepts. But most and least of all, we have pure nothing 100% fat free with barely any gimmicks actually being used in "it's"
Your packaging methods seem a bit suspect. I hope you haven't been ruining nothing by packaging "it" compressed. We generally ship nothing near a package rather than in one. Part of the reason for
this is that it is almost impossible to get everything out of the package, some people are likely to complain if there is dust or hair in the shipment. "Hair in nothing? That's a dust bunny." some
have said, we don't want to repeat that fiasco. In case you may be familiar with this incident, the parties involved have been sacked.
I am however, quite interested in your "essence of nothing" idea. What do you actually put in "it" to give "it" an essence? I am fascinated, please inform.
Thank you for your "no can do" attitude. We may need your assistance in helping us track down a very volatile order of nothing puns that really are not funny yet are quite beloved. If you see them,
there should be a box next to them containing a lead lined vacuum flask, a roll of yttrium-barium-copper oxide tape, a pressurized tank of liquid nitrogen, oven mitts, a hydrogen powered rocket and
directions to the "event horizon" of the nearest black hole. There may also be a device resembling a standard aviation gyroscope, please don't touch that. If you see these items, please contact us
and we'll negotiate safe transfer. Be warned, the puns will severely damage your credibility if used improperly, that is to say, if they are uttered.
Thanks again, please continue your search for nothing, but let us know the next time you change nothing around.
On Jul 9, 2007, at 11:12 PM, Bryan L. wrote:
I can now say I've been at nothing for a while. In my case this would probably mean more than just a while, though, when you consider the nothing that came before this website.
Great site! I can honestly think of nothing else to say. =:)
Bryan L.
To which I replied:
I'm not sure what you mean by most of what you said in your e-mail. However, I do hope you meant nothing by it. You truly seem to appreciate nothing. But some who act like they love nothing are
really hiding something. I don't think this is the case with you, but just to be sure, I'd like to find out nothing else you have to say. I only ask this because you did say "else". So you can
understand why I need to be sure.
You started out saying something - kind words about nothing and the nothing.net website. This is forgivable. Indeed, we need to forgive ourselves every day as part our daily generalized therapeutic
negation. Not that we live monastic lives of self denial, but we strive for the nullific existence of nonacceptance. This, of course can not only be achieved, but it can also not quite be achieved.
But the unreachable goal we often seek is "it's" own reward.
As I drink unfiltered air from a brick, I am reminded that this - like many things - is unlikely, but nothing really is possible.
Thank you for your pure, true and hopeful interest in nothing.
Even though I haven't been talking about nothing as much as I usually do, the people are still speaking. Here are a couple of comments regarding nothing.net. I'm not sure if the second one was meant
to mean something or not. I chose to believe that it has no meaning.
On Jul 6, 2007, at 6:24 PM, No body wrote:
After reading this site, I absolutely understood nothing!
On Jul 6, 2007, at 8:30 AM, mark cryer wrote:
since nowhere means not any ware and everything must exist somewhere then surely only nothing can exist in the middle of no ware!
I'd like to thank everyone for their nothing centric e-mails. I haven’t had a chance to reply or post for a little while. I am even more ashamed to say that something was keeping me from doing
anything about nothing with this blog. I am further distressed about what I am preparing to say. I might be an artist. I’ve been hiding these tendencies for many years, and now I think it is time I
showed the world what my secret life has been like.
Xymyl Gallery
We can still do nothing together, “it” will just seem a little different for a while.
On Jun 19, 2007, at 11:36 PM, Jay wrote:
HI! I'd like a T that says nothing, and nothing says nothing like black. Black is NOTHING ,while white implies light, wich is SOMETHING. I can't find a black shirt on Amazon. Should I use your order
form? I don't have a printer ( printer = nothing ). Can I send you a reasonable hand-drawn order-form for a black T with "nothing" on it? I mean "nothing" on the T, NOT THE ORDER FORM! lol. I've
worked for an architect for ten years, and my lettering is quite good, so you wouldn't have to worry about sloppy writing. I've got $20 burning a hole in my pocket ( or maybe in my cyberspace
account, and who knows where it goes if it burns through that? ;)
So, what do you think? Can I send you a hand written order-form, or do you only accept printed forms?
There is nothing to thank you for, so thanks for NOTHING!
In Nothing-ness; Jay
To which I replied:
Hi Jay,
Yes, you may hand draw the form. And we would never worry about sloppy writing! In fact, you get extra points for a sloppier form that will waste our time. But it still needs to be decipherable.
Write an essay if you wish, and cleverly embed your request into the handmade document, so that the entire note must be read to be understood. We might even throw in nothing extra to show our
appreciation. Of course, most people seem to assume this anyway. There's really no way to know. The ambiguous nature of our rewards system is the key to you not knowing whether you're a satisfied
customer or not. And nothing gives us greater satisfaction than being able to assume our customers are satisfied. But when you buy something (like shirts) from us, you're not only supporting nothing,
you're getting something about nothing in return.
All of our outstanding nothing unthusiasts have some sort of memorial constructed in their honor and placed beneath our conference table, or near a nuclear facility, or a lead box or another
appropriate location. These are often found quickly and destroyed by the cleaners, but some have lasted for several months. The cleaners will be sacked if monument bunnies start to form. Do they
think we are paying them for nothing?!?! Sorry, I just got a little carried away there. We pay them to leave nothing behind.
As for your comment about money burning a hole in your pocket, I'm not one to take things too literally, and your description seems to support the metaphorical nature of your statement. Indeed, the
fact that you could type your request and include such details, speaks to the symbolism of your remarks. However, even a metaphorical hole in one's figurative pocket, or the "pocket" one's cyberspace
account, carries great meaning to all who know nothing. This is because holes can suggest a sort of gateway to nothing. And nothing is more beautiful than viewing nothing through a hole in the fabric
of reality.
Thank you, Jay, for your interest in nothing (and a shirt).
--Xymyl (KON)
On Jun 13, 2007, at 2:53 AM, Aryeh Lewis wrote:
Alright, now here's a question!! If nothing is a "no-brainer" as you mentioned earlier....... then why the hell are we all turning to YOU for the answers to nothing!
FYI "no reason" will not be tolerated as an answer, because then you would have to explain why you fooled us with this great web site. And if the answer to that is still no reason, then I am forced
to return to my original opinion, which was that you really ought to be sued!!!! :O
To which I responded:
I believe you are referring to when I said, "Whether or not you know anything, knowing nothing should be a "no brainer"."
You seem to have ignored the "should be" in this sentence. I stand by that statement. What is more, to many people knowing nothing is a "no brainer". Knowing that you (yes you) must have some
knowledge, I would assume that knowing why you (yes you as an individual) ask questions regarding nothing would be a "no brainer", in the sense that you would have an easy time figuring out why you
are asking me questions about nothing.
So I must put it to you, since "no reason" will not be tolerated as an answer and you are the only one who can know why you ask me questions about nothing, the burden of presentation lies squarely in
your court. Why do you ask me questions about nothing?
I hope you can think of a good answer, because it seems to me that you will commence legal proceedings toot sweet, if the answer is unsatisfactory.
I would like to give you some advice homonymous (or even more precisely, homophonic) with your own stated fear of my response. If you would like to understand why you seek my advice, I urge you to
know reason.
Perhaps, you know exactly why you keep asking questions about nothing. In fact, you said in a recent e-mail, "Your whole email truly helped me with nothing! I thank you! :)" If you were simply
looking for help with nothing, that certainly seems like no reason to me.
--Xymyl (KON)
Many people thank me for nothing, in the sense that they appreciate that I have taught them nothing, in the sense that they understand proper usage of the word “nothing” so they can properly express
nothing to others.
Nothing they wish to express may be an absolute, relative, or amorphous quality of something, or it may truly be nothing. It is often better if it is a new way of twisting the word nothing with the
meanings of that word or the lack of meaning that has been tagged with that word. Better still is when the humor and logic of the statement is only implied, so that to the uninitiated it appears
meaningless, when in reality the message is nothing short of pure infotainment to those in the know.
I am both pleased and disturbed by fans who believe that I invented nothing. Yes, I am the King Of Nothing. Even this descriptor is nothing made up by me, but I use it because it is true. Yes, I’ll
say it again, the word “nothing” was not created by me but I use it to communicate nothing to the world. Even though this word has vast imperfections - chief amongst them being that it is something,
a word - it is still the best darn word to use to let people know what you are referring to isn’t something.
Nothing (the word) has a very redeeming (or negating, depending on your conversational thrust) quality. That is its inherent humor. Nothing is ever good for a laugh, in the sense that the word
nothing is always good for a laugh.
I have always had a way with nothing, even no way, anyway, I always have known nothing. Even while learning things (and stuff) I never lost sight of the void of my desires. In civics class my job
aptitude test came out to a wash. I was not interested in anything. The teacher thought I was just randomly answering, but I did it exactly as he said. He was mad, but that’s because he didn’t know
I have heard or used every play on the word nothing, I have used some of them way too many times, but there is a certain negation achieved by repetition that both/neither enhances and/nor detracts
from nothing.
Being in the game for the long haul, one needs to be adaptable and willing to make sacrifices. If everyone who knows nothing suddenly decided never to repeat themselves, there would be nothing to
say. The delight all nothing unthusiasts would enjoy would be short lived, for the negation achieved would bear scrutiny, and thus conversation. This conversation may or may not be about nothing in
the sense that it is or isn’t about anything important or unimportant or in the sense that it isn’t or is about something at all, but it would hardly compare to the relative nothingness that once was
a conversation and now has become a parody of that conversation.
We enjoy all of the ways nothing can be expressed in these excitingly relativistic ways. Yet, we know that nothing real really isn’t anything at all. That’s what make’s “it” all the more or less
special. “It” isn’t, and we admire nothing about that.
On May 29, 2007, at 4:26 PM, adam watson wrote:
Have you guys had nothing go missing from the warehouses? The reason I ask is because I bought a container at the new IKEA here in Draper, Utah, and when I got home and opened it up, I was surprised
to find nothing in it. It said nothing on the packaging about the container having nothing in it, so I think it may have been a mistake. I know you had to recently expand your warehouses to store the
infinite supply of nothing, are you keeping nothing at an IKEA warehouse too?
I wanted to keep nothing that I found in the container, but I had to use the container to store things in, so there wasn't room for nothing anymore. The nothing I found is probably in the living room
somewhere, once you remove nothing from the container, it's hard to keep track of. If you need me to send you nothing since I didn' t actually pay for it, just let me know, it'll help if you could
describe nothing so I'll know that you did really lose nothing.
To which I responded:
Another good question. Since we have redundant overlapping warehouses encompassing the entire universe, then we by default get a little bit of everything in them as well. Actually, our warehouses
contain everything, but we try to downplay that because when you think about the vastness of the wide open space in the universe, everything is like a pile of dust somebody forgot to sweep up.
But, unfortunately, IKEA is part of "everything" so, yes, it is in one or more of our warehouses.
Don't be surprised that somebody shoved nothing in your container because we keep nothing anywhere. In fact, nothing is in atoms, in between molecules, pretty much anywhere you can't put anything,
you'll find nothing.
As for your offer to send us nothing back, you can keep "it", they obviously don't know how to handle their inventory at IKEA. Stick it to the man!
On May 29, 2007, at 4:17 PM, adam watson wrote:
Quick question,
In an earlier blog, you spoke of a reoccurring dream that you had as a child in the which you found yourself in a brown mesh tube (I'm paraphrasing) and you reached out and found nothing. Was this
your first encounter with nothing, or had you found nothing before either in consciousness or unconsciousness?
Amateur Psychiatrist
To which I replied:
Very good question. In unconsciousness and non-existence I found nothing for all eternity prior to my conception (which I don't like to think about). My first encounter with nothing in conscious
memory was (when I regained consciousness) following an extended period of staring in the direction of the wall and seeing only the purest brightest white. It wasn't white, but that is the only way
to describe it as anything. It didn't take long to realize that the best way to describe "it" was not to describe "it" as anything at all, because there was really no vision, thought or reality in my
comprehension until the moment had passed.
I still don't know how it happened, and I never tried to re-create the circumstances that brought my thoughts to nothing that day. As for infinity and nothing, they are opposite sides of neither a
coin nor the space that coin may have once occupied. And being that I always thought of life in infinite terms until that day (being frustrated by my eternal non-existence prior to the chance meeting
of two gametes which heralded my emergence as a viable entity on the world scene), this knowledge of nothing new made the comprehension of my eternal nothingness prior to my eternal existence (in
some form) with an as-yet undefined term of intellect punctuating my two eternities somehow more tolerable.
--Xymyl (KON)
On May 28, 2007, at 10:37 PM, Aryeh Lewis wrote:
My dear sir, I am as confused as a pig with no mud to roll in... as a cat with no litter... I am....... well I'm just confused.
I just don't understand how "it" is possible .... I mean.... you can't know nothing... and a whole lot of "general concepts" at the same time can you?? : l
To which I replied:
Now I feel like an inversely starving nothing burger patron who was just asked, "would you like something to go with that?"
I find myself wishing that you had said you were as confused as someone with nothing. But you didn't say that at all. Because that wouldn't be confusing at all. And yet you believe that knowing
nothing and being reasonably well informed are mutually exclusive. I know nothing, not in the sense that I don't know anything, but in the sense that I comprehend the reality of non-existence, the
ultimate void, the theoretical lack of contents in the hypothetical vacuum flask inside a lead box stationed between two black holes. Whether or not you get it, you should get "it". It's easy because
"it" isn't. Whether or not you know anything, knowing nothing should be a "no brainer".
So, yes, to know anything, you should probably also know nothing. You may start by knowing nothing in the sense that you don't know anything. After learning all you possibly can, you might pause to
consider the fact that nothing exists independently of anything in the sense that "it" doesn't. Your understanding of this concept (the reality of nothing) is something. If comprehending nothing is
something then it is clearly not a contradiction to know nothing and things. This, of course, does not make nothing a thing, just the understanding or conceptualizing of nothing would be considered a
I know that I didn't need to write such a wordy reply to explain this point, but I get the impression that with you "more is less". Here's to hoping something I wrote in this letter will help you
with nothing.
--Xymyl (KON)
On May 20, 2007, at 1:42 AM, adam watson wrote:
Hello, I was doing nothing tonight on the Internet and to avoid the temptation to look up naughty stuff, I decided to Google nothing instead. I was very pleased with what I found, which was nothing.
I was pleasantly surprised. Many people have told me "Nothing is more important than a college education" so I don't put as much effort into my school as I put into doing nothing. I'm bilingual so I
can also do nada, which helps out when there's nothing else to do. Nothing is too difficult for really determined people who try as hard as they can, but for me it nothing is simple. If you break it
down, in essence it's just "not" and "hing" and since nobody know what "hing" is and nothing is simply the lack of "hing" then nothing could be more clear. I must say thanks, thanks for nothing, if I
happen to get some money maybe I'll order nothing from you guys. Nothing would make me happier, at least I think nothing would.
P.S. I know you guys are truly dedicated to nothing, but I was wondering if you had tried anything? Just curious. Thanks.
To which I responded:
Thank you for using nothing in a positive way. Using nothing to avoid internet naughtiness and outmoded educational constructs is highly commendable. We certainly agree with your stand. I think that
more young people should stand up and proudly say that nothing is far better than dirty pictures on the internet.
As for the schooling, if they are really teaching you nothing, it can't be all bad.
I like your explanation of nothing, not because it is makes sense, but because it takes the "thing" out of nothing.
You asked whether we have ever tried anything. No, we have not tried anything in the sense that we have not just randomly tried ANYthing. We have tried things and usually have done quite well. We
certainly are not anti-thing, but I know sometimes our pro-nothing stance pumps us up to the point that we speak out against things. Of course, some things suck. These things are terrible and nothing
is always better than anything like that.
We have at times had associates who tried a little bit of everything. We disagreed with this carefree lifestyle because it left little time for nothing, plus, these people were mostly disease rich
and hygiene poor. Needless to say, we often found ourselves saying phrases such as, "what IS that thing?" or "could you do something about that smell?" or another phrase that indicated that some
thing needed to be done to protect us. And we're not about always having to take everything into our own hands.
I think you are on the right track. You seem to have a very balanced view toward everything. Most importantly, you have your priorities in the right order, with nothing fighting for the top ten
--Xymyl (KON)
On May 17, 2007, at 4:38 AM, Nicholas wrote:
Would you consider expanding nothing into not here? That way nothing could be considered no where and make nothing a bit more elusive for the non-initiated seeker of nothing knowledge. Not to mention
the economic boom such an expansion could bring.
If you should adopt my very detailed nowhere plan I would have to charge my normal consultation fee based upon a sliding scale of zero increments. I expect that the cost for this endeavor should fall
between zero and nothing dollars.
Yours truly,
Nicholas Parrella
To which I responded:
Thanks for the suggestion, and we appreciate your desire to obscure nothing from the comprehension of the general public but we are all about education. We WANT people to know nothing, we don't want
to confuse them. Okay, we want to confuse some of them, but we certainly don't want to confuse all of them or ourselves. This idea of yours is so confusing that if we tried to somehow incorporate it
with nothing as part of our campaign, it may negate the quality of nothing in the sense that it may assign a value to nothing thus making it seem to be the opposite of something in a very literal
way, thus making education about nothing pretty much pointless.
I fail to see how this idea of yours, while intriguing, could have a positive economic impact.
So, I regret to inform you that we will not be incorporating your nothing/not here fusion scheme. We will, however, respect your fee scale as we understand(?) it and pay you zero dollars for your
time. Expect a check.
Please keep up your interest in nothing, and try not to thwart the others who want to know nothing too.
--Xymyl (KON)
On May 25, 2007, at 4:21 AM, Dead Psycho wrote:
for nothing to really exist you need something, else there would be nothing to call nothing. so if in the absence of something you have nothing then you must really have something to call nothing.
confused yet?
let me continue just to clarify.
say i had something (i don't know what because i haven't thought of that yet, nothing came to mind). and i lost it. would i really have nothing instead. or would i have the absence of something which
is actually something itself which you could call nothing, but as nothing is actually something nothing really exists.
have fun.
all the best
Dead psycho
To which I replied:
Hi Dead Psycho, thanks for your comments.
I made a valiant attempt to be confused by your assertion that nothing requires something in order to exist. Unfortunately I was unable to find any way to become confused by your platform for debate,
leaving me no choice but to totally own you. Of course the ownership of which I speak is merely metaphorical and will hopefully not have any bearing on aspects of your life beyond the scope of this
First of all, you tear down your entire argument in your initial statement which says, "for nothing to really exist you need something, else there would be nothing to call nothing."
As for the existence of nothing, we have held firmly to the fact that nothing exists in the sense that "it" doesn't. Yes, there are many other uses of the *word* nothing (which is something), which
have to do with something, but that doesn't make nothing something. It is true that there must be something (intelligent life) to comprehend nothing, but comprehension is far from necessary for
nothing to exist.
As for the lack of something leaving us nothing to call nothing, we have already established that the lack of something would really leave us nobody to call nothing nothing, or nothing to call
nothing nothing. Which I might point out shows (even with your own flawed logic) that nothing would still exist in the absence of something. However, this is really the only place (this issue of
perception) in which your case holds some small merit. You seem to believe that people must comprehend and label nothing in order for "it" to exist. Yes, in order for people to comprehend nothing
there must be 1) People, and 2) something to use as a label to draw a line between the concepts of nothing and anything. But this idea is based on the assumption that perception changes a thing or
indeed changes nothing.
There are documented cases where perception changes nothing but only in the sense that it does not change anything except the person who has developed the perception. Certainly, nothing (the real
nothing that exists in the sense that "it" doesn't) has not been changed in the slightest by perception. Although it is true to say that at times something has been changed by perception, this
doesn't tie it to my previous statement because something and nothing are not necessarily related, as you might expect due to the fact that they are not usually opposites and certainly not mutually
exclusive. Something may be able to displace nothing (this is a point of great debate), nothing may be able to replace something (this is also hotly debated), but nothing cannot negate something in
any equation. See my "Volley Nothing" blog posting on February 3rd 2007 for further details. http://xymyl.blogspot.com/2007/02/q-volley-nothing.html for more information regarding acceptable use of
nothing as an opposite of something.
