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Accounting method

In the field of analysis of algorithms in computer science, the accounting method is a method of amortized analysis based on accounting. The accounting method often gives a more intuitive account of the amortized cost of an operation than either aggregate analysis or the potential method. Note, however, that this does not guarantee such analysis will be immediately obvious; often, choosing the correct parameters for the accounting method requires as much knowledge of the problem and the complexity bounds one is attempting to prove as the other two methods.

The accounting method is most naturally suited for proving a O(1) bound on time. The method as explained here is for proving such a bound.

The method

Preliminarily, we choose a set of elementary operations which will be used in the algorithm, and arbitrarily set their cost to 1. The fact that the costs of these operations may in reality differ presents no difficulty in principle. What is important, is that each elementary operation has a constant cost.

Each aggregate operation is assigned a "payment". The payment is intended to cover the cost of elementary operations needed to complete this particular operation, with some of the payment left over, placed in a pool to be used later.

The difficulty with problems that require amortized analysis is that, in general, some of the operations will require greater than constant cost. This means that no constant payment will be enough to cover the worst case cost of an operation, in and of itself. With proper selection of payment, however, this is no longer a difficulty; the expensive operations will only occur when there is sufficient payment in the pool to cover their costs.


A few examples will help to illustrate the use of the accounting method.

Table expansion

It is often necessary to create a table before it is known how much space is needed. One possible strategy is to double the size of the table when it is full. Here we will use the accounting method to show that the amortized cost of an insertion operation in such a table is O(1).

Before looking at the procedure in detail, we need some definitions. Let T be a table, E an element to insert, num(T) the number of elements in T, and size(T) the allocated size of T. We assume the existence of operations create_table(n), which creates an empty table of size n, for now assumed to be free, and elementary_insert(T,E), which inserts element E into a table T that already has space allocated, with a cost of 1.

The following pseudocode illustrates the table insertion procedure: function table_insert(T,E) if num(T) = size(T) U := create_table(2 × size(T)) for each F in T elementary_insert(U,F) T := U elementary_insert(T,E)

Without amortized analysis, the best bound we can show for n insert operations is O(n2) — this is due to the loop at line 4 that performs num(T) elementary insertions.

For analysis using the accounting method, we assign a payment of 3 to each table insertion. Although the reason for this is not clear now, it will become clear during the course of the analysis.

Assume that initially the table is empty with size(T) = m. The first m insertions therefore do not require reallocation and only have cost 1 (for the elementary insert). Therefore, when num(T) = m, the pool has (3 - 1)×m = 2m.

Inserting element m + 1 requires reallocation of the table. Creating the new table on line 3 is free (for now). The loop on line 4 requires m elementary insertions, for a cost of m. Including the insertion on the last line, the total cost for this operation is m + 1. After this operation, the pool therefore has 2m + 3 - (m + 1) = m + 2.

Next, we add another m - 1 elements to the table. At this point the pool has m + 2 + 2×(m - 1) = 3m. Inserting an additional element (that is, element 2m + 1) can be seen to have cost 2m + 1 and a payment of 3. After this operation, the pool has 3m + 3 - (2m + 1) = m + 2. Note that this is the same amount as after inserting element m + 1. In fact, we can show that this will be the case for any number of reallocations.

It can now be made clear why the payment for an insertion is 3. 1 goes to inserting the element the first time it is added to the table, 1 goes to moving it the next time the table is expanded, and 1 goes to moving one of the elements that was already in the table the next time the table is expanded.

We initially assumed that creating a table was free. In reality, creating a table of size n may be as expensive as O(n). Let us say that the cost of creating a table of size n is n. Does this new cost present a difficulty? Not really; it turns out we use the same method to show the amortized O(1) bounds. All we have to do is change the payment.

When a new table is created, there is an old table with m entries. The new table will be of size 2m. As long as the entries currently in the table have added enough to the pool to pay for creating the new table, we will be all right.