This idea that one thing or indeed nothing needs something to bring it (or "it") into existence is a just a rip off of the old chicken and egg argument. I will reiterate that the word nothing
required something or someone to coin that word, but that word is unnecessary if nothing exists in the sense that everything doesn't. As for the chicken or the egg argument, "Which came first, the
chicken or the egg?" I had an answer for that simple argument when I was an 8 year old chicken farmer. The birds that are today called chickens were derived primarily from Red Jungle Fowl and at
times interbred with other fowl. The fact that selective breeding has been used to develop the various strains of chickens that exist throughout the world means that there was a point of divergence
from it's ancestral form at which it began to be called "chicken". This chicken would have arrived via the egg which was produced from parents that were closer to the original characteristics of the
main progenitors. Being that the resultant chicken would be markedly different from its parents and would arrive via the egg, we have to conclude that the egg came first because that first chicken
had parents that were either Red Jungle Fowl or an intermediate form. Sorry for the wordiness, just wanted to make it clear.
As for your question regarding the absence of something, yes, you could say that you have the absence of something, which would be a true opposite of something and thus is also something. To say you
have nothing when you lose something is more of a figure of speech than a literal statement, but it can be appropriate to use such a term if all or almost all of the specific category of something to
which you were referring had been mislaid.
Of course, nothing isn't.
--Xymyl (KON)
On May 24, 2007, at 4:23 PM, Aryeh Lewis wrote:
I've been reading your blog for quite some time now, ever since you called me a maniac for laughing manically (naturally... because that kind of caused me to laugh manically once again).
Anyways, I just would like to let you know that I have, as of today, come to the final conclusion that your entire company ought to be sued for the biggest load of hypocrisy I've ever seen!!!
I mean... you're not actually an expert in nothing! You're an expert in making people laugh hysterically, and double antangeras, and heck! You could even run for president with your languistic
skills! You're not a nothing expert at ALL! You're in fact, an expert on so many somethings that I must say... you really deserve to be put in prison for stealing all that money from all those fools
who believed you to actually know nothing!
That said, I think I'ld like to order nothing and a book about "it." :)
To which I replied:
Thank you for your kind words and your insults.
Yes, it would appear that I am no expert on nothing because it often appears that I do indeed know everything. I want to point out that I do not know everything. I have a good grasp of many general
concepts and I am awesome. But this doesn't make me an expert on everything, just awesome, that's all.
Just because I happen to know something and be awesome doesn't preclude my ability to know nothing. In fact, nothing is actually enhanced by my knowledge and awesomeness, because how could "it" be?
I hope that this clears up the confusion you had regarding my non-expert status. But I hope that the restoration of my credibility in your eyes does not dissuade you from buying nothing from me.
Thanks again.
--Xymyl (KON)
On May 18, 2007, at 8:20 PM, Richard wrote:
Dear Sir,
I just watched the Matrix Reloaded, and it occured to me that it is nothing.
Nothing was said. Nothing occurred. Often characters (if there were
any) acted, and those actions changed nothing and meant nothing. For
Neo, who was nothing, nothing seemed to be able to stop him from his
relentless pursuit of nothing. It was great the way he saved Trinity
in the end so that together they could do nothing.
I really empathised with the characters. They seemed to feel nothing
and when I watched the film I felt nothing too.
My question is: what exactly was being 'reloaded' if the result was
nothing? If nothing squared equals nothing, that is indisputable
evidance for something.
Any thoughts?
To which I replied:
Thank you for your thoughtful dissection of Matrix Reloaded. Your willingness to wait four years to watch the movie or to write this e-mail and hold onto it for up to four years before sending it to
us shows that you have the potential to really sit around doing nothing professionally one day. No hurry.
The answer to your question is that there were many things being reloaded, 1) the same effects from the original film were "reloaded", 2) the same theatre goers were "reloaded" into theaters to watch
the movie, 3) many of the same actors were "reloaded" to make the whole reloading process that much more visceral, 3) the same story lines were also "reloaded" so that it was almost like the same
movie was just reloaded into the can. Naturally, all of this reloading resulted in a certain negation akin to white noise and as such would be commonly characterized by a metaphorical "nothing" which
is an appropriate topic for this forum.
As for your statement about nothing squared equalling nothing, I'm not quite sure where you are trying to take that. But no, that is not evidence for something (well, it could be evidence of math
wasting everyones valuable time, but that can be proved in many other ways). But we also don't need any evidence for something besides its existence. Sometimes, even that seems excessive.
--Xymyl (KON)
On May 9, 2007, at 11:52 AM, Steve H. wrote:
I mailed you a check for $20 for a black, size large, nothing T-shirt and I have not received it. I mailed the check in mid-April. Could you please let me know if you received the check and when i
can expect to get my T-Shirt? Thanks
To which I replied:
Hi Steve,
We got your check and we will send out your shirt as promised. As our site says, 4 to 8 weeks to delivery.
We know that is a long time to wait for a shirt, but we have enjoyed spending the last few weeks hanging out with your check. That's right, we haven't even brought it to the bank yet, the extra time
we take to cash your check or send your shirt is our way of saying thank you for your purchase.
Remember, we think nothing is more important than a satisfied customer, and we were doing nothing important at the time we got your check.
In all seriousness, we have been considering running our operation like a real business, actually making money, processing orders at greater speed, etc, we even put tee shirts up for sale on Amazon.
Sure you pay a little more, but then we send your tee-shirt within two days. Perhaps one day we will just be another big corporation that wastes peoples time at random without caring. In many ways
that would be very cool. But for now we are still just thinking about you and staring at your check. It is a very nice check with what appears to be a simple herringbone pattern, but on closer
inspection you can see that the pattern isn't connected but then you start to think that the pattern... I'm sorry, I'm deliberately stalling for time.
We'll send that right out to you.... well, give us just a couple more days.
Xymyl (KON)
To which Steve responded:
Hey, thanks for nothing! Well, for the time being at least. I am encouraged by your thoughtful note however and will look forward to receiving nothing from you real soon. In the meantime you may
continue to gaze longingly at the herringbone pattern on my check. However you will find it even more rewarding if you cash it. Of course that would constitute my giving you something in return for
nothing. Aside from a good laugh and a T-shirt.
Best regards,
Steve H.
On May 10, 2007, at 5:25 PM, Bane93@aol.com wrote:
it threatens your entire existence be careful
To which I replied:
Hi Noman,
Thanks for the heads up!
We aren't too worried though, as we have been barraged by anything and everything for all eternity. And as far as that goes, something doesn't necessarily threaten our existence as much as it
threatens us WITH existence. We already know that we exist. We know that even we are something, some might even go so far as to say that to them we are everything. We remain extant in spite of
something, anything and everything with a stated purpose. That purpose, of course, isn't.
Thank you for your outstanding efforts to protect nothing important from something insignificant!
--Xymyl (KON)
On May 4, 2007, at 10:43 PM, Sean Craft wrote a letter with many questions. These questions have been placed into this blog in an easy to read question and answer format. Enjoy:
X: Hello Sean, Thank you for your interest in nothing. We are always eager to help meager young minds to comprehend nothing. We have taken our time to pore over your questions and feel that we have
adequately answered them. If you feel that we have not answered them in a satisfactory way then we suggest that you go do something with yourself.
SC: I'm just wondering, You wouldn't happen to have any empty spaces for nobodies, who normally would be doing nothing, but really need nothing else to do, would you?
X: Yes and no.
SC: Is nothing really nothing or could it be something more than nothing as some philosophers have thought?
X: Yes and no.
SC: Is it hard to package nothing?
X: Yes and no.
SC: Sorry for all the questions, but I really have nothing else to be doing at the moment.
X: Oh, that's alright. We love to talk about nothing. Answering your questions has really been a joy for me personally as it has brought back many memories of my childhood. Treasured moments from the
past, which I will cherish forever, mostly dreams, but still great times. One in particular was when I was floating in a void of space, then slowly lowered down a dark brown mesh tube. All I could
see were my hands reaching out in front of me feeling for anything, but nothing was there. The dream seemed to be on a perpetual loop. It started with an infinite memory of the dream repeating, as
though the dream had always been happening to me even before my existence. Eventually the dream just ended, apparently before it had ever began.
SC: What are your beliefs on who it was or how nothing was created?
X: None.
SC: I can imagine that nothing could deter you from pursuing your studies of nothing, but if something could, deter you from nothing, what would that something be?
X: First of all, nothing couldn't even stop us from studying nothing. But if anything could, it would be teaching others about nothing.
SC: I trust you will divulge nothing of our conversations to noone else unless they have nothing to do with nothing.
X: If you mean that you hope I'll post this on the blog, yes, I will.
SC: Nothing is a hard topic to study is it not?
X: Yes and no.
SC: Once more, I'm sorry for being so inquisitive as to the value, study, and pursuit of nothing.
X: Once again I must remind you that we enjoy sharing our knowledge about nothing. Do not be ashamed of your interest in nothing, or for your need of assistance with nothing. Although the first three
questions were very difficult to answer, I felt it was worth the extra effort to give you an in-depth generalization. That's why I didn't try very hard to answer them. On the other hand, those same
questions were very simple and straightforward. Being that they were so easy to answer made me pause and really ruminate on them until I came up with the same exact answer I already had.
Thank you again.
--Xymyl (KON)
For several weeks a fan named Sea Bass has been writing us e-mails explaining how he has nothing to say. We had faith that he would one day really have nothing to say and today our dreams came true
On Apr 16, 2007, at 7:52 AM, Sea Bass finally wrote:
And I responded:
Sea Bass,
Thank you for your lack of words, finally, you really do have nothing to say! We have been waiting for this moment since your first e-mail on march 20th. We consider this your graduation. We will be
carving your name on a piece of rice and throwing it beneath our conference table, we guarantee that it will remain on display for a minimum of two days.
Xymyl (KON)
On Apr 15, 2007, at 9:59 PM, Nicole Davis wrote:
I have nothing to do. Now because of this site, I know nothing makes sense, and I wish more people would strive for nothing and finally we will all realize that we have nothing to worry about.
Thank you for nothing. I llaughed unntil the tears ran from my face annd then I had nothing left.
I have nithing else to say.
N. Davis
A fan of nothing
To which I replied:
My heart skipped a beat when I read your letter which was posted all around the office and even incorporated into our greeting cards. Thank you for getting my heart to shut up for a moment. That
moment of silence helped me to see nothing slightly more clearly than nearly ever before. Actually it did nothing for me, but I say that to people way too often.
I'm very glad to see that you have nothing to do and that nothing finally makes sense to you. I'm sorry that you are worrying about nothing though. Although that is our intention with certain people
we certainly don't want everyone to worry about nothing. So we will tell you what a disturbed fan recently told us, "if you say nothing, I'm gonna kill you!! haha". For us, the most frightening part
of the threat was the maniacal laughter they typed in. Hopefully, this will strike you with terror as well, so you can stop worrying about nothing and get on with your life.
Thank you for emptying your tear ducts as a tribute to nothing. We will erect a small shrine the size of a dust particle in your honor, which will be swept around the office for a minimum of two
Thank you for your heart pausing interest in nothing.
Xymyl (KON)
On Apr 10, 2007, at 6:04 AM, Aryeh Lewis wrote:
but... but... but there's no lack of nothing!!
What's so special about the nothing we get when we sign your order forms?
(if you say nothing, I'm gonna KILL you!! haha)
To which I responded:
I would say nothing, but you claim that I will be killed for that simple, correct answer. So I will say another simple and true answer. When we sell nothing you get the instructions. We make no
claims that this is an exhaustive resource, but it certainly gets you thinking about nothing and how easy it is to make nothing a major part of your daily life. It includes sections about safety,
nothing to do, nothing to say, nothing to eat, nothing to wear, troubleshooting, and even information on how to install the NO/OS.
As for your comment about there being no lack of nothing, I must submit that there is also no surplus of nothing. If you want to get all technical about it there is also no supply of nothing. But we
have recently expanded our warehouses to cover infinite space. That means that now all of our warehouses overlap, thus creating the most redundant access to nothing imaginable. When someone buys
nothing, they don't want to buy nothing from the guy on the corner who doesn't know nothing, who doesn't have access to the largest empty warehouses in or out of the universe, who can't explain what
nothing isn't, or how "it" does or doesn't work. They want nothing to be reliable and predictable. They want to know that they are dealing with the one distribution house that has been authorized to
distribute nothing universally. They want to know that nothing they buy is no gimmick, but that "it" really isn't something, everything or anything. There is only one place to get nothing like that,
right here.
As for your question in an earlier e-mail question regarding how many e-mails we get each day, we get a different amount of e-mail each day. Often times there are far too many to answer, but other
times there are too few to answer. I have to admit that nothing is not as popular as "it" once was. We have had millions of viewers over the years, but people seem to like to see change, so they
eventually stop coming. If nothing was going to be about anything (which "it" isn't) "it" would be about not changing anything. Still, we get several hundred unique visitors a day, which is nothing
to sneeze at. I don't understand that phrase, but it had the word nothing in it so I decided to use it.
Thank you for your interest in nothing and e-mail.
Xymyl (KON)
On Apr 11, 2007, at 12:12 PM, Tom McLeod wrote:
if I order a nothing T-Shirt will I really get one? I don't want to waste money.
To which I responded:
Yes, you'll get a shirt! In fact, to prove they are real we have them available on Amazon now. Just follow this link: http://www.amazon.com/exec/obidos/ASIN/B000PCOB6C
There are just a few available right now, but here at nothing.net we like to see how close to "0" we can get. As soon as we run out there will be more available.
Thanks for your interest in nothing (and shirts).
Xymyl (KON)
On Mar 29, 2007, at 9:58 AM, Mike Kust wrote:
Is ordering that t-shirt thing real? Just a question. If not it should that would be AWESOME
To which I responded:
Yes, you can order a tee-shirt by printing, filling out and mailing our easy to use insecure store form. In just a few days we will start selling our tee-shirts and a few other select nothing related
items on Amazon. We know that the true nothing loyalists appreciate the irony that can be achieved by ordering nothing (or in fact anything) with our gloriously outdated means of transacting. But we
also appreciate that it is so cheap to do e-business these days that we can no longer pretend that our budget does not allow for a little online shopping.
We are proud to do all of this for nothing, but there is nothing wrong with getting a little something back.
Xymyl (KON)
On Mar 31, 2007, at 12:52 AM, john cash wrote:
i think in most popular religions theres a possibility of getting punished for doing nothing
but no possibility of getting rewarded doing nothing , am i right?
To which I replied:
Although I cannot claim to be an expert on all the worlds religions, I am the worlds foremost expert on nothing. As such, I would have to say that while there is some possibility of being punished
for doing nothing, I think it is much more likely that you would be simply thought little of. Certainly, many religions of various origins would frown upon doing nothing to save someone who was
drowning, or doing nothing to help a person in great need.
In these instances, It is a simple matter of knowing that something is right and then not doing it.
However, if someone is offered an opportunity to assist in a bank robbery and they decide to sit on the beach watching people drown, that would be a bit of a toss-up for the majority of religions out
You are right about one thing, most religions (as well as most other organizations one might join) would not reward doing nothing. Many, for the simple fact that there is rarely any way of knowing
that someone was doing nothing, so even the organizations that would love to reward the doing of nothing have to settle for rewarding the people who do less bad stuff. That isn't so bad for all of us
who just love to spend a day doing nothing, because doing nothing is its own reward.
Nothing is more important to me right now than making sure this issue is fully explained to you, so I hope what I already said is good enough.
Xymyl (KON) Stating the obvious and signing out.
On Mar 24, 2007, at 11:00 PM, jon dieks wrote:
well thank you again for a swift response to my e-mail but once again i found myself sadend by your response and relised i was was going nowhere with all this. but since i was going nowhere i figured
i might as well talk to no one about this. no one said" i should talk to nobody about nothing". i decided that might be a good idea so i asked nobody about nothing. and nobody said " nothing is
possible" and that sparked my hope(because i was starting to get depressed about the whole situation). so i inquired as to how . nobody told me to close my eyes and think of nothing, then open them.
and guess what!!!!!!!! NOTHING HAPPEND!!!! so nobody said to me that even though i may be something or somebody. that nothing always happens. that i should be thankful for nothing. so i will thank
nobody for nothing. and thank you for nothing. i sure hope you haven't cornerd the market on nothing. as i like to get in on this action of nothing. i thought about burning stuff and building my
electro magnetic magnet but relized it takes a whole lot of something to get nothing. so well i may still dream of nothing,hope for nothing ,or want nothing, i've relised that i am getting nothing
out of life. which makes me so happy(nothing excites me). but as i end this i've noticed that for someone who has a website about nothing you sure know alot about something.
To which I responded:
I am glad you were able to find nobody to help you with nothing. And what wasn't said by anyone was also said by you, and it is true. When we are willing to listen to nobody then nothing is possible.
I am positively (yes positively) thrilled that no one was able to restore your hope in nothing. And even though this may seem to slightly contradict the facts (and lack thereof) that I have shared
with you in previous correspondences, I must say, your story is nothing if not a success.
As for your desire to embark on a nothing venture, we do have what is referred to as a nonopoly on nothing. This means that we have the exclusive right to sell nothing. You may wonder how this could
be, and ask, "Isn't everything regulated by the world governments?" Of course, as you mouth these words you will be struck by the fact that nothing isn't regulated by the world governments in some
way or another. This means that we stand alone as the one entity with control over nothing. Yes, we are the only group with full authority over nothing. We plan to keep it that way. But this doesn't
mean that nothing can't be yours, just that "it" isn't.
As for knowing a lot about something, we have to, we are surrounded by it. But to us something isn't a hinderance, it is simply an obstacle to be overcome to get back to "the cold dead void" of
nothing, where we feel most comfortable. We feel less and less and/or more and more significant with each piece of trivia we accidentally learn while educating the world about nothing. Since the
blessing is thus a curse and the curse a blessing, the knowledge we take in often times achieves its own negation due to the knowledge (or lack thereof) that is imparted as a product of its use.
People sometimes say that we take nothing seriously. I would say to everyone that nothing should be taken too seriously, way too seriously. And can nothing really be taken too seriously? No. But why
not try?
Thank you for your thanks for nothing. Your infinite journey has begun.
--Xymyl (KON)
On Mar 22, 2007, at 5:08 PM, jon dieks wrote:
although relativley dismayed at the prospect of never being nothing.as in my e-mail wanting to be nothing.i but i've found a new hope. it all relates to my consciousness as the expression " i think
there for i am" so even if my physical self cannot in effect become nothing but ash molicules atoms or energy if i lost the abillity to be conscious i would not know i was ash molicules atoms energy
protons nucleons or any other ions or eons i would ineffect be come nothing. because i wouldn't be able to think of something. and as we all know notyhing is perfect. so one glorious day nothing may
be mine.
To which I responded:
I'm also deeply saddened at the thought that you cannot be nothing. If you were nothing I could sell you for 5 dollars.
I can see that you are very driven to succeed at nothing, especially at becoming nothing. You remind me of nobody specific. I remember distinctly talking to him about something, and he got mad, I had
to say nothing louder than I had ever yelled before to explain to him that I was kidding. But I'll tell you this, If you have even none of the dedication he had to nothing, you won't get far. And I
don't need to tell you that that's not saying much.
The phrase you are referring to, "I think, therefore I am" is a truncated form of a phrase less popular (but nonetheless popularized) by René Descartes, "I doubt, therefore I think, therefore I am".
Indeed, the shortened and long versions of this phrase are not truly negated by their inversion. And the subject "I" exists, at least as forensic evidence. On the shorty tip, the inverse of our
famous phrase of discussion would perhaps be "I think not, therefore I am not" or "I think not, therefore I am nothing" or perhaps "I do not think, therefore I am nothing" this interpretation is
easily put to rest when we realize that trees, rocks, and even earthworms, "think not" and yet "are". Also, if "I think not" was used as it often is, as a statement of doubt, then the philosophical
crux of your gist would have been overridden by itself, being that "I think not, therefore I am" is almost identical in meaning to our original source phrase. Of course, you know that I could go on
and on, but suffice it to say that fulfilling the broader meaning of a phrase during an attempt at its nullification is not going to accomplish nothing. Certainly, the existence of the subject "I"
proves being. He could have just said, "I". That would have been good enough for me.
It is interesting to note that not all of René D's thoughts were completely abolished at his demise. He had apparently wanted to be "am" for as long as possible, so he had the faith that if he wrote
down his doubts and other whimsical musings, they would be preserved for us so that in a sense he would still be with us, to impart doubts about the very things he believed in. Of course, in a very
real sense, he's dead. But give him props 'cause dude's still am'n it old school!
If you find a way to doubt after your own death then you will have still earned your pi meson (sized) shrine as promised. Of course Descartes was also a math addict, he was all about precise
convolution. In a very real way the guy was "all up in everything's business" and I most certainly mean that in a way, but he wasted peoples time, and he's still doing it every day at school even
though he's long gone.
To summarize, no thought does equal nothing in relation to thought itself, but if those thoughts are preserved in, oh I don't know, perhaps an e-mail, then in a sense those thoughts still exist. So
you are speaking of an existence devoid of future thought. Many people who are still alive and going through their daily activities never come up with even 1 original thought. They live each day as
though nothing has happened to them. And that's not far away from the truth.
I have faith in you, the same type of faith as a man who doubts. I believe in your ability to create an electromagnet so powerful that you will be able to destroy all the data on every computer in
the world that may contain any record of your existence and/or the existence of your thoughts. I believe in your ability to burn down your house so thoroughly that not a minute trace of your
existence will be left at that location. I believe that you can find every copy of your social security number that you've ever put on a medical form or given to an employer or prospective employer
and pour kerosene on them and ignite them without once being noticed. I believe you can track down and destroy every school you ever went to. I believe you can expunge your FBI files. Clearly I could
go on and on, suffice it to say, I believe you can get rid of every shred of information that says you exist, and when this is accomplished, I believe you will send me an e-mail telling me you did
it. The irony will be worth it, because all of that would have been for nothing.
--Xymyl KON) | {"url":"https://xymyl.blogspot.com/2007/","timestamp":"2024-11-13T22:53:58Z","content_type":"text/html","content_length":"299160","record_id":"<urn:uuid:5652869c-495f-482c-91bf-ca59a6f4146c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00513.warc.gz"} |
SSC Logical Reasoning - Logical Sequences Questions with Answers
Common sense is the collection of prejudices acquired by age eighteen.
"Die hard fro your name, your name should be enough to recognise you"
Thanks m4 maths for helping to get placed in several companies. I must recommend this website for placement preparations. | {"url":"https://m4maths.com/placement-puzzles.php?SOURCE=SSC&TOPIC=Logical%20Reasoning&SUB_TOPIC=Logical%20Sequences","timestamp":"2024-11-06T01:04:48Z","content_type":"text/html","content_length":"93125","record_id":"<urn:uuid:63ea4f25-8f0e-41d7-a4ba-ebe31d2068d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00082.warc.gz"} |
Decision Structures
Decision Structure
Decisions in a program are used when the program has conditional choices to execute a block of code.
Decision structures evaluate multiple expressions and return True or False as the result. You need to determine which action to take and which statements to execute if the result is True or False.
The Python Boolean type has only two possible values: True and False.
We are going to work on three types of decision making structures.
1. If statements
2. If-else statements 3. Nested if-else statements
When working with decision conditions, we need relational operators. A relational operator checks whether its operands satisfy a specified relationship and produces a Boolean value based on its
assessment. It returns True or False.
For example, the value of the expression 3 < 9 is True because the < operator is the "less than" operator. The expression x < y has the value True if the value of x is smaller than the value of y and
False otherwise.
The following table lists several commonly used Boolean operators in Python.
Let's look at some examples of Comparison Operators.
x = 24
y = 69
print('x > y is', x>y)
# Output: x > y is False
print('x < y is', x<y)
# Output: x < y is True
print('x >= y is',x>=y)
# Output: x >= y is False
print('x <= y is',x<=y)
# Output: x <= y is True
print('x == y is',x==y)
# Output: x == y is False
print('x != y is',x!=y)
# Output: x != y is True
Check the following links for more relational operators and images.
https://creativecomputing.ca/11/11_1_1_Relational_Operators.html https://towardsdatascience.com/python-operators-from-scratch-a-beginners-guide-8471306f4278 | {"url":"https://www.labs.cs.uregina.ca/165/decision/index.html","timestamp":"2024-11-04T05:52:37Z","content_type":"text/html","content_length":"12324","record_id":"<urn:uuid:fd523920-ed28-42f8-8a4a-ec606dd0487e>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00424.warc.gz"} |
How to Do E in Excel - Learn Excel
If you are looking for a quick and efficient way to find the value of e using Microsoft Excel, this post is for you. The mathematical constant e is a popular number used in many mathematical and
statistical calculations. With Excel’s built-in functions and formulas, finding the value of e is quick and simple. In this post, we will guide you through the step-by-step process of finding e in
Excel, so you can use it for your own calculations and analysis.
If you are looking to use the mathematical constant e in your Microsoft Excel calculations, you’re in luck. Excel has powerful built-in functions and formulas that make calculations using e quick and
What is E and Why is it Important?
Before we dive into calculating e in Excel, it’s important to understand what e is and why we use it. The mathematical constant e is often referred to as Euler’s number and is equal to approximately
2.71828. E is used in many mathematical and statistical calculations such as compound interest, growth rates, and exponential decay.
Using the EXP function to Find e
The easiest way to find the value of e in Excel is by using the EXP function. Here’s how:
1. Start by opening a blank Excel sheet and select a cell where you want to display the value of e.
2. Type the formula =EXP(1) into the cell and press Enter. The cell will display the value of e, which is approximately 2.71828.
3. You can now use this value in other calculations and formulas.
Let’s say you want to calculate the growth of an investment with a starting principal of $1000 and a growth rate of e^2. You can use the following formula in Excel: =1000*EXP(2). This will give you
the value of the investment after one period, which is approximately $7353.41.
Using the POWER Function to Approximate e
If you don’t want to use the exact value of e, you can use the POWER function to approximate it. Here’s how:
1. Select a cell where you want to display the approximate value of e.
2. Type the formula =POWER(1+(1/n),n) where n is a large number such as 1000.
3. Press Enter and the cell will display an approximation of e.
Let’s say you want to approximate the value of e in Excel using the POWER function. You can use the formula =POWER(1+(1/1000),1000). This will give you an approximate value of e, which is
approximately 2.71828.
We hope this post has helped you learn how to find the value of e in Excel. Once you know how to find e, you can use it in a variety of mathematical and statistical calculations. Excel’s built-in
functions and formulas make it easy to work with this important constant.
Using e in Excel Formulas
Now that you know how to find the value of e in Excel, you can start using it in your formulas. Here are some examples of how e can be used in Excel formulas:
Compound Interest
Compound interest is calculated using the formula A=P(1+r/n)^(nt) where A represents the final amount, P represents the principal, r represents the interest rate, n represents the number of times
interest is compounded per year, and t represents the number of years. If you want to use e in the formula, you can use the following formula:
A = P*e^(rt)
In this formula, e^(rt) is the compounding factor. This formula can be used to calculate the future value of an investment that has a continuous interest rate.
Growth Rates
Growth rates can be calculated using the formula r = ln(P/F)/t where r represents the growth rate, P represents the initial price, F represents the final price, and t represents the time period. If
you want to use e in the formula, you can use the following formula:
r = ln(F/P)/t = ln(e^(rt))/t = r
In this formula, e^(rt) is the growth factor. This formula can be used to calculate growth rates that have continuous compounding.
E is an important mathematical constant that is used frequently in various calculations. Excel provides various functions and formulas to calculate e as well as ways to use it in your calculations.
We hope this article has shown you how to use e in Excel formulas and given you a better understanding of why e is an important constant.
Here are some frequently asked questions about using e in Excel:
What is the value of e in Excel?
The value of e in Excel is approximately 2.71828. You can find the value of e using the EXP function or the POWER function.
What is e used for in Excel?
E is used in many mathematical and statistical calculations such as compound interest, growth rates, and exponential decay. It helps provide a more accurate result when compared to using other values
as constants.
Can e be used in Excel formulas?
Yes, e can be used in Excel formulas. Excel provides various functions and formulas to calculate e as well as ways to use it in your calculations. The formula to use e in Excel is A = P*e^(rt).
How do you approximate e in Excel?
You can use the POWER function to approximate e in Excel. The formula to approximate e using the POWER function is =POWER(1+(1/n),n) where n is a large number such as 1000.
What is the difference between e and ln in Excel?
E is a mathematical constant that is used in many calculations, while ln is the function used to find the natural logarithm of a number. The natural logarithm of a number is the logarithm to the base
of e. In Excel, you can use the LN function to find the natural logarithm of a number.
Featured Companies
• Explore the world of Microsoft PowerPoint with LearnPowerpoint.io, where we provide tailored tutorials and valuable tips to transform your presentation skills and clarify PowerPoint for
enthusiasts and professionals alike.
• Your ultimate guide to mastering Microsoft Word! Dive into our extensive collection of tutorials and tips designed to make Word simple and effective for users of all skill levels.
• Boost your brand's online presence with Resultris Content Marketing Subscriptions. Enjoy high-quality, on-demand content marketing services to grow your business.
Other Categories | {"url":"https://learnexcel.io/e-excel/","timestamp":"2024-11-08T02:11:09Z","content_type":"text/html","content_length":"139228","record_id":"<urn:uuid:5a4b5e9b-5109-4100-a2a0-874933d822ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00050.warc.gz"} |
Classification of unitary vertex subalgebras and conformal subnets for rank-one lattice chiral CFT models
Classification of unitary vertex subalgebras and conformal subnets for rank-one lattice chiral CFT models
Sebastiano Carpi
Tiziano Gaudio
Robin Hillier
October 24, 2018
We provide a complete classification of unitary subalgebras of even rank-one lattice vertex operator algebras. As a consequence of the correspondence between vertex operator algebras and conformal
nets, we also obtain a complete classification of conformal subnets of even rank-one lattice conformal nets. | {"url":"https://lqp2.org/node/1537","timestamp":"2024-11-13T11:17:17Z","content_type":"text/html","content_length":"16069","record_id":"<urn:uuid:44ab5797-fc5e-4cfc-8e65-7f389cdfffb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00159.warc.gz"} |
Mastering Reinforcement Learning: A Comprehensive Guide
Written on
Chapter 1: Introduction to Reinforcement Learning
Reinforcement Learning (RL) is an innovative approach that enables an agent to learn how to navigate an environment through trial and error. If you're curious about how to leverage RL to excel at a
game, continue reading...
The core objective is to train an RL agent that explores the game environment by interacting with it. The agent discovers which actions in various states will lead to the highest long-term rewards.
Once it has developed an optimized policy—essentially a strategy for decision-making—the agent becomes adept at maximizing rewards, allowing it to tackle the game like an expert.
The comprehensive process of training an agent using RL can be broken down into several key steps:
1. Define the objective
2. Initialize the environment
3. Engage with the environment
4. Discover an optimal policy
5. Implement the optimal policy
Section 1.1: Defining the Objective
The initial step in deploying RL involves clarifying the agent's objective and grasping the environment it will operate in. This includes outlining the specific goals and desired outcomes while
understanding the state space, action space, and reward function of the environment.
Following this, it's essential to create a simulated environment for the agent's interactions. This can be achieved through custom environments or simulation tools like OpenAI Gym, which accurately
reflect real-world scenarios and provide essential data for the agent's decision-making.
Defining clear objectives and constructing an effective simulated environment are pivotal for successful RL application.
Section 1.2: Initializing the Agent’s Policy
Next, the agent's policy is initialized with random values. This policy will evolve as the agent engages with the environment and learns the optimal strategy via trial and error.
Reinforcement Learning in 3 Hours | Full Course using Python - YouTube: This video provides a comprehensive introduction to reinforcement learning concepts, guiding viewers through the intricacies of
building a reinforcement learning model using Python.
Section 1.3: Engaging with the Environment
In RL, the agent learns through its interactions with the environment by taking actions in different states via exploration and exploitation.
The agent investigates the environment by experimenting with various actions and observing the resulting rewards. This process aids the agent in gathering valuable information while simultaneously
exploiting its existing knowledge by opting for actions likely to yield the highest rewards. Striking a balance between exploration and exploitation is crucial for the agent's learning process.
Exploration involves the agent testing new actions to collect information and enhance its understanding, while exploitation focuses on leveraging current knowledge to maximize rewards.
Section 1.4: Discovering an Optimal Policy
As the agent engages with the environment, it updates its policy and value estimates based on the rewards observed from various states. The ultimate goal is to identify an optimal policy that guides
the agent in making decisions that maximize cumulative rewards.
An optimized policy is vital for the agent's success in RL, allowing it to make informed decisions that enhance its performance.
Finding an optimal policy involves two intertwined processes: Policy Evaluation (PE) and Policy Improvement (PI). PE estimates the value of a given policy, while PI updates the policy to enhance its
effectiveness. Collectively, these processes are known as Generalized Policy Iteration.
To achieve an optimal policy, the agent must navigate the exploration-exploitation dilemma. A common strategy is the ε-greedy policy, which randomly chooses between exploration and exploitation. This
method allows the agent to explore various parts of the environment while also exploiting its knowledge to maximize expected rewards.
Chapter 2: Applying the Optimized Policy
Once the agent has trained adequately and developed an optimized policy, it can now use this policy to engage with the game and fulfill its designated tasks.
It’s important to remember that both the environment and the objectives may evolve over time. Consequently, you might need to retrain the agent to adapt to new circumstances or enhance its
End To End Machine Learning Project Implementation With Dockers, GitHub Actions And Deployment - YouTube: This video covers the complete implementation process for a machine learning project,
including deployment strategies and best practices.
Section 2.1: End-to-End Code Implementation for CartPole-v1
For the CartPole challenge, the agent will be trained using Q-Learning.
Defining the Objective:
The CartPole-v1 task requires the agent to maintain balance by applying forces to move the cart left or right.
• Action Space: The agent can either:
□ 0: Push cart to the left
□ 1: Push cart to the right
• Observation Space:
• Rewards: The agent earns a reward of +1 for each time step the pole remains upright, with a threshold of 500 for version 1.
Initializing the Agent's Policy:
A policy maps states to actions, shaping the agent's behavior within the environment. The goal of RL is to identify an optimized policy that maximizes cumulative rewards over time.
To do this, the agent’s policy is initialized with random values reflective of the state and action spaces.
# Create the environment
env = gym.make('CartPole-v1')
# Define the state and action spaces
state_space = env.observation_space.shape[0]
action_space = env.action_space.n
# Initialize the agent's policy
policy = np.random.rand(state_space, action_space)
The CartPole task features four continuous states, which need to be discretized for effective Q-Learning application.
Discretizing the States:
Since Q-learning typically relies on discrete states, continuous states must be divided into manageable categories. This approach enables the algorithm to function efficiently.
The following function bins the continuous space into discrete segments based on predefined boundaries:
def convert_state_discrete(obs):
# Creating 10 bins for cart position, pole angle, cart velocity, and pole angular velocity
cart_position_bins = pd.cut([-2.4, 2.4], bins=10, retbins=True)[1][1:-1]
pole_angle_bins = pd.cut([-2, 2], bins=10, retbins=True)[1][1:-1]
cart_velocity_bins = pd.cut([-1, 1], bins=10, retbins=True)[1][1:-1]
angle_rate_bins = pd.cut([-3.5, 3.5], bins=10, retbins=True)[1][1:-1]
# Discretizing the state
disc_state = int("".join(map(lambda feature: str(int(feature)), [np.digitize(x=[obs[0]], bins=cart_position_bins)[0],
np.digitize(x=[obs[1]], bins=pole_angle_bins)[0],
np.digitize(x=[obs[2]], bins=cart_velocity_bins)[0],
np.digitize(x=[obs[3]], bins=angle_rate_bins)[0]])))
discrete_states = list(map(int, str(disc_state)))
return tuple(discrete_states)
Thus, we modify the policy initialization:
policy = np.zeros((10,10,10,10) + (env.action_space.n,))
Engaging with the Environment to Find an Optimal Policy:
Here, we implement the Q-Learning algorithm to locate the optimal policy. The agent employs an ε-greedy policy to balance exploration and exploitation.
# Define the learning rate, exploration rate, and number of episodes
gamma = 0.01
alpha = 0.1
epsilon = 0.1
num_episodes = 10000
# Train the agent
for episode in range(num_episodes):
state = env.reset()
state = convert_state_discrete(state)
done = False
while not done:
if np.random.uniform(0,1) < epsilon:
action = env.action_space.sample()
action = np.argmax(policy[state])
next_state, reward, done, _ = env.step(action)
next_state = convert_state_discrete(next_state)
# Update the policy
policy[state][action] += alpha * (reward + gamma * np.max(policy[next_state]) - policy[state][action])
state = next_state
print('Training Completed')
Applying the Optimized Policy:
After sufficient training, the agent is equipped with an optimized policy and is ready to perform its tasks effectively.
num_episodes = 20
for episode in range(num_episodes):
state = env.reset()
steps = 1
done = False
episode_reward = 0
while not done:
steps += 1
state = convert_state_discrete(state)
action = np.argmax(policy[state])
state, reward, done, _ = env.step(action)
episode_reward += reward
print("Reward: ", reward, "steps: ", steps, "episode: ", episode, "Total episode Reward: ", episode_reward)
It’s essential to acknowledge that environments and goals can evolve, necessitating retraining of the agent to adapt to new challenges or to enhance its performance.
The graph illustrates the agent's improved performance after the application of the optimized policy. Performance can be further enhanced by:
• Providing the agent with additional experience through interaction to foster learning.
• Implementing more sophisticated algorithms such as Deep Q-Learning, A2C, or PPO.
• Employing a suitable exploration strategy to enable the agent to gather more valuable insights about the environment.
• Tuning hyperparameters, including the learning rate, discount factor, exploration rate, or epsilon.
Hyperparameter Tuning for Q-Learning
1. Learning Rate (α): This parameter controls the update magnitude of the agent’s policy, typically set between 0 and 1. A high learning rate may lead to rapid convergence but can also overshoot the
optimal policy.
2. Discount Factor (γ): This dictates the significance of future rewards versus current rewards, also set between 0 and 1. A high value prioritizes future rewards, while a lower value emphasizes
immediate rewards.
3. Exploration Rate (ε): This influences the ε-greedy policy, determining the likelihood of the agent taking random actions instead of the action with the highest expected reward.
4. Number of Episodes: This refers to the total interactions the agent has with the environment before concluding the training.
• Sutton, R. S., & Barto, A. G. (2018). Reinforcement Learning: An Introduction.
• David Silver’s Reinforcement Learning Lecture Series. | {"url":"https://panhandlefamily.com/mastering-reinforcement-learning-guide.html","timestamp":"2024-11-15T03:16:01Z","content_type":"text/html","content_length":"21443","record_id":"<urn:uuid:09bb70f8-83ad-4b5d-a239-30f81b5251d5>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00204.warc.gz"} |
7 Times Table
Welcome to the 2nd Grade Math Salamanders 7 Times Table Worksheets.
Here you will find our selection of free multiplication worksheets to help your child learn their 7 times multiplication table.
Multiplication is introduced as a concept around Grade 2.
Unlike addition and subtracting, multiplication is a concept that does not come naturally to many children. Quite a lot of time may need to be dedicated in developing children's understanding of what
multiplcation is and how it works. Time spent doing this is time well spent, so that children become more confident with their understanding of multiplication before they continue their journey into
the multiplication table and solving multiplication problems.
The multiplication learning in 2nd Grade underpins future learning of the multiplication table, and the standard multiplication algorithm learnt in future grades.
During 2nd grade, children should be learning the following multiplication skills:
• understand what multiplication is and how it works;
• know the multiplication table to 5x5;
• solve simple multiplication problems.
Multiplication Facts - 7 Times Table
The following worksheets are all about learning and consolidating multiplication facts and counting up by 7s.
The sheets in this section are at a similar level of difficulty from each other.
Each sheet is designed to help your child learn and understand their tables in a slightly different way, from learning about grouping to counting up and writing out their table facts.
An answer sheet is available for each worksheet provided.
Using the sheets in this section will help your child to:
• learn your table facts for the seven times tables;
• understand multiplication as repeated addition;
• understand multiplication as skip counting;
• solve a range of simple multiplication challenges.
Want to test yourself to see how well you have understood this skill?.
• Try our NEW quick quiz at the bottom of this page.