We cannot expect the first \frac{m}{2} entries to help pay for the new table. Those entries already paid for the current table. We must then rely on the last \frac{m}{2} entries to pay the cost 2m. This means we must add \frac{2m}{m/2} = 4 to the payment for each entry, for a total payment of 3 + 4 = 7.

Test method

A test method is a definitive procedure that produces a test result.

A test can be considered as technical operation that consists of determination of one or more characteristics of a given product, process or service according to a specified procedure. Often a test is part of an experiment.

The test result can be qualitative (yes/no), categorical, or quantitative (a measured value). It can be a personal observation or the output of a precision measuring instrument.

Usually the test result is the dependent variable, the measured response based on the particular conditions of the test or the level of the independent variable. Some tests, however, involve changing the independent variable to determine the level at which a certain response occurs: in this case, the test result is the independent variable.

Importance of test methods

In software development, engineering, science, manufacturing, and business, it is vital for all interested people to understand and agree upon methods of obtaining data and making measurements. It is common for a physical property to be strongly affected by the precise method of testing or measuring that property. It is vital to fully document experiments and measurements and to provide needed definitions to specifications and contracts.

Using a standard test method, perhaps published by a respected standards organization, is a good place to start. Sometimes it is more useful to modify an existing test method or to develop a new one. Again, documentation and full disclosure are very necessary.

A well-written test method is important. However, even more important is choosing a method of measuring the correct property or characteristic. Not all tests and measurements are equally useful: usually a test result is used to predict or imply suitability for a certain purpose. For example, if a manufactured item has several components, test methods may have several levels of connections:

  • test results of a raw material should connect with tests of a component made from that material
  • test results of a component should connect with performance testing of a complete item
  • results of laboratory performance testing should connect with field performance

These connections or correlations may be based on published literature, engineering studies, or formal programs such as quality function deployment. Validation of the suitability of the test method is often required.

Content of a test method

Quality management systems usually require full documentation of the procedures used in a test. The document for a test method might include:

  • Descriptive title
  • Scope over which class(es) of materials or articles may be evaluated
  • Date of last effective revision and revision designation
  • Reference to most recent test method validation
  • Person, office, or agency responsible for questions on the test method, updates, and deviations.
  • The significance or importance of the test method and its intended use.
  • Terminology and definitions to clarify the meanings of the test method
  • A listing of the types of apparatus and measuring instrument (sometimes the specific device) required to conduct the test
  • Safety precautions
  • Required calibrations and metrology systems
  • Environmental concerns and considerations
  • Sampling procedures: How samples are to be obtained, and Number of samples (sample size).
  • Conditioning or required environmental chamber: temperature, humidity, etc., including tolerances
  • Preparation of samples for the test and test fixtures
  • Detailed procedure for conducting the test
  • Calculations and analysis of data
  • Interpretation of data and test method output
  • Report: format, content, data, etc.

Test Method Validation

Test methods are often scrutinized for their validity, applicability, and accuracy. It is very important that the scope of the test method be clearly defined, and any aspect included in the scope is shown to be accurate and repeatable through validation.

Test method validations often encompass the following considerations:

  • Accuracy and precision: Demonstration of accuracy may require the creation of a reference value if none is yet available.
  • Repeatability and Reproducibility, sometimes in the form of a Gauge R&R.
  • Range, or a continuum scale over which the test method would be considered accurate. Example: 10 N to 100 N force test.
  • Measurement resolution, be it spatial, temporal, or otherwise.
  • mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, , , and in the 19th century for computing invariants of algebraic forms. It is based on treating the form as if it were a power of a degree one form.

    Symbolic notation

    The symbolic method uses a compact but rather confusing and mysterious notation for invariants, depending on the introduction of new symbols a, b, c, ... (from which the symbolic method gets its name) with apparently contradictory properties.