7 Times Tables Printable Display Sheet
Here is a printable display sheet for the seven times table to use as a quick reference or part of a display.
There is also a blank version for the children to fill in.
Seven Times Tables Worksheets - Counting by 7s
Looking for more 7 Times Tables worksheets like these?
Try the Seven Times Table practice worksheets at the Math Salamanders!
Using the link below will open the Math Salamanders main site in a new browser window.
Take a look at some more of our worksheets similar to these.
Multiplication Facts Learning
Multiplication to 5x5
The following webpages involve learning multiplication facts to 5x5.
The sheets in this section are for children who already have a solid grounding in what multiplication is and how it works. They involve the skill of adding multiplying numbers to 5x5.
Using the sheets in this section will help your child to:
• multiply numbers to 5x5;
• work out times table facts to 5x5 where the total is given but one of the other numbers is missing.
All the multiplication worksheets in this section will help your child to develop their speed and accuracy at multiplying.
Using Arrays as a model of understanding Multiplication
The following webpages involve using arrays as a model to help to develop an understanding of what multiplication is.
These sheets are particularly useful to visual learners, and those children who like to see how things work visually.
Using the sheets in this section will help your child to:
• understand multiplication as repeated addition;
• know how multiplication relates to adding groups or sets of objects;
• know that multiplication can be done in any order (5x3 is the same as 3x5).
All the sheets in this section will help your child to develop their multiplication understanding.
Our quizzes have been created using Google Forms.
At the end of the quiz, you will get the chance to see your results by clicking 'See Score'.
This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.
For incorrect responses, we have added some helpful learning points to explain which answer was correct and why.
We do not collect any personal data from our quizzes, except in the 'First Name' and 'Group/Class' fields which are both optional and only used for teachers to identify students within their
educational setting.
We also collect the results from the quizzes which we use to help us to develop our resources and give us insight into future resources to create.
For more information on the information we collect, please take a look at our Privacy Policy
We would be grateful for any feedback on our quizzes, please let us know using our Contact Us link, or use the Facebook Comments form at the bottom of the page.
This quick quiz tests your knowledge of the 7 Times Table.
How to Print or Save these sheets
Need help with printing or saving?
Follow these 3 easy steps to get your worksheets printed out perfectly!
How to Print or Save these sheets
Need help with printing or saving?
Follow these 3 easy steps to get your worksheets printed out perfectly!
Whether you are looking for a free Homeschool Math Worksheet collection, banks of useful Math resources for teaching kids, or simply wanting to improve your child's Math learning at home, there is
something here at the Math Salamanders for you!
The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.
We welcome any comments about our site on the Facebook comments box at the bottom of every page.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. | {"url":"https://www.2nd-grade-math-salamanders.com/7-times-table.html","timestamp":"2024-11-15T02:52:51Z","content_type":"text/html","content_length":"53593","record_id":"<urn:uuid:244b3793-4243-437c-8681-fe9c8824fdfa>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00830.warc.gz"} |
How do you determine all values of c that satisfy the mean value theorem on the interval [1,9] for f(x)=x^-4? | HIX Tutor
How do you determine all values of c that satisfy the mean value theorem on the interval [1,9] for #f(x)=x^-4#?
Answer 1
$c = \sqrt[4]{\frac{17496}{728}} \cong 2.2141$
Based on the mean value theorem, as #f(x) = x^(-4) # is continuous in #[1,9]# then there must be at least one point #c in [1,9]# for which:
#c^(-4) = 1/8int_1^9 x^(-4)dx#
As f(x) is strictly decreasing in the interval [1,9], there can be only one point in the interval satisfying the theorem.
#1/8int_1^9 x^(-4)dx = 1/8[-1/(3x^3)]_1^9 =1/8(1/3 - 1/2187)=1/8((729-1)/2187)=728/17496#
So the value #c# that satisfies the mean value theorem is:
#c^(-4) = 728/17496#
#c=root(4)(17496/728) ~=2.2141#
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To determine all values of ( c ) that satisfy the Mean Value Theorem on the interval ([1, 9]) for ( f(x) = x^{-4} ), you first find the average rate of change of ( f(x) ) over that interval. Then,
you find the derivative of ( f(x) ) and evaluate it at ( c ), the value guaranteed by the Mean Value Theorem.
The Mean Value Theorem states that if a function ( f(x) ) is continuous on the closed interval ([a, b]) and differentiable on the open interval ((a, b)), then there exists at least one number ( c )
in ((a, b)) such that ( f'(c) = \frac{f(b) - f(a)}{b - a} ).
In this case, the function is ( f(x) = x^{-4} ), and its derivative is ( f'(x) = -4x^{-5} ).
The average rate of change of ( f(x) ) over ([1, 9]) is:
[ \frac{f(9) - f(1)}{9 - 1} = \frac{1/6561 - 1}{8} = -\frac{3280}{6561} ]
So, to find the value of ( c ), we set:
[ f'(c) = -4c^{-5} = -\frac{3280}{6561} ]
Solving for ( c ), we get:
[ c^5 = \frac{6561}{3280} ]
[ c = \sqrt[5]{\frac{6561}{3280}} ]
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
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
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-determine-all-values-of-c-that-satisfy-the-mean-value-theorem-on-the--25-8f9af9fc7c","timestamp":"2024-11-07T17:31:29Z","content_type":"text/html","content_length":"581850","record_id":"<urn:uuid:ea56a7c9-0814-4040-a174-1521692ab236>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00551.warc.gz"} |
Vector Graphics
gCoord Points Vectors PV Planes
The code in this section is designed to allow you to store and draw lines, arcs and splines which may represent three-dimensional objects.
People used to worry about speed differences in using integers or floating-point arithmetic for graphics. Modern processors and graphics boards mean that whatever is most natural can often be
used, so floating-point arithmetic is used here and rounding errors dealt with by specifying a "Tolerance". This Tolerance is the core of the whole system on these pages and, consequently,
discussed in depth later. Using floating-point arithmetic allows you to debug and see real values. Conversion from double to integer still needs some thought though, because current floating
point processors clear the processor pipeline when setting the rounding mode. Methods Floor and Round provided in Global.h work around this.
The Graphics Primitive structures are all prefixed with "g" (for "graphic") and a number representing the number of Dimensions the structure will handle (2 or 3). So the three-dimensional Point
structure is a g3Point. Two and three dimensional structures are presented for each concept and abuse the whole idea of object orientation by defining the structures in terms of the data they
will hold rather than the object they represent. At first this seems bizarre, but when you are actually working with the classes you'll be surprised at how rarely you think about the object
you're dealing with: you're almost always more concerned with the maths defining the object. So, for example, there is a class which contains a Vertex and a Vector... The word "Point" is used
instead of "Vertex" so that the class for a for "PointVector" can be called "g3PV"... This class could represent a Position Vector, or a Line, or an axially-aligned Box (Point is one corner,
Vector points to opposite corner). The maths for each possibility is likely to be in the primitive, so a g3PV will have a "Hit" function to to see if a point lies on the line; a "HitBox" function
to see if it lies in the Box etc. That's why these structures are referred to as "Primitives": you can derive "proper" classes from them, but at this level, they are simple open boxes to hold
things in.
Start with the gCoord structure and work along the sections in the top menu.
Each class has a CString Report() const; method that is great for TRACE when debugging, or passing to APIs. It's also valuable for regression testing: collect a file full of report strings and
remember the MD5 checksum of the file. Next time you run the test you only need to check the MD5 against the one you remembered to see if anything changed. Once you have a difference, compare the
two text files to see where things started to differ. This is good over slow connections because you can have a huge number of tests and just send a pile of MD5 strings for comparison with the
server's MD5s. | {"url":"http://lumpylumpy.com/Electronics/Computers/Software/Cpp/Graphics/Vector/index.php","timestamp":"2024-11-05T03:01:39Z","content_type":"text/html","content_length":"8630","record_id":"<urn:uuid:005023d4-e1a4-4554-8a26-88e4c67a866f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00830.warc.gz"} |
Formula function reference
Learn the operators and functions you can use to create formulas in ThoughtSpot.
ThoughtSpot allows you to create derived columns in worksheets using formulas. You create these columns by building formulas using the Formula Assistant. An individual formula consists of n
combinations of operators and functions.
This reference lists the various operators and functions you can use to create formulas.
Aggregate functions (group aggregate)
Use the following functions to aggregate data.
Returns the average of all the values of a column.
Returns the average of all the columns that meet a given criteria.
average_if(city = "San Francisco", revenue)
Returns the number of rows in the table containing the column.
Returns the number of rows in the table containing the column that meets the specified condition.
count_if(region =’west’, region)
Takes a measure and one or more attributes. Returns the average of the measure, accumulated by the attribute(s) in the order specified.
cumulative_average (revenue, order date, state)
Takes a measure and one or more attributes. Returns the maximum of the measure, accumulated by the attribute(s) in the order specified.
cumulative_max (revenue, state)
Takes a measure and one or more attributes. Returns the minimum of the measure, accumulated by the attribute(s) in the order specified.
cumulative_min (revenue, campaign)
Takes a measure and one or more attributes. Returns the sum of the measure, accumulated by the attribute(s) in the order specified.
cumulative_sum (revenue, order date)
Takes a measure and optional attributes and filters. Used to specify columns and filters to include or ignore in your query. Commonly used in comparison analysis.
This formula takes the following form:
group_aggregate (<aggregation(measure)>, <groupings>, <filters>)
Define lists using curly brackets, { }. Optional list functions query_groups or query_filters specify the lists or filters used in the original search. Use + (plus) and - (minus) to add or
exclude specific columns for query groups.
group_aggregate (sum (revenue), {ship mode, date}, {} )
group_aggregate (sum (revenue), {ship mode , date}, {day_of_week (date) = 'friday'} )
`group_aggregate (sum (revenue), query_groups(), query_filters() ) `
group_aggregate (sum (revenue), query_groups() + {date}, query_filters() )
Takes a measure and one or more attributes. Returns the average of the measure grouped by the attribute(s).
group_average (revenue, customer region, state)
Takes a measure and one or more attributes. Returns the count of the measure grouped by the attribute(s).
group_count (revenue, customer region)
Takes a measure and one or more attributes. Returns the maximum of the measure grouped by the attribute(s).
group_max (revenue, customer region)
Takes a measure and one or more attributes. Returns the minimum of the measure grouped by the attribute(s).
group_min (revenue, customer region)
Takes a measure and one or more attributes. Returns the standard deviation of the measure grouped by the attribute(s).
group_stddev (revenue, customer region)
Takes a measure and one or more attributes. Returns the sum of the measure grouped by the attribute(s).
group_sum (revenue, customer region)
Takes a measure and one or more attributes. Returns the unique count of the measure grouped by the attribute(s).
group_unique_count (product , supplier)
Takes a measure and one or more attributes. Returns the variance of the measure grouped by the attribute(s).
group_variance (revenue, customer region)
Returns the maximum value of a column.
Returns the maximum value among columns that meet a criteria.
max_if( (revenue > 10) , customer region )
Returns the value of the measure from the row that has the 50th percentile value.
Returns the minimum value of a column.
Returns the minimum value among columns that meet a criteria.
min_if( (revenue < 10) , customer region )
Takes a measure, two integers to define the window to aggregate over, and one or more attributes. The window is (current - Num1…Current + Num2) with both end points being included in the window.
For example, “1,1” will have a window size of 3. To define a window that begins before Current, specify a negative number for Num2. Returns the average of the measure over the given window. The
attributes are the ordering columns used to compute the moving average.
moving_average (revenue, 2, 1, customer region)
Takes a measure, two integers to define the window to aggregate over, and one or more attributes. The window is (current - Num1…Current + Num2) with both end points being included in the window.
For example, “1,1” will have a window size of 3. To define a window that begins before Current, specify a negative number for Num2. Returns the maximum of the measure over the given window. The
attributes are the ordering columns used to compute the moving maximum.
moving_max (complaints, 1, 2, store name)
Takes a measure, two integers to define the window to aggregate over, and one or more attributes. The window is (current - Num1…Current + Num2) with both end points being included in the window.
For example, “1,1” will have a window size of 3. To define a window that begins before Current, specify a negative number for Num2. Returns the minimum of the measure over the given window. The
attributes are the ordering columns used to compute the moving minimum.
moving_min (defects, 3, 1, product)
Takes a measure, two integers to define the window to aggregate over, and one or more attributes. The window is (current - Num1…Current + Num2) with both end points being included in the window.
For example, “1,1” will have a window size of 3. To define a window that begins before Current, specify a negative number for Num2. Returns the sum of the measure over the given window. The
attributes are the ordering columns used to compute the moving sum.
moving_sum (revenue, 1, 1, order date)
Returns the value of the measure from the row that has a rank_percentile less than or equal to N. If there is no rank_percentile below N, the value of the measure of the first row above N will be
percentile (sales , 99 , 'asc' )
percentile (sales , 99, 'desc' )
Returns the rank for the current row. Identical values receive an identical rank. Takes an aggregate input for the first argument. The second argument specifies the order, 'asc' | 'desc'.
rank (sum (revenue) , 'asc' )
rank (sum (revenue) , '`desc' )
Returns the percentile rank for the current row. Identical values are assigned an identical percentile rank. Takes an aggregate input for the first argument. The second argument specifies the
order, 'asc' | 'desc'.
rank_percentile (sum (revenue) , 'asc' )
rank_percentile (sum (revenue) , 'desc' )
Returns the standard deviation of all values of a column.
Returns a standard deviation values filtered to meet a specific criteria.
stddev_if( (revenue > 10) , (revenue/10.0) )
Returns the sum of all the values of a column.
Returns sum values filtered by a specific criteria.
sum_if(region=’west’, revenue)
unique count
Returns the number of unique values of a column.
Returns the number of unique values of a column provided it meets the specified condition.
unique_count_if( (revenue > 10) , order date )
Returns the variance of all the values of a column.
Returns the variance of all the values of a column provided it meets a criteria..
variance_if( (revenue > 10) , (revenue/10.0) )
Conversion functions
Use these functions to convert data from one data type into another data type.
ThoughtSpot does not support date data type conversion.
Returns the input as a boolean data type (true` or false).
to_bool (0) = false
to_bool (married)
Accepts a date represented as an integer or text string, and a second string parameter that can include strptime date formatting elements.
Replaces all the valid strptime date formatting elements with their string counterparts and returns the result.
Does not accept epoch-formatted dates as input.
Does not accept datetime values. Only accepts month, day, and year.
to_date (date_sold, '%Y-%m-%d')
Returns the input as a double data type.
to_double ('3.14') = 3.14
to_double (revenue * .01)
Returns the input as an integer.
to_integer ('45') + 1 = 46
to_integer (price + tax - cost)
Returns the input as a text string. To convert a date data type to a string data type, specify the date format you want to use in the second argument (for example, to_string ( <date column> ,
"%Y-%m-%d" )). Use strftime for the date format.
to_string (45 + 1) = '46'
to_string (revenue - cost)
to_string (date,'%m/%d/%y')
Date functions
Returns the result of adding the specified number of days to the given date.
add_days (01/30/2015, 5) = 02/04/2015
add_days (invoiced, 30)
Returns the result of adding the specified number of minutes to the given date, datetime, or time.
add_minutes ( 01/30/2015 00:10:20 , 5 ) = 01/30/2015 00:15:20
add_minutes ( invoiced , 30 )
Returns the result of adding the specified number of months to the given date.
add_months ( 01/30/2015, 5 ) = 06/30/2015
add_months ( invoiced_date , 5 )
Returns the result of adding the specified number of seconds to the given date/ datetime/ time.
add_seconds ( 01/30/2015 00:00:00, 5 ) = 06/30/2015 00:00:05
add_seconds ( invoiced_date , 5 )
Returns the result of adding the specified number of weeks to the given date.
add_weeks ( 01/30/2015, 2 ) = 02/13/2015
add_weeks ( invoiced_date , 2 )
Returns the result of adding the specified number of years to the given date.
add_years ( 01/30/2015, 5 ) = 01/30/2020
add_years ( invoiced_date , 5 )
Returns the date portion of a date.
Returns the number (1-31) of the day of the month for a date.
day (01/15/2014) = 15
day (date ordered)
Returns the number of the day in a quarter for a date. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result. The default is
day_number_of_quarter (01/30/2015) = 30
day_number_of_quarter (01/30/2015, 'fiscal') = 91 // May 1 is start of fiscal year
Returns the number (1-7) of the day in a week for a date. Monday is 1, and Sunday is 7.
day_number_of_week(01/15/2014) = 3
day_number_of_week (shipped)
Returns the number (1-366) of the day in a year from a date. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result.
day_number_of_year (01/30/2015) = 30
day_number_of_year ( 01/30/2015, 'fiscal' ) = 275 // May 1 is start of fiscal year
day_number_of_year (invoiced)
Returns the day of the week for the given date.
day_of week (01/30/2015) = Friday
day_of_week (serviced)
Subtracts the second date from the first date and returns the result in number of days, rounded down if not exact.
diff_days (01/15/2014, 01/17/2014) = -2
diff_days (purchased, shipped)
Subtracts the second date from the first date and returns the result in number of seconds.
diff_time (01/30/2014, 01/31/2014) = -86,400
diff_time (clicked, submitted)
Returns the hour of the day for the given date.
Returns true if the date is a Saturday or a Sunday.
is_weekend (01/31/2015) = true
is_weekend (emailed)
Returns the month from the date.
month (01/15/2014) = January
month (date ordered)
Returns the number (1-12) of the month from a given date. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result.
month_number (09/20/2014) = 9
month_number ( 09/20/2014, 'fiscal' ) = 5 // May 1 is start of fiscal year
month_number (purchased)
Returns the month (1-3) number for the given date in a quarter. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result.
month_number_of_quarter (02/20/2018) = 2
In the following example, May 1st is the start of the fiscal year.
month_number_of_quarter (02/20/2018,'fiscal' ) = 1
Returns the current date and time in your locale’s standard date and time format. For example, if your locale is English (United States), it returns MM/dd/yyyy hh:mm:ss (04/27/2022 12:34:00).
Returns the number (1-4) of the quarter associated with the date. You can add an optional second parameter to specify 'fiscal' or 'calendar' dates.
quarter_number ( 04/14/2014) = 2
quarter_number ( 04/14/2014, 'fiscal' ) = 4 // May 1 is start of fiscal year
quarter_number ( shipped )
Returns MMM yyyy for the first day of the month. Your installation configuration can override this setting so that it returns a different format, such as MM/dd/yyyy. Speak with your ThoughtSpot
administrator for information on doing this.
start_of_month ( 01/31/2015 ) = Jan FY 2015
start_of_month (shipped)
Returns the date for the first day of the quarter for the date. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result.
start_of_quarter ( 04/30/2014) = Apr 2014
start_of_quarter ( 04/30/2014, 'fiscal') = Feb 2014 // May 1 is the start of the fiscal year
start_of_quarter (sold)
Returns the date for the first day of the week for the given date.
start_of_week ( 01/31/2020 ) = 01/27/2020
start_of_week (emailed)
Returns the date for the first day of the year for the date. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result. The default
is 'calendar'.
start_of_year (04/30/2014) returns Jan 2014
start_of_year (04/30/2014, 'fiscal') // May 1 is start of fiscal year
start_of_year (joined)
Returns the time portion of a date.
time (1/31/2002 10:32) = 10:32
time (call began)
Returns the current date in your locale’s standard date format. For example, if your locale is English (United States), it returns MM/dd/yyyy (04/27/2022).
Returns the week number for the date in a month.
week_number_of_month(03/23/2017) = 3
Returns the week number for the given date in a quarter. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result.
week_number_of_quarter (01/31/2020) = 5
week_number_of_quarter (05/31/2020, 'fiscal') = 5 // May 1 is start of fiscal year
Returns the week number for the date in a year. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result.
week_number_of_year (01/17/2014) = 3
week_number_of_year ( 01/17/2014, 'fiscal') = 38. // May 1 is start of fiscal year
Returns the year from a given date. You can add an optional second parameter to specify whether a 'fiscal' or 'calendar' year is used to calculate the result. The default is 'calendar'.
year (01/15/2014) = 2014
year (12/15/2013, 'fiscal' ) = 2014. // May 1 is start of fiscal year
year (date ordered)
Mixed functions
These functions can be used with text and numeric data types.
Returns true if the first value is equal to the second value.
2 = 2 = true
revenue = 1000000
Returns true if the first value is greater than the second value.
3 > 2 = true
revenue > 1000000
Returns true if the first value is greater than or equal to the second value.
3 >= 2 = true
revenue >= 1000000
Returns the larger of the values.
greatest (20, 10) = 20
greatest (q1 revenue, q2 revenue)
Returns the smaller of the values.
least (20, 10) = 10
least (q1 revenue, q2 revenue)
Returns true if the first value is less than the second value.
3 < 2 = false
revenue < 1000000
Returns true if the first value is less than or equal to the second value.
1 <= 2 = true
revenue <= 1000000
Returns true if the first value is not equal to the second value.
3 != 2 = true
revenue != 1000000
Number functions
* (multiply)
Returns the result of multiplying two numbers.
3 * 2 = 6
price * taxrate
+ (add)
Returns the result of adding two numbers.
1 + 2 = 3
price + shipping
− (subtract)
Returns the result of subtracting the second number from the first number.
/ (divide)
Returns the result of dividing the first number by the second number.
6 / 3 = 2
markup / retail price
Returns the first number raised to the power of the second number.
3 {caret} 2 = 9
width {caret} 2
Returns the absolute value of a number.
abs (-10) = 10
abs (profit)
Returns the inverse cosine, in degrees.
acos (0.5) = 60
acos (cos-satellite-angle)
Returns the inverse sine, in degrees.
asin (0.5) = 30
asin (sin-satellite-angle)
Returns the inverse tangent, in degrees.
atan (1) = 45
atan (tan-satellite-angle)
Returns the inverse tangent, in degrees.
atan2 (10, 10) = 45
atan2 (longitude, latitude)
Returns the cube root of a number.
cbrt (27) = 3
cbrt (volume)
Returns the rounded up integer value of a fraction.
ceil (5.9) = 6
ceil (growth rate)
Returns the cosine of an angle that is specified in degrees.
cos (63) = 0.45
cos (beam angle)
Returns the cube of a number, or the number to the 3rd power.
cube (3) = 27
cube (length)
Returns Euler’s number (~2.718) raised to a power specified by the number.
exp (2) = 7.38905609893
exp (growth)
Returns 2 raised to a power specified by the number.
exp2 (3) = 8
exp2 (growth)
Returns the rounded down integer value of a fraction.
floor (5.1) = 5
floor (growth rate)
Returns the natural logarithm of a number.
ln (7.38905609893) = 2
ln (distance)
Returns the base 10 logarithm of a number.
log10 (100) = 2
log10 (volume)
Returns the base 2 logarithm, or the binary logarithm, of a number.
log2 (32) = 5
log2 (volume)
Returns the remainder of a division of the first number by the second number.
mod (8, 3) = 2
mod ( revenue , quantity )
Returns the first number raised to the power of the second number.
pow (5, 2) = 25
pow (width, 2)
Returns a random number between 0 and 1.
random ( ) = .457718
random ( )
Returns the first number rounded to the second number (the default is 1).
round (35.65, 10) = 40
round (battingavg, 100)
round (48.67, .1) = 48.7
Returns the result of dividing the first number by the second.
If the second number is 0, returns 0 instead of NaN (not a number).
safe_divide (12, 0) = 0
safe_divide (total_cost, units)
Returns +1 if the number is greater than zero, -1 if less than zero, 0 if zero.
sign (-250) = -1
sign (growth rate)
Returns the sine of an angle that is specified in degrees.
sin (35) = 0.57
sin (beam angle)
Returns the distance, in km, between two points on Earth, as defined by their latitude and longitude.
The order of parameters is: lat1, long1, lat2, long2.
spherical_distance (
37.465191, -122.153617,
37.421962, -122.142174) = 4,961.96
spherical_distance (
start_latitude, start_longitude,
end_latitude, end_longitude)
Returns the square of a numeric value, or the number to the power of 2.
Returns the square root of a number, or the number to the power of 1/2.
Returns the tangent of an angle that is specified in degrees.
tan (35) = 0.7
tan (beam angle)
Returns true when both conditions are true, otherwise returns false.
(1 = 1) and (3 > 2) = true
lastname = 'smith' and state ='texas'
Not available for row-level security (RLS) formulas.
Conditional operator. Allows for multiple clauses.
if (cost > 500) then 'flag' else 'approve'
if ( item type in {'shirts', 'jackets', 'sweatshirts', 'sweaters'}) then 'tops' else if ( item type in {'shorts', 'pants'}) then 'bottoms' else 'all other apparel'
Returns the first value if it is not null, otherwise returns the second value.
Takes a column name and a list of values. It checks each column value against the list of values in the formula, and returns true if the column value matches one of the values in the formula.
state in { 'texas' , 'california' }
Returns true if the value is null.
Returns true if the condition is false, otherwise returns false.
not (3 > 2) = false
not (state = 'texas')
not in
Takes a column name and a list of values. It checks each column value against the list of values in the formula, and returns true if the column value does not match any of the values in the
state not in { 'texas' , 'california' }
Returns true when either condition is true, otherwise returns false.
(1 = 5) or (3 > 2) = true
state = 'california' or state ='oregon'
Text functions
Returns two or more values as a concatenated text string. Use single quotes around each literal string, not double quotes.
concat ( 'hay' , 'stack' ) = 'haystack'
concat (title, ' ', first_name , ' ', last_name)
Returns true if the first string contains the second string, otherwise returns false.
contains ('broomstick', 'room') = true
contains (product, 'trial version')
Accepts two text strings. Returns the edit distance (minimum number of operations required to transform one string into the other) as an integer. Works with strings under 1023 characters.
edit_distance ('attorney', 'atty') = 4
edit_distance (color, 'red')
Accepts two text strings and an integer to specify the upper limit cap for the edit distance (minimum number of operations required to transform one string into the other). If the edit distance
is less than or equal to the specified cap, returns the edit distance. If it is higher than the cap, returns the cap plus 1. Works with strings under 1023 characters.
edit_distance_with_cap ('pokemon go', 'minecraft pixelmon', 3) = 4
edit_distance_with_cap (event, 'burning man', 3)
Returns the portion of the given string of given length, beginning from the left-hand side of the string.
left ( 'persnickety' , 4 ) = 'pers'
left ( lastname , 5 )
Returns the portion of the given string of given length, beginning from the right-hand side of the string.
right ( 'persnickety' , 4 ) = 'kety'
right ( lastname , 5 )
Accepts a document text string and a search text string. Returns true if relevance score (0-100) of the search string with respect to the document is greater than or equal to 20. Relevance is
based on edit distance, number of words in the query, and length of words in the query which are present in the document.
similar_to ('hello world', 'hello swirl') = true
similar_to (current team, drafted by)
Accepts a document text string and a search text string. Returns the relevance score (0-100) of the search string with respect to the document. Relevance is based on edit distance, number of
words in the query, and length of words in the query which are present in the document. If the two strings are an exact match, returns 100.
similarity ('where is the burning man concert', 'burning man') = 46
similarity (tweet1, tweet2)
Accepts two text strings. Returns true if they sound similar when spoken, and false if they do not.
sounds_like ( 'read' , 'red' ) = true
sounds_like ( owner , promoter )
Accepts two text strings. Returns true if they are spelled similarly and false if they are not. Works with strings under 1023 characters.
spells_like ('thouhgtspot', 'thoughtspot') = true
spells_like (studio, distributor)
Returns the length of the text.
strlen ('smith') = 5
strlen (lastname)
Returns the numeric position of the first occurrence of the second string in the first string. If using an external cloud data warehouse connection, the position starts at 1, and 0 indicates not
found. If using ThoughtSpot’s internal in-memory database (Falcon), the position starts at 0, and -1 indicates not found.
In-memory database: strpos ('haystack_with_needles', 'needle') = 14
External database: strpos ('haystack_with_needles', 'needle') = 15
strpos (complaint, 'lawyer')
Returns the portion of the given string, beginning at the location specified (starting from 0), and of the given length.
substr ('persnickety', 3, 7) = snicket
substr (lastname, 0, 5)
These variables can be used in your expressions.
Returns a list of all the groups the current logged in user belongs to. For any row, if the expression evaluates to true for any of the groups, the user can see that row.
Returns the user with the matching name.
You cannot use these variables (ts_groups and ts_username) within an expression. For example, ts_groups = substr(rls_group_name, 0, 3) is valid, but substr(ts_groups,0,3) = rls_group_name is NOT
Connection passthrough functions
The following passthrough SQL functions are supported in connections for all cloud data warehouses:
Returns the boolean data type. The first argument takes the signature of the external function and runs it against the datasource. Subsequent arguments pass the values to the external function.
sql_bool_aggregate_op (
"booland_agg ({0})",
is_delivered )
Returns the boolean data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_bool_op (
"is_decimal ({0})",
itemCount )
sql_bool_op (
"boolor ({0}, {1})",
0 ) = True
Returns the date data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_date_aggregate_op (
"max ({0})" ,
orderdate )
Returns the date data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_date_op (
"previous_day ({0})",
ship_date )
Returns the timestamp data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
ql_date_time_aggregate_op (
"max ({0})",
delivery_time )
Returns the timestamp data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_date_time_op (
"timestamp_sub ({0}, {1})",
'INTERVAL 30 MINUTE')
Returns the double data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_double_aggregate_op (
"approx_percentile ({0}, {1})",
0.99 )
Returns the double data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_double_op (
"acosh ({0})",
quantity )
sql_double_op (
"radians ({0})",
180 ) = 3.141592654
Returns the int data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the external
sql_int_aggregate_op (
"approx_count_distinct ({0})",
sale_volume )
sql_int_aggregate_op (
"bitand_agg({0}) OVER ( [ partition by {1} ] )",
user_type )
Returns the int data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the external
sql_int_op (
"ceil ({0})",
itemCount )
sql_int_op (
"charindex ({0}, {1})",
"rty" ) = 4
Returns the string data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_string_aggregate_op (
"min ({0})",
username )
Returns the string data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_string_op (
"soundex ({0})",
"Marks" )
Returns the time data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_time_aggregate_op (
"max (time ({0}))",
delivery_time )
Returns the time data type. The first argument takes the signature of the external function to be executed against the datasource. Subsequent arguments take the values to be passed to the
external function.
sql_time_op (
"time_from_parts ({0}, {1}, {2})",
20 ) = 12:30:20 | {"url":"https://docs.thoughtspot.com/software/7.2/formula-reference","timestamp":"2024-11-03T10:45:55Z","content_type":"text/html","content_length":"226481","record_id":"<urn:uuid:26b912a2-6285-4e69-a5be-2e54d2ef69fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00404.warc.gz"} |
Reflection | What is called reflection? | Solved Examples on Reflection
We know, reflection is a known word and is related to the subject of physics and mathematics. When an object is place before the mirror, the image is formed behind the mirror. This is called
More detailed information about reflection:
When the object M is placed before the plane mirror, then
● Its image M’ if found behind the mirror at an equal distance from O.
● Mirror line is the perpendicular bisector of the line joining the object and its image.
● M’ is called the reflection (image) of M.
● ST is called the axis of reflection.
Solved examples on reflection:
1. Draw a line XZ and make point M outside the line. Now find M’ the image of M.
From M, draw a perpendicular to line XZ, intersecting at O.
Extend MO to OM’, such that MO = OM’
Therefore M’ is the image (reflection) of M and XZ acts as a mirror line.
2. Draw a line AB. Also, draw a line segment MN outside the line. Now find the image of MN.
From M and N, draw two perpendiculars to line AB intersecting at C and D such that CM = CM’ and DN = DN’.
Join M’ and N’.
Therefore, M’N’ is the reflection (image) of MN.
● Related Concepts
● Order of Rotational Symmetry
● Reflection of a Point in x-axis
● Reflection of a Point in y-axis
● Reflection of a point in origin
● Rotation
● 90 Degree Clockwise Rotation
● 90 Degree Anticlockwise Rotation
7th Grade Math Problems
8th Grade Math Practice
From Reflection to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question. | {"url":"https://www.math-only-math.com/reflection.html","timestamp":"2024-11-07T06:06:45Z","content_type":"text/html","content_length":"41575","record_id":"<urn:uuid:932d5569-db19-4dd7-a7f2-e10c50177fc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00269.warc.gz"} |
Frontiers | Probabilistic Spike Propagation for Efficient Hardware Implementation of Spiking Neural Networks
• ^1Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai, India
• ^2School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, United States
Spiking neural networks (SNNs) have gained considerable attention in recent years due to their ability to model temporal event streams, be trained using unsupervised learning rules, and be realized
on low-power event-driven hardware. Notwithstanding the intrinsic desirable attributes of SNNs, there is a need to further optimize their computational efficiency to enable their deployment in highly
resource-constrained systems. The complexity of evaluating an SNN is strongly correlated to the spiking activity in the network, and can be measured in terms of a fundamental unit of computation, viz
. spike propagation along a synapse from a single source neuron to a single target neuron. We propose probabilistic spike propagation, an approach to optimize rate-coded SNNs by interpreting synaptic
weights as probabilities, and utilizing these probabilities to regulate spike propagation. The approach results in 2.4–3.69× reduction in spikes propagated, leading to reduced time and energy
consumption. We propose Probabilistic Spiking Neural Network Application Processor (P-SNNAP), a specialized SNN accelerator with support for probabilistic spike propagation. Our evaluations across a
suite of benchmark SNNs demonstrate that probabilistic spike propagation results in 1.39–2× energy reduction with simultaneous speedups of 1.16–1.62× compared to the traditional model of SNN
1. Introduction
Spiking Neural Networks (SNNs), often referred to as the third generation of neural networks (Maass, 1997), are attracting a lot of attention due to several desirable characteristics, including their
ability to model temporal event streams, the possibility of training them using unsupervised, bio-inspired learning rules such as Spike Timing Dependent Plasticity (STDP) (Bi and Poo, 1998), and the
emergence of low-power SNN hardware platforms such as IBM TrueNorth (Akopyan et al., 2015) and Intel Loihi (Davies et al., 2018).
SNNs represent information as discrete spike events and follow an event-driven model of computation, where the work done (and hence, the time or energy consumed) is proportional to the number of
spike events. Further, they do not require multiplication to be performed when processing a spike, offering the prospect of reduced hardware complexity compared to conventional Artificial Neural
Networks (ANNs). Due to these differences, SNNs are not well-suited to commodity hardware platforms like graphics processing units (GPUs). Further, in contrast to hardware accelerators for ANNs,
which usually focus on exploiting regular data parallelism, hardware architectures for spiking networks (e.g., Furber et al., 2014; Neil and Liu, 2014; Akopyan et al., 2015; Roy et al., 2017) focus
more on features that enable efficient event-driven computation.
Despite being event driven, spiking networks still require a large number of memory accesses (Neil and Liu, 2014). When a neuron spikes, it is first necessary to identify its fanout neurons, i.e.,
the connectivity information needs to be fetched along with the weights of the corresponding synapses. Finally, the membrane potentials of the fanout neurons impacted by the spike are fetched and
updated. Recent data (Pedram et al., 2017) indicates that fetching data from memory is much more expensive than arithmetic computations. Consequently, developing techniques for reducing the number of
memory accesses in SNNs is critical for improving their energy-efficiency.
1.1. Probabilistic Spike Propagation
In this paper, we present a probabilistic method of spike propagation that can significantly reduce the number of memory accesses required for the evaluation of a rate-coded spiking neural network,
thus saving both run-time and energy. We realize the proposed probabilistic spike propagation mechanism through probabilistic synapses. Conventionally, the weight of a synapse connecting two neurons
in an SNN specifies the amount by which the membrane potential of the postsynaptic neuron is increased whenever the presynaptic neuron spikes. Alternatively, inspired by the ideas in Seung (2003) and
Kasabov (2010), we could view this weight as a measure of how likely it is that a spike will propagate across the synapse. A probabilistic synapse doesn't propagate all spikes generated by its
presynaptic neuron to the postsynaptic neuron. Instead, whenever a neuron spikes, only a subset of its outgoing synapses with weights above a certain randomly-chosen threshold propagate the spike. To
minimize the effect of this randomness on a network's accuracy while maximizing the time and energy savings, we develop techniques that generate the random thresholds and perform the synaptic updates
in an optimized manner. To summarize, the specific contributions of this work are as follows:
• We propose probabilistic spike propagation, an approach to reduce the cost of spike propagation in rate-coded SNNs. Probabilistic spike propagation reduces the number of memory accesses and
consequently reduces the time and energy consumed in evaluating a spiking network.
• We propose techniques that allow probabilistic spike propagation to be applied to existing SNNs and methods to optimize the tradeoff between energy and accuracy degradation.
• We evaluate the approach on a benchmark suite of six SNNs across five image classification datasets and characterize its performance. We also develop P-SNNAP, an SNN accelerator enhanced to support
probabilistic spike propagation, on which we evaluate the reductions in energy consumption and run-time.
The paper is organized as follows. First, we present a brief overview of SNN preliminaries in section 2, and motivate the need to optimize spike propagation. In section 3, we discuss the key concepts
of probabilistic spike propagation and in section 4, we present the P-SNNAP hardware architecture. We present the results of evaluating the proposed approach in section 6. In section 7, we present
related efforts and highlight the unique aspects of our work. Finally, section 8 concludes the paper.
2. SNN Preliminaries
The evaluation of a spiking neural network involves three phases, namely (a) spike injection, (b) spike generation, and (c) spike propagation, as illustrated in Figure 1. Although the illustration is
for the case of a simple fully connected network here, the algorithm remains unchanged for arbitrary connectivity patterns. As shown in the figure, the connection between different neurons is
referred to as a synapse and the neurons on either side of the connection are referred to as the presynaptic and postsynaptic neuron, respectively.
FIGURE 1
A more detailed description of SNN evaluation is presented in Algorithm 1. The first phase, spike injection, involves introducing input spikes that initiate activity in the network. These input
spikes can be directly obtained from event-based sensors, or can be generated from static inputs through conversion methods. There have been numerous efforts in developing neuromorphic or spiking
sensors (Vanarse et al., 2016) and spike based benchmark datasets (Orchard et al., 2015; Hu et al., 2016; Rueckauer and Delbruck, 2016). In many of these efforts, the inputs are presented as Poisson
spike trains (Diehl and Cook, 2015) or as analog stimuli directly applied to the membrane potentials of input layer neurons (Rueckauer et al., 2017).
The second phase, spike generation, is the process of evaluating each neuron and, based on a mathematical model of the neuron, determining whether it produces a spike. Neuron models with varying
levels of bio-fidelity have been proposed. In this work, we use the Integrate-and-Fire (IF) neuron model for illustration, but the approach is largely independent of the underlying neuron model. The
spike generation step, as described in Algorithm 1 (lines 8–15), typically involves fetching the state variables of the neuron model from memory and performing some arithmetic operations. In the case
of the IF model, the membrane potential v[m] is fetched and the bias is added to it (line 10). Next, it is checked for firing by comparing it with the threshold voltage v[th] (line 11). In case of
firing, the neuron ID is pushed into a spike queue (line 12), and the membrane potential is reset and written back to the memory (line 13). If every neuron in the network is evaluated at every
timestep, the above process will involve at least one memory access per neuron per timestep. Thus, the number of computations and memory accesses performed during spike generation are proportional to
the number of neurons.
The final phase, spike propagation, as described in Algorithm 1 (lines 16–23), is performed in the event of a neuron spiking. For every spike in the queue, the postsynaptic neurons connected to the
spiking neuron are identified (line 19) and all such neurons are updated with the respective synaptic weights (line 21). This process is referred to as a synaptic update. It involves fetching the
synaptic weight and the state of the postsynaptic neuron from memory, updating the state with the weight, and writing the neuron state back to memory. This amounts to at least two memory reads, one
arithmetic operation and one memory write per synapse per spike per timestep. The total number of computations and memory accesses for the propagation step is thus proportional to the amount of
spiking activity (number of spikes) and the number of synapses in the network. Overall, as the number of synapses in a network far outnumber neurons, the number of memory accesses associated with the
spike propagation step exceeds that of the other two phases and it accounts for the dominant share of memory accesses during SNN valuation.