    Example: the discriminant of a binary quadratic form

    These symbols can be explained by the following example from . Suppose that

    \displaystyle f(x) = A_0x_1^2+2A_1x_1x_2+A_2x_2^2

    is a binary quadratic form with an invariant given by the discriminant

    \displaystyle \Delta=A_0A_2-A_1^2

    The symbolic representation of the discriminant is

    \displaystyle 2\Delta=(ab)^2

    where a and b are the symbols. The meaning of the expression (ab)2 is as follows. First of all, (ab) is a shorthand form for the determinant of a matrix whose rows are a1, a2 and b1, b2, so

    \displaystyle (ab)=a_1b_2-a_2b_1

    Squaring this we get

    \displaystyle (ab)^2=a_1^2b_2^2-2a_1a_2b_1b_2+a_2^2b_1^2

    Next we pretend that

    \displaystyle f(x)=(a_1x_1+a_2x_2)^2=(b_1x_1+b_2x_2)^2

    so that

    \displaystyle A_i=a_1^{2-i}a_2^{i}= b_1^{2-i}b_2^{i}

    and we ignore the fact that this does not seem to make sense if f is not a power of a linear form. Substituting these values gives

    \displaystyle (ab)^2= A_2A_0-2A_1A_1+A_0A_2 = 2\Delta

    Higher degrees

    More generally if

    \displaystyle f(x) = A_0x_1^n+\binom{n}{1}A_1x_1^{n-1}x_2+\cdots+A_nx_2^n

    is a binary form of higher degree, then one introduces new variables a1, a2, b1, b2, c1, c2, with the properties


    What this means is that the following two vector spaces are naturally isomorphic:

    • The vector space of homogeneous polynomials in A0,...An of degree m
    • The vector space of polynomials in 2m variables a1, a2, b1, b2, c1, c2, ... that have degree n in each of the m pairs of variables (a1, a2), (b1, b2), (c1, c2), ... and are symmetric under permutations of the m symbols a, b, ....,

    The isomorphism is given by mapping a''a, b'b, .... to Aj. This mapping does not preserve products of polynomials.

    More variables

    The extension to a form f in more than two variables x1, x2,x3,... is similar: one introduces symbols a1, a2,a3 and so on with the properties


Kumon method

The Kumon method, developed by educator Toru Kumon, is a math and reading educational method that is practiced in Kumon's own learning centers. The Kumon Method advocates several values of learning that include speed, accuracy and mastery of material before a student is able to move on to the next lesson. As of 2009, over 4 million students were studying under the Kumon Method at more than 26,000 Kumon Centers in 46 countries.


In 1954, Toru Kumon, a Japanese high school mathematics teacher, began to teach his eldest son due to his problems in mathematics at school. Kumon developed the Kumon Method. In 1956, Kumon opened the first Kumon Center in Osaka, Japan with the help of parents that were very interested in the Method. In 1958, he founded the Kumon Institute of Education, after which Kumon Centers began to open around the world. Since 1956, some 20 million students have been enrolled in Kumon. Today, there are around 4 million Kumon students worldwide. At present there are 1500 Kumon Centers in the USA, and there are a total of 26,000 Kumon Centers in 44 countries. This list of countries includes the USA, Canada, Mexico, Brazil, Argentina, United Kingdom, Spain, Germany, South Africa, Australia, New Zealand, Japan, South Korea, the Philippines, Singapore, Malaysia, Indonesia, India, Thailand, and Hong Kong.

The programs

Kumon is a math and reading enrichment program. Students do not work together as a class, but progress through the curriculum at their own pace, moving on to the next level when they have achieved mastery of the previous level. Mastery is defined as speed (using a standard completion time) and accuracy. They take an achievement test at the end of each level.

Company value

The Kumon family, led by his wife Teiko, own 60 percent of the company. Forbesmagazine estimated in March 2009 that the entire company was currently worth over $650 million.