In Figure 2, we show the fraction of energy consumed by memory and compute operations during SNN evaluation on three different hardware platforms, viz. IBM TrueNorth (Akopyan et al., 2015), SNNAP (
Sen et al., 2017), and PEASE (Roy et al., 2017). It is observed that memory accounts for the predominant portion of energy consumption in each of these hardware platforms. Thus, techniques for
improving the energy efficiency of SNNs should focus on reducing memory energy. Further, as discussed above, spike propagation requires more memory accesses than the other phases in SNN evaluation.
Hence, to improve energy efficiency of SNN implementations, it is imperative to develop better spike propagation techniques that reduce memory accesses.
FIGURE 2
3. Probabilistic Spike Propagation
We propose a probabilistic approach to spike propagation for reducing the number of memory accesses during SNN evaluation, and consequently the total energy consumed. It can be applied to existing
spiking networks while causing minimal-to-zero degradation in recognition accuracy. This section first outlines the key concepts involved and subsequently describes the proposed approach in detail.
3.1. Key Concepts
Consider two neurons (labeled i and j, respectively) in an SNN, that are connected by a synapse. Neuron i is called presynaptic and neuron j postsynaptic if the output of i is used to drive the
membrane potential of j. The magnitude of the weight associated with the synapse is w[ij], which we will assume (without loss of generality) to be a real-valued number ∈ [0, 1]. This is a safe
assumption since the effects of the weights are evaluated relative to a threshold value, and so it is possible to normalize all weights to this range of magnitudes. Note that negative weights would
correspond to inhibitory synapses, which are modeled in exactly the same way as excitatory synapses, and only the magnitude of the weight matters for the discussion that follows.
The weight associated with the synaptic connection as well as the spiking activity of neuron i dictate the amount of impact it has on neuron j. We quantify this impact as the total potentiation of a
postsynaptic neuron due to a presynaptic neuron. Due to the temporal nature of spiking neural networks, the total potentiation should be measured after incorporating the spikes from neuron i across
all timesteps. Thus, considering a spiking pattern of S[i] for neuron i and a synaptic weight of w[ij], the total potentiation, M[ij], of postsynaptic neuron j by neuron i at time t is
$Mijd(t)=Ci(t)×wij (1)$
where C[i](t) is the total number of times neuron i has spiked until time t. We term this process of spike propagation as deterministic, and denote it using the superscript d. It should be noted here
that ${M}_{ij}^{d}\left(t\right)$ is only the impact neuron i has on the membrane potential of neuron j, while the spiking behavior of neuron j itself depends on the neuron model and its potentiation
by other presynaptic neurons.
Instead of potentiating neuron j by w[ij] every time neuron i spikes, if we apply a weight of ŵ[ij] for a random subset of the spikes, the total potentiation becomes
$Mijp(t)=Ĉi(t)×ŵij (2)$
Ŝ[i](t) is the random subset of spikes from neuron i that were propagated to neuron j and Ĉ[i](t) is the number of spikes in Ŝ[i](t) till time t. In other words, we propagate a spike from neuron i to
neuron j with a probability of p[ij], where
$pij=limt→∞Ĉi(t)Ci(t) (3)$
We term this process of spike propagation as probabilistic, and denote it by the superscript p.
We can define the average potentiation of neuron j by neuron i as follows:
$Mij(t)¯=Mij(t)Ci(t) (4)$
It should be noted that the average potentiation is defined when there is at least one spike from neuron i. For the deterministic approach, the average potentiation is equal to the synaptic weight
$Mijd(t)¯=wij (5)$
On the other hand, for the probabilistic approach corresponding to Equation (3), the average potentiation is a limit.
$limt→∞Mijp(t)¯=pij×ŵij (6)$
We hypothesize that, it is sufficient that the average potentiation for probabilistic spike propagation tends toward the average potentiation for the deterministic case, for the network to achieve
similar levels of accuracy with both the probabilistic and deterministic approaches. This can be achieved by carefully choosing values for p[ij] and ŵ[ij] in the probabilistic approach. One
interesting choice is to set p[ij] = w[ij] and ŵ[ij] = 1, which is simply an alternative interpretation of each weight as the probability of spike propagation. We highlight the effects of such a
probabilistic approach through an example below.
Consider the connectivity pattern illustrated in Figure 3. Neurons 1 and 2 are spiking sources, which are connected to neuron 3 through synapses. The behavior of this simple network with the
deterministic and probabilistic spike propagation approaches is visualized in Figure 4.
FIGURE 3
FIGURE 4
Figure 4. Behavior of the network in Figure 3: (A) spike patterns of neurons 1 and 2; (B) membrane potential of neuron 3; (C) consequent spike pattern of neuron 3; (D,E) average potentiation of
neuron 3, by neurons 1 and 2, respectively, with deterministic and probabilistic (p[ij] = w[ij];ŵ[ij] = 1) spike propagation.
The spike patterns of neurons 1 and 2 are shown in Figure 4A. The effects of these spikes on the instantaneous membrane potential of neuron 3 for both the deterministic and probabilistic approaches
are shown in Figure 4B. The resulting output spike pattern S[3] for neuron 3 is shown in Figure 4C. As we can see, the probabilistic spike propagation causes the behavior of neuron 3 to slightly
differ from that in the deterministic case, but the average potentiations $\overline{{M}_{13}\left(t\right)}$ and $\overline{{M}_{23}\left(t\right)}$, which are in shown in Figures 4D,E for the two
synapses, respectively show an interesting convergence.
Overall, when the spikes are propagated in a probabilistic fashion, the instantaneous membrane potential of neuron 3 may differ from that under deterministic propagation, but as more and more spikes
are generated by the presynaptic neuron, the average potentiation for the probabilistic case converges to the deterministic one, which is essentially the synaptic weight. We introduced randomness
into the process of spike propagation in a network that is otherwise deterministic, and allowed the temporal nature of the network to average it out.
The crux of our hypothesis is that even though the introduced randomness alters the instantaneous state of the network, the variations will average out over time and result in a network-level
equivalence with the deterministic evaluation scheme. In the following subsections, we describe how to take advantage of this randomness to develop efficient implementations of SNNs.
3.2. Accelerating Convergence
From Figure 4, we can infer that, given enough spikes, the average probabilistic potentiation converges to the average deterministic potentiation, which is the synaptic weight.
$limCi→∞Mijp(t)¯→wij (7)$
Clearly, the number of spikes required for convergence is an important issue to address. For better convergence, we would need to process more spikes. The number of spikes is directly related to the
number of timesteps for which the network is evaluated. Alternatively, probabilistic spike propagation can be viewed as Monte Carlo sampling for approximating the value of w[ij]. The number of
Bernoulli trials required, which in this case is the number of spikes, for an approximation with low relative error is inversely proportional to the probability of success (Asmussen and Glynn, 2007).
As most weights in neural networks are observed to be small in value, the probabilities of propagation is going to be small for most synapses. And thus, the number of spikes required for convergence
of network behavior is going to be large, which directly means that the probabilistic approach will require that the networks be run for more timesteps. Thus, it is desirable to drive up the
probabilities of propagation, which will reduce the required number of spikes and consequently bring down the number of timesteps required to converge to the same levels of network performance.
One solution is to let ${p}_{ij}=\frac{{w}_{ij}}{{w}_{i}^{max}}$ and ${ŵ}_{ij}={w}_{i}^{max}$, where ${w}_{i}^{max}$ is the maximum weight of all the outgoing synapses of neuron i. For most of the
neurons, ${w}_{i}^{max}$ is lower than 1, which increases the probability of propagation.
However, the number of outgoing synapses per neuron in modern networks could be in the thousands and, ${w}_{i}^{max}$, in most cases, tends to be an outlier in the weight distribution. In Figure 5,
we see the ratio of ${w}_{i}^{max}$ to the median outgoing weight ${w}_{i}^{med}$ for all the neurons in a layer of a fully connected network trained on the MNIST classification task. We observe that
${w}_{i}^{max}$ is roughly around 5× ${w}_{i}^{med}$ for most neurons, which means half the outgoing synapses of these neurons in that layer will have spike propagation probabilities of <0.2, which
will require a higher number of timesteps for the convergence of the probabilistic approach.
FIGURE 5
In order to overcome this, we group outgoing synapses of a neuron into synaptic clusters. Synaptic clusters are simply spatial groupings of the outgoing synapses from a neuron. For each synapse, we
use the maximum weight of its corresponding cluster as the normalizer, as ${p}_{ij}=\frac{{w}_{ij}}{{w}_{{i}^{b}}^{max}}$ and ${ŵ}_{ij}={w}_{{i}^{b}}^{max}$, where b is the cluster to which the
synapse between neurons i and j belongs. This prevents the synaptic weights from getting dominated by outlier maximum weights, increasing their spike propagation probabilities and accelerating
Figure 6 presents histograms for spike propagation probabilities across synapses in the same layer as Figure 5. B denotes the number of synaptic clusters in Figure 6. When the outgoing synapses of
each neuron are grouped into eight clusters, we see that the histogram is skewed toward higher probabilities, unlike when there is no clustering (B = 1).
FIGURE 6
It is important to note that, for a given number of timesteps, the probability of spike propagation controls a trade-off relationship between cost and performance. The lower the probability, the
lower the number of synaptic updates and poorer network performance. The higher the probability, the higher the number of synaptic updates and better network performance. We explore this trade-off in
greater detail in section 6.
3.3. Realizing the Probabilistic Synapse
A probabilistic synapse can be realized by generating a uniformly distributed random number ${r}_{b}\in \left[0,{w}_{{i}^{b}}^{max}\right]$ and comparing with w[ij]. The probability of success of
this Bernoulli trial is
$rb∈[0,wibmax]→P(wij>rb)=wijwibmax (8)$
which is equal to the desired probability of propagation presented in the previous discussion. Hence, the spike can be propagated on every synapse that has a weight w[ij] greater than r[b].
While this implements the probabilistic synapse, it requires fetching of the weight for each synapse from memory prior to the decision of propagation. This can be cheaper than the deterministic
approach, as for the synapses that we don't propagate the spikes on, the post-synaptic neurons need not be updated. It should be noted that this random skipping of synapses can cause pipeline
inefficiencies in a hardware implementation. In the probabilistic spike propagation process described in Algorithm 2, we overcome this limitation and show how to reduce the memory accesses further.
It involves a preprocessing step of organizing synapses into multiple synaptic clusters and sorting the synapses in each cluster by their weights. Along with storing the weights of all the outgoing
synapses in sorted order, we also store their indices in rank order (line 5).
Consider that neuron i has spiked. Assume that the weights of the outgoing synapses of neuron i in synaptic cluster b are ranked from 1 to ${N}_{{i}^{b}}^{max}$ (line 2). As described in Equation
(8), for each synaptic cluster, we generate one random number r[b] (line 3). Since the weights are stored in sorted order, every synaptic update requires reading the index (line 5) and weight (line
7), but as soon as the comparison fails for one synapse, all the remaining synapses in the cluster can be skipped (line 10). The index j of the postsynaptic neuron can be determined with the indices
of the presynaptic neuron and the synapse (line 6), based on the underlying connectivity pattern.
3.3.1. Optimization: Determining the Termination Point
We define the termination point of the spike propagation from neuron i, ${t}_{p}\in \left[1,{N}_{{i}^{b}}^{max}\right]$, as the number of synapses with w[ij] > r[b]. It is the rank of the smallest
weight in the synaptic cluster that is greater than r[b]. In the straightforward method of determining t[p] described above, note that we need both the actual weight value and the index of the target
neuron, potentially requiring twice the number of memory accesses.
An alternate approach is to use a cumulative histogram of the outgoing weights of each neuron in each cluster, indicated by ${C}_{{i}^{b}}$. The cumulative histogram is a discrete function that gives
the number of values below the input i.e., ${C}_{{i}^{b}}\left({r}_{b}\right)$ gives the number of outgoing synapses of neuron i in synaptic cluster b with weights lesser than r[b]. Therefore, the
termination point t[p] is essentially ${N}_{{i}^{b}}^{max}-{C}_{{i}^{b}}\left({r}_{b}\right)$. Thus, by generating and storing a cumulative histogram of the form shown in Figure 7 in a preprocessing
step, as shown in Algorithm 3, we can determine t[p] through a single memory access (line 4). Consequently, we can perform synaptic updates without fetching the synaptic weights.
FIGURE 7
Another way to look at this is as a way of discretizing the space of random number generation for r[b]. For discrete values of r[b], we can store the termination point t[p] directly and sample from
these points. The number of discrete points correspond to the number of bins of the cumulative histogram and is a parameter of concern. It controls the trade-off between memory overhead and
performance. The higher the number of bins, the better the fidelity of the termination point. The lower the resolution, the lower the memory overhead. In this work, we have implemented this approach
in the hardware architecture and have studied its implications on accuracy and cost. The trade-off between accuracy and memory overhead has been studied in section 6.
In summary, the proposed probabilistic spike propagation approach reduces the number of synaptic updates, and consequently the number of memory accesses in SNNs by introducing randomness into the
process of spike propagation.
4. Hardware
To evaluate the system-level impact of probabilistic spike propagation, we develop P-SNNAP, an SNN accelerator based on SNNAP (Sen et al., 2017). The overall architecture is shown in Figure 8 and the
individual components are described in detail below.
FIGURE 8
4.1. Overview
The P-SNNAP architecture consists of three different modules—the Spike Neural Processing Element (SNPE), the Eval unit and the global controller. It also contains three types of on-chip memories—the
spike memory, the weight memory, and the state memory, for storing spikes, weights, and neuronal state variables, respectively.
In a deterministic SNN evaluation, as performed in SNNAP, the neurons in every layer are evaluated at each timestep before moving on to the next timestep. However, it involves loading the neuronal
state variables and weights for each layer into the on-chip memory repeatedly at every timestep. To avoid these repeated off-chip memory accesses and increase the reuse of loaded weight values, we
evaluate one layer for the total number of timesteps before moving on to the next layer. The spikes generated at each timestep during the evaluation of one layer are stored in the spike memory and
subsequently fetched during the evaluation of the next layer. Since a large number of modern deep networks are strictly feed-forward, this layer-wise evaluation scheme can be applied to reduce the
required buffering. Specifically, all networks evaluated as part of this work are feed-forward networks. We note that this optimization is orthogonal to our proposal and is applied to both
deterministic and probabilistic SNN evaluation.
4.1.1. Eval Unit
The Eval unit, similar to the Leak-and-Spike unit in SNNAP, is the module that performs neuron evaluation. It fetches the membrane potentials from the state memory, increments it by the bias value
and compares it with the threshold. If the membrane potential exceeds the threshold, a spike is generated and communicated to the controller. The updated membrane potentials are written back to the
state memory.
4.1.2. Controller
The Controller orchestrates the functioning of the accelerator. It has two phases of operation—the first phase controls the SNPEs and the second phase controls the Eval unit. For each timestep, the
controller goes through both phases. In the first phase, the controller fetches the spikes generated by the previous layer from the spike memory and sends them to the SNPEs. Once all the spikes are
sent and the SNPEs finish their operations, the controller moves on to the second phase, in which the controller receives spikes from the Eval unit as it evaluates all the neurons in the layer and
writes the spikes to the spike memory. Once all the neurons of the current layer are evaluated, the current timestep is completed and the controller moves on to the next timestep for the layer.
4.1.3. SNPEs
Spike propagation is realized by an array of Spike Neural Processing Elements (SNPEs). The propagation along different outgoing synapses of neurons are parallelized and balanced across the 16 lanes
of the SNPE array. Each lane has an SNPE coupled with two blocks of on-chip memory, one each for membrane potentials and weights. When a layer is evaluated, the controller fetches the spikes from the
previous layer and sends them to the SNPEs. On receiving a spike, an SNPE uses the index of the spiking neuron to iterate through its outgoing synpases. For each synapse, the SNPE calculates the
index of the post-synaptic neuron. This calculation depends on the connectivity pattern of the layer being evaluated. Next, for each post-synaptic neuron, its membrane potential and the weight of the
corresponding synapse are fetched. The membrane potential is updated and written back.
4.1.3.1. Mapping Synaptic Clusters to Lanes
Both the lanes of SNPEs in the architecture and the synaptic clusters in the probabilistic approach group outgoing synapses of neurons. Despite the similarity, the grouping is done with different
goals. While deciding the number of lanes, the primary concerns are inference speed and the required logic area and size of the on-chip memories, at the hardware level. On the other hand, while
deciding the number of synaptic clusters, the concerns are computational effort and accuracy.
When the number of synaptic clusters and lanes are chosen to be equal, a simple direct mapping is possible—the outer loop in Algorithm 3 is unrolled completely and each SNPE processes one cluster. It
is also possible for the number of clusters and lanes to be different. When the number of synaptic clusters is less than the number of lanes, multiple lanes operate on a single synaptic cluster. When
the number of synaptic clusters are more than the number of lanes, each lane will have to process more than one cluster, i.e., the outer loop in Algorithm 3 is unrolled partially and each SNPE will
process multiple clusters.
The weight memory in each SNPE lane stores all the information required to perform probabilistic spike propagation, including the discretized cumulative histograms for the corresponding mapped
synaptic clusters, the sorted synaptic indices and the maximum weight values.
4.1.3.2. Asynchronous Spike Processing
In the deterministic propagation of spikes, since the outgoing synapses of the spiked neuron are distributed equally among the lanes, each lane ends up performing an equal number of synaptic updates,
which means that all the SNPEs take an equal amount of time, as shown in Figure 9A. In contrast, in the probabilistic propagation of spikes, even though the lanes have been assigned an equal number
of synapses, the termination point t[p] that each lane comes up with is random and thus, they perform different number of synaptic updates and end up taking unequal amounts of time, as illustrated in
Figure 9B. Before the controller can serve the next spike, a number of SNPEs would have been idle. These bubbles in the compute pattern in turn leads to under-utilization of SNPEs and compute
FIGURE 9
Figure 9. Timing diagrams illustrating the need for asynchronous spike serving for the probabilistic spike propagation. (A) Synchronous SNPE-deterministic spike propagation, (B) Synchronous
SNPE-probabilistic spike propagation, (C) Asynchronous SNPE-probabilistic spike propagation.
To address the aforementioned challenge, P-SNNAP implements asynchronous spike processing. Each SNPE is equipped with a queue as shown in Figure 8. The controller fills up the queues with spikes. As
soon as an SNPE has finished propagating a spike, it can move on to the next spike from the queue, as shown in Figure 9C. This allows the probabilistic approach to be faster and have better compute
5. Experimental Methodology
In this section, we describe the experimental methodology and benchmarks used to evaluate the proposed concepts.
5.1. Benchmarks
The benefits of the proposed approach have been studied across a range of image classification networks trained on MNIST, SVHN, CIFAR10, CIFAR100, and ImageNet datasets, as listed in Table 1. The
networks were trained as conventional analog (non-spiking) deep networks using backpropagation and converted to spiking networks using the Keras-based ANN-to-SNN conversion and simulation framework
developed by Rueckauer et al. (2017).
TABLE 1
We refer to the deterministic evaluation of all synapses in a network as the baseline (BSL) approach in section 6. On the other hand, for the Probabilistic Spike Propagation (PSP) approach in section
6, we empirically choose between deterministic or probabilistic spike propagation at a layer-granularity for each network in the benchmark suite, with the goal of iso-timesteps operation. The
probabilistic approach is beneficial only for layers with large numbers of synaptic connections and high activity. For instance, the CIFAR10-AllConv network in our benchmark suite is the All-CNN-C
network from Springenberg et al. (2014), that was converted into a spiking network. In PSP, layers 2, 4, 5, and 7 of this network were evaluated with probabilistic spike propagation, while the
remaining layers were evaluated with deterministic spike propagation. The savings achieved by this configuration are reported in section 6.
5.2. P-SNNAP Details
The P-SNNAP engine was designed at the Register Transfer Level and synthesized to the Nangate 15 nm technology using Synopsys Design Compiler. CACTI (Thoziyoor et al., 2008) was used to model the
memory blocks. A simulator was implemented in the dynamic, high level language Julia (Bezanson et al., 2017) to simulate the proposed spike propagation methods on P-SNNAP. The hardware simulator
profiles the memory accesses and number of cycles and uses the values obtained from hardware synthesis and CACTI to estimate energy consumption. The compute logic in P-SNNAP occupies a total area of
0.1 mm^2. The compute power consumption is 28.6 mW. A version of SNNAP without support for probabilistic spike propagation was implemented to act as the baseline in our comparisons. We observe that
the probabilistic approach incurs a compute logic area overhead of 12% and compute logic power overhead of 23.5%. These hardware additions facilitate significant improvements in time and energy
consumed to evaluate SNNs, as discussed in the following section.