Mathematics curriculum

  • Level 7A: Counting to 10
  • Level 6A: Counting to 30
  • Level 5A: Line drawing, number puzzles to 50
  • Level 4A: Reciting and writing numbers up to 220
  • Level 3A: Adding with numbers up to 5
  • Level 2A: Adding with numbers up to 10, subtracting with numbers up to 9
  • Level A: Horizontal addition and subtraction of larger numbers
  • Level B: Vertical addition and subtraction
  • Level C: Multiplication, division
  • Level D: Long multiplication, long division, introduction to fractions
  • Level E: Fractions
  • Level F: Four operations of fractions, decimals
  • Level G: Positive/negative numbers, exponents, introduction to algebra
  • Level H: Linear/simultaneous equations, inequalities, functions, graphs high school level math
  • Level I: Factorization, square roots, quadratic equations, Pythagorean theorem
  • Level J: Advanced algebra
  • Level K: Functions: Quadratic, fractional, irrational, exponential
  • Level L: Logarithms, basic limits, derivatives, integrals, and its applications
  • Level M: Trigonometry, straight lines, equation of circles.
  • Level N: Loci, limits of functions, sequences, differentiation
  • Level O: Advanced differentiation, integration, applications of calculus, differential equations.
  • Level X is an elective level; also, there are no tests available yet on this level.
    • Level XT—Triangles: Sine, cosine theorems, application of trigonometry in area
    • Level XV—Vectors: Vectors, inner products, equations of lines, planes, and figures in 3-space.
    • Level XM—Linear algebra: Matrices, determinant, mapping, linear transformation
    • Level XP—Probability: Permutations, combinations, probability, independent trials, expected value
    • Level XS—Statistics: Binomial and normal distributions, probability density functions, confidence intervals, hypothesis testing

Reading curriculum

  • Prereading skills
  • Phonics
  • Vocabulary building
  • Grammar and punctuation
  • Reading comprehension

Variation of programs throughout the world

The Kumon language program varies regionally. For example, the Chinese reading program in Mainland China is different from the Chinese reading program in Hong Kong and in Singapore, and the English program in the U.S., Canada, and the Philippines varies from the English program in the United Kingdom. Additionally, Kumon Korea has other subjects, such as science, calligraphy, Korean, and Chinese characters, which are not available elsewhere.

The Math program also differs. The math program for most countries goes up to Level O and Level X. However, in Japan, the math program is available up to Level V.

From Yahoo Answers

Question:In algebra, the definition of rule is a description identifying the members of a set. The definition of a roster is how you specify a set when listing all members of a set. I kind of get the gist of this, but do you think you could explain it in greater detail and/or give examples? I'm kind of confused on how to identify this (aka, if I take a quiz over it sometime. Which may be in the near future. Like tomorrow.) Help is appreciated!

Answers:In roster form, you actually list all the elements. A = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} is in roster form. The rule for this set is that A is the letters of the alphabet.

Question:1. The set of counting numbers between -3 and 5. Roster and rule? 2. The set of divisors of 36 greater than -5 but less than or equal to 18. Roster and rule? Thank you for answering. :)

Answers:In roster form, the elements are listed out as such. In rule method, the rule for building the set is specified 1) roster form : {1,2,3,4} rule method { x : x belongs to natural numbers and -3 < x < 5 } 2) roster form : { 1,2,3,4,6,9,12,18 } rule method {x : x/6 belongs to N and -5 < x <= 18}

Question:I am in complete panic mode. I finished grade 9 in Canada, then I came to the Philippines for grade 10 and I had to start right away in June. The day we got there, I was *then* told I had a 10-day quarantine so I missed a week of school. It is a tiny private school. But June 23 was my first day, and then came math class. My teacher puts this on the board: Fill-in the column accordingly: Words 1. Greater than 5 Inequality Interval Notation Graph 2. Numbers 3 through 10 3. Numbers between -1 and 3 And so on. You have to write something in each column for every question. I came from a public school and I have seriously never seen this stuff before. I was almost in tears during math class. I tried to explain to my teacher that I have never learned this ever in my life, and he said "Yes you have. We use American textbooks. Just do it and give it your best shot." It was a test and I had no idea what to do so I basically failed. The next part was: II. Descriptive Notation Roster Interval Notation Set Builder Not. 6. All real nos. except 0 7. Negative even numbers ... 9. Numbers 5 units of 1 So, what I'm asking is, can somebody PLEASE explain this all to me? This is all foreign to me and I have no idea what to do. I am definitely going back to Canada as soon as this year is over, my private school costs $5,000 a year and there is no way my dad would ever let me come back without finishing. Please. I need an explanation from the very beginning. My teacher also refused to help me with my problem and I'm not finished signing up for Kumon so I need help fast. I don't need you to give me the answer, I just need this stuff explained to me so I can survive class until I get to my heavenly Kumon. (the teacher was using the signs: <> ) Please please please help me out. I also asked my Dad about it, but he went back to Canada about 5 hours ago. (Which also has me in a downer mood.) Sorry... I posted the question twice.