In our implementation, the on-chip memory in the accelerator was sufficient to hold the largest layer in the suite of benchmarks. The on-chip memory can be reduced if needed by employing the
layer-wise evaluation scheme at a finer granularity and dividing layers into multiple blocks of neurons and evaluating one block at a time. The memory overhead of the probabilistic approach is due to
the tables of cumulative histograms and these tables are sparsely accessed at the rate of 1 read per lane per spike. Hence, these cumulative histogram tables can reside in off-chip DRAM and fetched
on demand if on-chip memory is constrained.
6. Results
In this section, we present results of our experiments that evaluate the benefits of probabilistic spike propagation (PSP) in SNNs.
6.1. Accuracy vs. Synaptic Updates
In this subsection, we study the trade-off between classification accuracy of a network and the number of synaptic updates performed during its evaluation. Specifically, we record the classification
accuracy and number of synaptic updates (averaged across all test inputs) at each timestep for both the deterministic and probabilistic propagation schemes. The results for the CIFAR10
all-convolutional network are presented in Figure 10. We observe that, for both approaches, accuracy saturates with increasing timesteps, and hence with increasing synaptic updates. We also observe
that the proposed probabilistic approach requires significantly fewer synaptic updates than the baseline to achieve roughly iso-accuracy. In Figure 11, we visualize the accuracy degradation of PSP as
a function of synaptic updates (normalized to a fraction of the baseline) for the other networks in the benchmark suite. The accuracy degradation and synaptic update fraction were calculated with
respect to the respective final values of BSL. The final accuracy values of the BSL networks are noted in the legend. PSP causes very minimal accuracy degradations of <0.1% in the networks trained on
the MNIST, SVHN, CIFAR10, and CIFAR100 tasks. The ImageNet-VGG16 network was evaluated on subset of 1,000 images of the ImageNet validation set and an accuracy degradation of 0.6% was observed.
FIGURE 10
FIGURE 11
6.2. Reductions in Synaptic Updates, Energy, and Run-Time
The benefits of PSP in terms of the reduction in the number of synaptic updates, total energy, and execution time on the P-SNNAP architecture are presented in Figure 12. The BSL and PSP cases were
evaluated for iso-timesteps and the corresponding number of synaptic updates, energy and execution time were measured. We observe that PSP achieves 2.4–3.69× reduction in average number of synaptic
updates per inference across all benchmarks. It should be noted that the reduction in synaptic updates for a specific network depends on the distribution of weights, which is why there is some
variability across the benchmark suite. These benefits translate to 1.39–2× reduction in average total energy per inference. Clearly, the bulk of the energy benefits can be attributed to the
reduction in memory accesses. As a result of the asynchronous spike serving, the probabilistic spike propagation approach also achieves a 1.16–1.62× speedup on the P-SNNAP architecture.
FIGURE 12
6.3. Number of Synaptic Clusters vs. Accuracy
As discussed in section 3.2, increasing the number of synaptic clusters causes the number of synapses affected by outlier weights to go down, and their probabilities of propagation go up.
We observe that this improves the classification accuracy for iso-synaptic updates. In the extreme case, with 1 synapse per cluster, probabilistic propagation becomes identical to the deterministic
approach. This dictates that the trade-off relationship between number of synaptic updates and accuracy has a sweet spot on the possible number of synaptic clusters.
The all-convolutional CIFAR10 network has been studied to explore this relationship in more detail. The network is evaluated with different number of synaptic clusters and the accuracy and average
number of synaptic updates per inference image are measured. The contour plot in Figure 13 visualizes this surface. Each line in the contour represents the accuracy degradation for different number
of synaptic clusters at a particular level of computational effort, or, number of synaptic updates. We observe that across our benchmark suite, the most favorable trade-off is achieved when the
number of synaptic clusters is set to 8 or 16.
FIGURE 13
It should be noted that this sweet spot is dependent, at a high level, on the number of synapses per cluster, which is decided by the size of the network. Ideally, the number of synaptic clusters
could be determined at a per-neuron granularity. However, in this work, we have chosen it to be a network-level hyperparameter to reduce the overall search space.
6.4. Resolution of the Cumulative Histogram
The number of bins used in the cumulative histogram impacts the fidelity of the random number r[b], as it affects the value of the termination point t[p] determined from the cumulative histogram.
Therefore, it directly affects the degradation in classification accuracy. At the same time, reducing the number of bins reduces the memory footprint. It should be noted that, the number of accesses
to determine t[p] is only one per lane per spike, irrelevant of the number of bins used in the cumulative histogram.
We specifically study the effect of the number of bins on the classification accuracy of CIFAR10 all-convolutional network. Figure 14 plots the corresponding degradation in recognition accuracy as a
function of the number of bins. As expected, we observe that the accuracy degradation reduces as the number of bins is increased.
FIGURE 14
A cumulative histogram of 50 bins causes a memory overhead of 23.7% in the CIFAR10-AllConv network. While this can be considered to be significant, we note the following
• The memory overhead is much lower in larger models like CIFAR100-VGG16 (10.9%) and ImageNet-VGG16 (1.1%).
• Although the memory footprint is larger, the total number of memory accesses with PSP is substantially lower.
7. Related Works
The focus of this work is to improve the energy efficiency of spiking neural networks by utilizing a probabilistic approach to spike propagation for reducing the number of memory accesses. It can be
directly applied to pre-trained spiking networks, without any structural or behavioral modifications. We now relate this to previously proposed approaches for improving SNN implementations and
highlight the unique aspects of our approach.
7.1. Custom Hardware Architectures
There have been several custom hardware accelerators designed expressly to implement spiking networks (Neil and Liu, 2014; Akopyan et al., 2015; Cheung et al., 2016; Smaragdos et al., 2017; Davies et
al., 2018). They employ specialized compute and communication units to match the computational and communication pattern in SNNs. Our approach is complementary to such techniques, and can potentially
be realized on these hardware architectures with some memory overheads.
7.2. Stochastic Techniques
Stochastic computation techniques apply randomness to the process of computation itself (Shanbhag et al., 2010). Variants of this approach have been applied to spiking neural networks (Rosselló et
al., 2012; Ahmed et al., 2016; Smithson et al., 2016). These are mostly orthogonal to the ideas we discuss, since a different (stochastic) hardware architecture for elementary compute units can also
be incorporated into our approach which introduces randomness in the process of spike propagation.
7.3. Specialized Neuron Models and Encoding Schemes
Ahmed et al. (2016) considered a probabilistic model of the neuron itself, wherein the spike generation mechanism is stochastic in nature but spike propagation is deterministic. Bayesian spiking
neurons (Deneve, 2008; Paulin and Van Schaik, 2014) apply probabilistic techniques for the neuron models to perform Bayesian inference. The idea of interpreting synaptic weights as probabilities of
spike propagation has also been explored in previous efforts (Seung, 2003; Kasabov, 2010; Neftci et al., 2016). However, these works are primarily algorithmic efforts focused on developing new
functionality or new training schemes and don't leverage the randomness to improve energy efficiency. We, on the other hand, demonstrate how randomness can be introduced in the spike propagation of
existing spiking networks without changing their intrinsic spiking behavior, while exploiting their time averaging capabilities. We further develop techniques to leverage this randomness for
improving the energy efficiency of SNNs. Park et al. (2019) demonstrated neural information coding schemes that improve the energy efficiency of SNN evaluation. This is orthogonal to the direction
our work, which improves energy efficiency of existing rate coding networks.
7.4. Pruning and Approximate Computing
Pruning is a technique used to reduce memory footprint of neural networks. Rathi et al. (2018) propose a pruning algorithm that works in parallel with STDP SNN learning algorithm on shallow networks.
Kundu et al. (2021) propose a pruning algorithm that compresses an ANN during training, converts the network into an SNN, and then retrains the network using a surrogate-gradient based supervised
sparse learning. These works prune the networks statically and result in a sparse network model. While these sparse networks can be very lightweight, they lack memory regularity. Developing hardware
implementation for these sparse and irregular networks is a niche of its own. Probabilistic spike propagation can be viewed as a stochastic online pruning scheme. Without requiring any retraining, or
losing memory regularity, probabilistic spike propagation is able to leverage temporality of SNNs and dynamically reduce memory accesses.
Approximate computing is well-known in the area of signal processing and neural network hardware, but has seen limited application to spiking networks. One example is Sen et al. (2017), where neurons
are progressively trimmed from evaluation as time progresses. Another is Krithivasan et al. (2019), where spike propagations are reduced by dynamically bundling spike events across time. Our approach
is parallel to these, and could possibly be combined to further reduce computations.
7.5. Emerging Technologies
Finally, there are approaches that rely on the use of new and emerging technologies, such as spin-based computing (Sengupta et al., 2016; Zhang et al., 2016; Srinivasan et al., 2017; Chen et al.,
2018; Sahu et al., 2018), photonics (De Lima et al., 2017; Chakraborty et al., 2019; Xiang et al., 2019), and memristors (Afifi et al., 2009; Serrano-Gotarredona et al., 2013; Al-Shedivat et al.,
2015). These works develop hardware implementations leveraging intrinsic characteristics of these technologies to exhibit properties of spiking networks like leakage, stochasticity, or learning.
While this work is focused on the contemporary generation of CMOS computing, our approaches should be applicable to emerging computing technologies.
8. Conclusions
In this work, we introduce probabilistic spike propagation as a new approach for improving the energy efficiency of spiking neural networks. The proposed approach reduces the number of memory
accesses during the spike propagation phase in SNNs by casting spike propagation as a probabilistic process. We show that the temporal nature of SNNs allows the network to regain any accuracy loss
caused by this approach. We successfully apply the technique on pre-trained spiking networks without any network modifications or retraining and demonstrate significant reductions in the number of
synaptic updates performed during evaluation while maintaining near iso-accuracy performance levels. We further develop a new hardware architecture, P-SNNAP, to realize probabilistic spike
propagation in hardware and show that the proposed approach achieves considerable execution time and energy savings when compared to deterministic spike propagation.
Data Availability Statement
Publicly available datasets were analyzed in this study. This data can be found at: http://yann.lecun.com/exdb/mnist/; http://ufldl.stanford.edu/housenumbers/; https://www.cs.toronto.edu/~kriz/
cifar.html; http://www.image-net.org/.
Author Contributions
AN implemented the experimental framework. All authors contributed to the conception of the ideas, design of the experiments, analysis of the results, and development of the manuscript.
This work was partially supported by grants from Xilinx and Center for Computational Brain Research, IIT Madras.
Conflict of Interest
SS was employed at IBM Thomas J. Watson Research Center.
The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Afifi, A., Ayatollahi, A., and Raissi, F. (2009). “Implementation of biologically plausible spiking neural network models on the memristor crossbar-based CMOS/nano circuits,” in 2009 European
Conference on Circuit Theory and Design (Antalya: IEEE), 563–566. doi: 10.1109/ECCTD.2009.5275035
Ahmed, K., Shrestha, A., Qiu, Q., and Wu, Q. (2016). “Probabilistic inference using stochastic spiking neural networks on a neurosynaptic processor,” in IJCNN '16 (Vancouver, BC: IEEE), 4286–4293.
doi: 10.1109/IJCNN.2016.7727759
Akopyan, F., Sawada, J., Cassidy, A., Alvarez-Icaza, R., Arthur, J., Merolla, P., et al. (2015). TrueNorth: design and tool flow of a 65 mW 1 million neuron programmable neurosynaptic chip. IEEE
Trans. Comput. Aided Des. Integr. Circuits Syst. 34, 1537–1557. doi: 10.1109/TCAD.2015.2474396
Al-Shedivat, M., Naous, R., Cauwenberghs, G., and Salama, K. N. (2015). Memristors empower spiking neurons with stochasticity. IEEE J. Emerg. Select. Top. Circuits Syst. 5, 242–253. doi: 10.1109/
Asmussen, S., and Glynn, P. W. (2007). “Chapter 6,” in Stochastic Simulation: Algorithms and Analysis, Vol. 57 (New York, NY: Springer Science & Business Media), 158–205.
Bezanson, J., Edelman, A., Karpinski, S., and Shah, V. B. (2017). Julia: a fresh approach to numerical computing. SIAM Rev. 59, 65–98. doi: 10.1137/141000671
Bi, G.-Q. and Poo, M.-M. (1998). Synaptic modifications in cultured hippocampal neurons: dependence on spike timing, synaptic strength, and postsynaptic cell type. J. Neurosci. 18, 10464–10472. doi:
Chakraborty, I., Saha, G., and Roy, K. (2019). Photonic in-memory computing primitive for spiking neural networks using phase-change materials. Phys. Rev. Appl. 11:014063. doi: 10.1103/
Chen, M.-C., Sengupta, A., and Roy, K. (2018). Magnetic skyrmion as a spintronic deep learning spiking neuron processor. IEEE Trans. Magn. 54, 1–7. doi: 10.1109/TMAG.2018.2845890
Cheung, K., Schultz, S. R., and Luk, W. (2016). NeuroFlow: a general purpose spiking neural network simulation platform using customizable processors. Front. Neurosci. 9:516. doi: 10.3389/
Davies, M., Srinivasa, N., Lin, T. H., Chinya, G., Cao, Y., Choday, S. H., et al. (2018). Loihi: A neuromorphic manycore processor with on-chip learning. IEEE Micro 38, 82–99. doi: 10.1109/
De Lima, T. F., Shastri, B. J., Tait, A. N., Nahmias, M. A., and Prucnal, P. R. (2017). Progress in neuromorphic photonics. Nanophotonics 6, 577–599. doi: 10.1515/nanoph-2016-0139
Deneve, S. (2008). Bayesian spiking neurons I: inference. Neural Comput. 20, 91–117. doi: 10.1162/neco.2008.20.1.91
Diehl, P. U., and Cook, M. (2015). Unsupervised learning of digit recognition using spike-timing-dependent plasticity. Front. Comput. Neurosci. 9:99. doi: 10.3389/fncom.2015.00099
Furber, S. B., Galluppi, F., Temple, S., and Plana, L. A. (2014). The SpiNNaker project. Proc. IEEE 102, 652–665. doi: 10.1109/JPROC.2014.2304638
Hu, Y., Liu, H., Pfeiffer, M., and Delbruck, T. (2016). DVS benchmark datasets for object tracking, action recognition, and object recognition. Front. Neurosci. 10:405. doi: 10.3389/fnins.2016.00405
Kasabov, N. (2010). To spike or not to spike: a probabilistic spiking neuron model. Neural Netw. 23, 16–19. doi: 10.1016/j.neunet.2009.08.010
Krithivasan, S., Sen, S., Venkataramani, S., and Raghunathan, A. (2019). “Dynamic spike bundling for energy-efficient spiking neural networks,” in 2019 IEEE/ACM International Symposium on Low Power
Electronics and Design (ISLPED) (Lausanne: IEEE), 1–6. doi: 10.1109/ISLPED.2019.8824897
Kundu, S., Datta, G., Pedram, M., and Beerel, P. A. (2021). “Spike-thrift: towards energy-efficient deep spiking neural networks by limiting spiking activity via attention-guided compression,” in
Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (Waikoloa, HI), 3953–3962. doi: 10.1109/WACV48630.2021.00400
Maass, W. (1997). Networks of spiking neurons: the third generation of neural network models. Neural Netw. 10, 1659–1671. doi: 10.1016/S0893-6080(97)00011-7
Neftci, E. O., Pedroni, B. U., Joshi, S., Al-Shedivat, M., and Cauwenberghs, G. (2016). Stochastic synapses enable efficient brain-inspired learning machines. Front. Neurosci. 10:241. doi: 10.3389/
Neil, D., and Liu, S.-C. (2014). Minitaur, an event-driven FPGA-based spiking network accelerator. IEEE Trans. VLSI 22, 2621–2628. doi: 10.1109/TVLSI.2013.2294916
Orchard, G., Jayawant, A., Cohen, G. K., and Thakor, N. (2015). Converting static image datasets to spiking neuromorphic datasets using saccades. Front. Neurosci. 9:437. doi: 10.3389/fnins.2015.00437
Park, S., Kim, S., Choe, H., and Yoon, S. (2019). “Fast and efficient information transmission with burst spikes in deep spiking neural networks,” in 2019 56th ACM/IEEE Design Automation Conference
(DAC) (San Fransico, CA: IEEE), 1–6. doi: 10.1145/3316781.3317822
Paulin, M. G., and Van Schaik, A. (2014). Bayesian inference with spiking neurons. arXiv [Preprint] arXiv: 1406.5115.
Pedram, A., Richardson, S., Horowitz, M., Galal, S., and Kvatinsky, S. (2017). Dark memory and accelerator-rich system optimization in the dark silicon era. IEEE Des. Test 34, 39–50. doi: 10.1109/
Rathi, N., Panda, P., and Roy, K. (2018). Stdp-based pruning of connections and weight quantization in spiking neural networks for energy-efficient recognition. IEEE Trans. Comput. Aided Des. Integr.
Circuits Syst. 38, 668–677. doi: 10.1109/TCAD.2018.2819366
Rosselló, J. L., Canals, V., and Morro, A. (2012). “Probabilistic-based neural network implementation,” in The 2012 International Joint Conference on Neural Networks (IJCNN) (Brisbane, QLD), 1–7.
doi: 10.1109/IJCNN.2012.6252807
Roy, A., Venkataramani, S., Gala, N., Sen, S., Veezhinathan, K., and Raghunathan, A. (2017). “A programmable event-driven architecture for evaluating spiking neural networks,” in 2017 IEEE/ACM
International Symposium on Low Power Electronics and Design (ISLPED) (Taipei: IEEE), 1–6. doi: 10.1109/ISLPED.2017.8009176
Rueckauer, B., and Delbruck, T. (2016). Evaluation of event-based algorithms for optical flow with ground-truth from inertial measurement sensor. Front. Neurosci. 10:176. doi: 10.3389/
Rueckauer, B., Lungu, I.-A., Hu, Y., Pfeiffer, M., and Liu, S.-C. (2017). Conversion of continuous-valued deep networks to efficient event-driven networks for image classification. Front. Neurosci.
11:682. doi: 10.3389/fnins.2017.00682
Sahu, U., Goyal, K., Saxena, U., Chavan, T., Ganguly, U., and Bhowmik, D. (2018). “Skyrmionic implementation of spike time dependent plasticity (STDP) enabled spiking neural network (SNN) under
supervised learning scheme,” in 2018 4th IEEE International Conference on Emerging Electronics (ICEE) (Bengaluru: IEEE), 1–6. doi: 10.1109/ICEE44586.2018.8937850
Sen, S., Venkataramani, S., and Raghunathan, A. (2017). “Approximate computing for spiking neural networks,” in Design, Automation & Test in Europe Conference & Exhibition (DATE) (Lausanne: IEEE),
193–198. doi: 10.23919/DATE.2017.7926981
Sengupta, A., Banerjee, A., and Roy, K. (2016). Hybrid spintronic-CMOS spiking neural network with on-chip learning: devices, circuits, and systems. Phys. Rev. Appl. 6:064003. doi: 10.1103/
Serrano-Gotarredona, T., Masquelier, T., Prodromakis, T., Indiveri, G., and Linares-Barranco, B. (2013). STDP and STDP variations with memristors for spiking neuromorphic learning systems. Front.
Neurosci. 7:2. doi: 10.3389/fnins.2013.00002
Seung, H. (2003). Learning in spiking neural networks by reinforcement of stochastic synaptic transmission. Neuron 40, 1063–1073. doi: 10.1016/S0896-6273(03)00761-X
Shanbhag, N. R., Abdallah, R. A., Kumar, R., and Jones, D. L. (2010). “Stochastic computation,” in Proceedings of DAC '10 (New York, NY: ACM Press), 859. doi: 10.1145/1837274.1837491
Smaragdos, G., Chatzikonstantis, G., Kukreja, R., Sidiropoulos, H., Rodopoulos, D., Sourdis, I., et al. (2017). BrainFrame: a node-level heterogeneous accelerator platform for neuron simulations. J.
Neural Eng. 14:066008. doi: 10.1088/1741-2552/aa7fc5
Smithson, S. C., Boga, K., Ardakani, A., Meyer, B. H., and Gross, W. J. (2016). “Stochastic computing can improve upon digital spiking neural networks,” in 2016 IEEE International Workshop on Signal
Processing Systems (SiPS) (Dallas, TX: IEEE), 309–314. doi: 10.1109/SiPS.2016.61
Springenberg, J. T., Dosovitskiy, A., Brox, T., and Riedmiller, M. (2014). Striving for simplicity: the all convolutional net. arXiv [Preprint] arXiv:1412.6806.
Srinivasan, G., Sengupta, A., and Roy, K. (2017). “Magnetic tunnel junction enabled all-spin stochastic spiking neural network,” in Proceedings of DATE (Lausanne: IEEE), 530–535. doi: 10.23919/
Vanarse, A., Osseiran, A., and Rassau, A. (2016). A review of current neuromorphic approaches for vision, auditory, and olfactory sensors. Front. Neurosci. 10:115. doi: 10.3389/fnins.2016.00115
Xiang, S., Zhang, Y., Gong, J., Guo, X., Lin, L., and Hao, Y. (2019). STDP-based unsupervised spike pattern learning in a photonic spiking neural network with VCSELs and VCSOAs. IEEE J. Select. Top.
Quant. Electron. 25, 1–9. doi: 10.1109/JSTQE.2019.2911565
Zhang, D., Zeng, L., Zhang, Y., Zhao, W., and Klein, J. O. (2016). “Stochastic spintronic device based synapses and spiking neurons for neuromorphic computation,” in 2016 IEEE/ACM International
Symposium on Nanoscale Architectures (NANOARCH) (Beijing: IEEE), 173–178.
Keywords: spiking neural networks, hardware acceleration, energy efficiency, memory, probabilistic spike propagation
Citation: Nallathambi A, Sen S, Raghunathan A and Chandrachoodan N (2021) Probabilistic Spike Propagation for Efficient Hardware Implementation of Spiking Neural Networks. Front. Neurosci. 15:694402.
doi: 10.3389/fnins.2021.694402
Received: 13 April 2021; Accepted: 07 June 2021;
Published: 15 July 2021.
Copyright © 2021 Nallathambi, Sen, Raghunathan and Chandrachoodan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use,
distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in
accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Abinand Nallathambi, ee16s032@ee.iitm.ac.in | {"url":"https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2021.694402/full","timestamp":"2024-11-06T11:38:34Z","content_type":"text/html","content_length":"586979","record_id":"<urn:uuid:7627e063-9dff-4a1a-b7ae-4872a3e64e8d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00857.warc.gz"} |
Epiperimetric Inequality and the Regularity of Free-Boundaries
1. Epiperimetric Inequality and the Regularity of Free-Boundaries
Epiperimetric Inequality and the Regularity of Free-Boundaries
Epiperimetric Inequality and the Regularity of Free-Boundaries
Luca Spolaor, MIT
In this talk I will present a new method for studying the regularity of minimizers of some variational problems, including in particular some classical free-boundary problems. Using as a model case
the so-called Obstacle problem, I will explain what regularity of the free-boundary means and how we obtain it by using a new tool, called (Log) -epiperimetric inequality. This technique is very
general, and much like Caffarelli's 'improvement of flatness' for regular points, it allows for a uniform treatment of singularities in many different free-boundary problems. Moreover it is able to
deal with logarithmic regularity, which in the case of the Obstacle problem is optimal due to an example of Figalli-Serra. If time permits I will explain how such an inequality is linked to the
behavior of a gradient flow at infinity. This is joint work with M. Colombo and B. Velichkov | {"url":"https://www.math.princeton.edu/events/epiperimetric-inequality-and-regularity-free-boundaries-2018-12-12t200000","timestamp":"2024-11-08T15:20:12Z","content_type":"text/html","content_length":"31884","record_id":"<urn:uuid:94b402b1-1d6b-40b1-a6f5-4240a2f66e08>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00649.warc.gz"} |
Klaus Tschira
Mathematics of Planet Earth, UNESCO, the International Mathematical Union (IMU), the International Commission on Mathematical Instruction (ICMI), and IMAGINARY are announcing the winners of the
second international competition for exhibition modules for the Open Source Exhibition Mathematics of Planet Earth (MPE). This project aims to showcase ways in which the mathematical sciences are
useful for understanding our planet and addressing the challenges of sustainable development and global changes.Devamını oku | {"url":"https://www.imaginary.org/tr/node/894?page=4","timestamp":"2024-11-05T21:44:17Z","content_type":"text/html","content_length":"33144","record_id":"<urn:uuid:2d379c76-ea90-472d-9490-02075ebbded7>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00513.warc.gz"} |
What is 63 Fahrenheit to Celsius? - ConvertTemperatureintoCelsius.info
63 Fahrenheit to Celsius
63 degrees Fahrenheit is a temperature commonly used in the United States, but for many people around the world who are more familiar with the Celsius scale, it can be difficult to understand what
this means in terms of temperature. Fortunately, converting Fahrenheit to Celsius is a simple process that can be done with some basic math.
To convert 63 degrees Fahrenheit to Celsius, you can use the following formula:
C = (F – 32) * 5/9
Where C is the temperature in Celsius and F is the temperature in Fahrenheit.
Plugging in the value of 63 for F, the formula looks like this:
C = (63 – 32) * 5/9
C = (31) * 5/9
C = 155/9
C ≈ 17.22
So, 63 degrees Fahrenheit is approximately equal to 17.22 degrees Celsius.
Understanding the conversion between Fahrenheit and Celsius is important, as it allows for easy comparison between the two temperature scales. While the Fahrenheit scale is commonly used in the
United States, the Celsius scale is used in most other parts of the world. Knowing how to convert between the two scales can be useful for travel, cooking, and understanding weather forecasts.
It’s also important to note that the Celsius scale is based on the freezing and boiling points of water, with 0 degrees Celsius being the freezing point and 100 degrees Celsius being the boiling
point, at standard atmospheric pressure. In contrast, the Fahrenheit scale is based on a scale devised by Daniel Gabriel Fahrenheit in the early 18th century, with 32 degrees Fahrenheit being the
freezing point of water and 212 degrees Fahrenheit being the boiling point, at standard atmospheric pressure.
Understanding these differences can provide context for the temperature scales and help in interpreting temperature readings in different parts of the world. It’s also worth noting that the Celsius
scale is often considered more intuitive and easier to use for scientific and everyday purposes due to its alignment with the properties of water.
In conclusion, 63 degrees Fahrenheit is approximately equal to 17.22 degrees Celsius. Knowing how to convert between Fahrenheit and Celsius can be useful for a variety of practical purposes, and
understanding the differences between the two temperature scales can provide valuable context for interpreting temperature readings. Whether for travel, cooking, or simply understanding the weather,
having a grasp of Fahrenheit to Celsius conversion is a valuable skill to have. | {"url":"https://converttemperatureintocelsius.info/what-is-63-fahrenheit-in-celsius/","timestamp":"2024-11-05T23:18:00Z","content_type":"text/html","content_length":"72572","record_id":"<urn:uuid:652677cb-ef98-4645-a037-c17a0f44717b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00160.warc.gz"} |
How Big of a Sample Size do you need for Factor Analysis
Determining an appropriate sample size for Factor Analysis is tricky. You can’t easily use a power calculation because Factor Analysis ordinarily does not have a traditional research hypothesis. The
page cites various recommendations for sample size (along with references!).
Karen Grace Martin. How Big of a Sample Size do you need for Factor Analysis? The Analysis Factor. Available in html format | {"url":"http://new.pmean.com/sample-size-factor-analysis/","timestamp":"2024-11-06T15:13:31Z","content_type":"text/html","content_length":"2180","record_id":"<urn:uuid:83f2abff-b059-476d-b8d1-52b232cbcfd7>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00772.warc.gz"} |
4.1 Which inequality measure should be used? | Poverty and Inequality with Complex Survey Data
Which inequality measure should be used?
The variety of inequality measures begs a question: which inequality measure should be used? In fact, this is a very important question. However, the nature of it is not statistical or mathematical,
but ethical. This section aims to clarify and, while not proposing a “perfect measure”, to provide the reader with some initial guidance about which measure to use.
The most general way to analyze if one distribution is more equally distributed than another is by the Lorenz curve. When \(L_A(p) \geqslant L_B(p), \forall p \in [0,1]\), it is said that \(A\) is
more equally distributed than \(B\). Technically, we say that \(A\) (Lorenz) dominates \(B\).7 Krämer (1998Krämer, Walter. 1998. “Measurement of Inequality.” In Handbook of Applied Economic
Statistics, edited by Amman Ullah and David E. A. Giles, 1st ed., 39–62. Statistics: A Series of Textbooks and Monographs 155. New York: Marcel Dekker.) and Mosler (1994Mosler, Karl. 1994.
“Majorization in Economic Disparity Measures.” Linear Algebra and Its Applications 199: 91–114. https://doi.org/https://doi.org/10.1016/0024-3795(94)90343-3.) provide helpful insights to how
majorization, Lorenz dominance, and inequality measurement are connected. On the topic of majorization, Hardy, Littlewood, and Pólya (1934Hardy, G. H., J. E. Littlewood, and G. Pólya. 1934.
Inequalities. 2nd ed. Cambridge University Press.) is still the main reference, while Marshall, Olkin, and Arnold (2011Marshall, Albert W., Ingram Olkin, and Barry C. Arnold. 2011. Inequalities:
Theory of Majorization and Its Applications. 2nd ed. Springer Series in Statistics. Springer.) provide a more modern approach. In this case, all inequality measures that satisfy basic properties8
Namely, Schur-convexity, population invariance, and scale invariance. will agree that \(A\) is more equally distributed than \(B\).
When this dominance fails — i.e., when Lorenz curves do cross — Lorenz ordering is impossible. Then, under such circumstances, the choice of which inequality measure to use becomes relevant.
Each inequality measure is a result of a subjective understanding of how to rank the fairness of a distribution. As Dalton (1920, 348Dalton, Hugh. 1920. “The Measurement of the Inequality of
Incomes.” The Economic Journal 30 (September). https://doi.org/10.2307/2223525.) puts it, “the economist is primarily interested, not in the distribution of income as such, but in the effects of the
distribution of income upon the distribution and total amount of economic welfare, which may be derived from income.” The importance of how economic welfare is defined is once again expressed by
Atkinson (1970Atkinson, Anthony B. 1970. “On the Measurement of Inequality.” Journal of Economic Theory 2 (3): 244–63. https://ideas.repec.org/a/eee/jetheo/v2y1970i3p244-263.html.), where an
inequality measure is directly derived from a class of welfare functions. Even when a welfare function is not explicit, such as in the Gini index, there is an implicit, subjective judgement of the
impact of inequality on social welfare.
The idea of what is a fair distribution is a matter of Ethics, a discipline within the realm of Philosophy. Yet, as Fleurbaey (1996, Ch.1Fleurbaey, Marc. 1996. Théories Économiques de La Justice.
Économie Et Statistiques Avancées. Paris: Economica.) proposes, the analyst should match socially supported moral values and theories of justice to the set of technical tools for policy evaluation.
Although this can be a useful principle, a more objective answer is needed. By knowing the nature and properties of inequality measures, the analyst can further reduce the set of applicable
inequality measures. For instance, choosing from the properties listed in Frank Alan Cowell (2011, 74Cowell, Frank Alan. 2011. Measuring Inequality. 3rd ed. London School of Economics Perspectives in
Economic Analysis. New York: Oxford University Press.), if we require group-decomposability, scale invariance, population invariance, and that the estimate falls within \([0,1]\), we might resort to
the Atkinson index.
Even though the discussion can go deep in technical and philosophical aspects, this choice also depends on the public. For example, it would not be surprising if a public official has not encountered
the Atkinson index; however, they might be familiar with the Gini index. The same goes for publications: journalists have been introduced to the Gini index and can find it easier to compare and,
therefore, write about. Also, we also admit that the Gini index is relatively straightforward, if compared to many other measures.
In the end, the choice is mostly subjective and there is no consensus of which method offers the “ideal” inequality measure.9 In fact, there is much discussion about the properties of an ideal
inequality measure. We must remember that this choice is only problematic for relative inequality measures if Lorenz curves cross; otherwise, the choice among relative inequality measures is a less
substantial issue. | {"url":"https://www.convey-r.org/4.1-which-inequality-measure-should-be-used.html","timestamp":"2024-11-12T20:16:50Z","content_type":"text/html","content_length":"33746","record_id":"<urn:uuid:453cca14-5c87-454a-a8b3-6112292bb1db>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00550.warc.gz"} |
The Finite Difference Method of Discretization in CFD
Key Takeaways
• The numerical method of solving differential equations by approximating them with difference equations is called the finite difference method.
• The finite difference method can easily obtain high-order approximations.
• The finite difference method requires a structured grid.
Different discretization methods such as the finite element method, finite volume method, and finite difference method are used in CFD modeling
In engineering problems, we often come across linear and non-linear differential equations. These governing equations must be solved, and the analytical method of solving them can not be used due to
their complexity. However, it is possible to use computational fluid dynamic techniques to obtain computer-based solutions for these complex equations. Different discretization methods, such as the
finite element method, finite volume method, and finite difference method, can be used in the CFD modeling of these engineering problems. In this article, we will explore the finite difference method
of discretization in detail.
Solving Complex Equations With CFD
Laws of physics are associated with various physical phenomena like heat transfer and fluid flow. They need to be represented as mathematical equations to simulate these phenomena over various length
scales. Take fluid dynamics, for example: governing equations are obtained by applying the fundamental laws of mechanics to the fluid. The conservation of mass equation, the conservation of momentum
equation, and the conservation of energy equation form a set of coupled, non-linear partial differential equations–collectively called the Navier-Stokes equations, which describe fluid flow behavior.
The complicated or coupled nature of partial differential equations makes solving them analytically difficult. Numerical techniques have been developed to solve such complex differential equations.
In physical problems related to fluid flow, heat transfer, and aerodynamics, CFD is prominent in finding accurate solutions. In CFD problem-solving, a physical phenomenon is mathematically modeled
and solved using numerical techniques.
Numerical techniques are incorporated in CFD exclusively for solving differential equations. The fundamental principle of including numerical methods for solving partial differential equations is to
bring the idea of discretization, one of the key aspects of a numerical solution strategy. Discretization converts the governing equations into a set of simple equations.
There are different methods to discretize the complex governing equations, called discretization methods. A few popular discretization methods utilized in CFD tools are the finite element method,
finite volume method, and finite difference method.
We will take a deep dive into the finite difference method in the upcoming section.
The Finite Difference Method
The numerical method of solving differential equations by approximating them with difference equations is called the finite difference method. For the purpose of discretization, the derivatives in
the governing equations use the truncated Taylor series expansion in the finite difference method. The implementation of this method is easy and is one of the widely-used approaches to solving
partial differential equations numerically.
The finite difference method is based on the secant line approximation of the derivatives. In the place of actual derivatives, the finite difference method uses the difference approximation. Use of
the finite difference numerical method results in the generation of a set of algebraic equations that can be solved for the dependent variables. The set of algebraic equations is solved at the
discrete grid points in the physical domain under consideration.
To formulate the finite difference scheme, the first step is to discretize the domain into grid points. At each of these grid points, the derivatives are written in the form of differences. The
differences can be the central difference, backward difference, or forward difference. The differences relate the values of the variable at each grid point to its neighboring points. After completing
this process on all the grid points in the physical domain under consideration, a set of equations are obtained. By numerically solving this set of algebraic equations, the solution for the partial
differential equation is determined.
Here is a table that clearly lays out the advantages and disadvantages of the finite difference method.
Classification of Difference Formulae
The difference formulae can be classified into two based on:
1. The geometrical relationship of the neighboring grid points. They are central, forward, and backward differences.
2. The accuracy of the expression. The forward and backward differences are first-order accurate, whereas the central difference is second-order accurate.
The fundamental philosophy of the finite difference method is to replace the derivates in the governing equations with algebraic differences. The complete set of CFD simulation software from Cadence
can support you when solving physical phenomena such as heat transfer, fluid flow, and aerodynamics.
Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you. | {"url":"https://resources.system-analysis.cadence.com/blog/msa2022-the-finite-difference-method-of-discretization-in-cfd","timestamp":"2024-11-07T09:27:08Z","content_type":"text/html","content_length":"210697","record_id":"<urn:uuid:9afca9f7-4544-43ac-992f-61e369f0be16>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00857.warc.gz"} |
How do you find areas bounded by polar curves using calculus? | Socratic
How do you find areas bounded by polar curves using calculus?
1 Answer
If the region is bounded by a polar curve $r = r \left(\theta\right)$ from $\theta = {\theta}_{1}$ to ${\theta}_{2}$, then its area $A$ can be found by the double-integral
$A = {\int}_{{\theta}_{1}}^{{\theta}_{2}} {\int}_{0}^{r \left(\theta\right)} r \mathrm{dr} d \theta$.
I hope that this was helpful.
Impact of this question
1780 views around the world | {"url":"https://socratic.org/questions/how-do-you-find-areas-bounded-by-polar-curves-using-calculus","timestamp":"2024-11-05T13:06:15Z","content_type":"text/html","content_length":"31298","record_id":"<urn:uuid:e8da36e0-a272-46b8-86cc-61e764793f0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00146.warc.gz"} |
Hello all!
Today I’ll be doing another Project Euler problem.
Problem 2:
Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.
There are many ways to do this problem, but I’ve chosen to create a series of functions that I can run to get Fibonacci numbers until some upper bound, and then the sum of even numbers in a list. I
find that abstracting calculations like these into functions is a very easy way to make programs feel and look more organized as well as increasing reusability and convenience. A good example would
be that I may need a Fibonacci sequence in a future problem, and using a function like this allows me to just paste this code into a future program and I can recall on it using fib(max) when needed.
I’m going to begin by creating a function to find the Fibonacci numbers up to some upper bound. I’ll create the function fib(max) where max is the upper bound of the number we’ll be getting from the
Fibonacci sequence. The best way I can come up with to create a Fibonacci sequence is to create a list and use fib[-1] and fib[-2] to grab the latest two numbers in the sequence. Next, I’ll add in a
while loop that iterates through the Fibonacci sequence and places the numbers into the list fib[ ], to create the next Fibonacci number. This loop then breaks when we hit the upper bound we placed
earlier. After this the function will return the newly filled list fib[ ].
def fib(max):
fib = [1, 2]
while(int(fib[-1] + fib[-2]) <= max):
return fib
Next, I need to create a function that will add the even numbers of a list together. I’ll call this function even_sum(list) and to create it I’m going to use something new I learned called List
Comprehension. You can learn more about List Comprehension here but I’ll provide a quick overview. List Comprehension allows a smaller and more efficient way to create a list by placing a for loop
inside of square brackets. We can also place operators such as if statements inside the brackets. Using this new tool my next function will look like this:
def even_sum(list):
return sum([i for i in list if i%2==0])
Now that we have both of our functions we can just combine the two and run them through each other like so:
This now returns us with the sum of all even Fibonacci numbers up until 4 million!
Thanks for reading and have a wonderful day!
~ Corbin
I would like to give special thanks to Christian Ferko for teaching me about List Comprehension. | {"url":"https://maker.godshell.com/archives/tag/functions","timestamp":"2024-11-12T02:14:23Z","content_type":"text/html","content_length":"47172","record_id":"<urn:uuid:b7bba82c-57ea-4d78-9436-53710c1331a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00110.warc.gz"} |
Search for:
This work presents six research-based elements that align with building algebraic fluency from conceptual understandings in the teaching and learning of algebra. The six elements are: symbol sense,
processes/relationships of algebra, process as an object, anticipating solution strategies, anticipating solution formats, and relationships among representations.
more » « less
Free, publicly-accessible full text available November 9, 2025 | {"url":"https://par.nsf.gov/search/author:%22McGathey,%20N.%22","timestamp":"2024-11-09T15:36:16Z","content_type":"text/html","content_length":"287896","record_id":"<urn:uuid:0f2112be-6fab-478c-8f42-efb120e92250>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00736.warc.gz"} |