Answers:Start with different sets of numbers. The natural numbers are the set of positive whole numbers. The symbol for the set of natural numbers is . The integers are the set of whole numbers (so that's the natural numbers and the negative whole numbers). We write . The rational numbers are the set of numbers that can be written as a fraction (NOTE: this includes the integers as all numbers can be written as being over 1.) So these numbers are the integers and all the fractions. We write . The real numbers are all the numbers. So this is the rational numbers, but also all the numbers in between. The real numbers consist of the whole number line. We write . The above sets of numbers are special sets, so they have their own symbols ( , , , ) but you can write any collection of numbers as a set. We always write sets in curly brackets, { }. We could just as easily write the set of natural numbers: = {1, 2, 3, ...} where the dots show that it goes up in the same manner (in this case ones) infinitely. But for your example above, numbers 3 through to 10, we would write {3, 4, 5, 6, 7, 8, 9, 10} or {3, 4, 5, ..., 10}. The numbers between -1 and 3 would be {0, 1, 2} assuming that we aren't including the -1 and the 3. Now, the real numbers without 0 would be \{0} where the \ sign means "without". So it means the real numbers without the set {0}. Other things you may come across are union, and intersection, you use the union sign to "add" sets together. So if you wanted the natural numbers and -5, for example, you could write {-5}. or {1, 2, 3, ...} {-5} the intersection of 2 sets is the stuff in both sets so if you had the following 2 sets: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} then {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} = {2, 4, 6, 8, 10}. now we can have a go at negative even numbers. We know that an even number is divisible by 2. So any even number can be written 2n where n is a whole number, or an integer in fact. First we find the set of even numbers {2n : n } where the means that n is an element of , the integers. Then we can find the set of positive even numbers: {2n : n } so the set of negative even numbers is {-2n : n } Inequalities: < means less than, for example: 3 < 5 > means greater than, for example: 5 > 3 means less than or equal to. So 3 5, but 5 5 as well. means greater than or equal to. So 5 3, but 5 5 as well. When using equalities we can solve equations, for example: 2x + 4 = 10 then 2x = 6 x = 3 we can solve inequalities in the same way: 2x + 4 < 10 2x < 6 x < 3 the only difference being that if you divide by a negative number then you have to swap the inequality round: -2x + 5 < 3 -2x < -2 x > 1 this actaully makes sense though, because when you get to -2x < -2 then instead of dividing by -2, you can add "2x + 2" to each side: 2 < 2x then divide by 2: 1 < x so x > 1. The symbol means "infinity" but I can't think what to tell you about that at the moment! If you haven't seen this stuff before, then it's a lot to take in and learn. I'm sorry that your teacher is so unhelpful, but maybe you can find a nicer teacher and explain your problem to them?

Question:I need a simple example of how to use this method, just to understand how it works. Example with ODE would be great, anyone know such sites or something? Thanks!

Answers:The finite element method is much more complicated than you think. It's possible however to give simple examples with the finite difference method. The basic idea in the finite difference method is to convert the differential equations into difference equations. e.g. d^2 y /dx^2 = [y(n+1) - 2y(n) + y(n-1)] / x^2 Then you solve for y at each of hte node points.

From Youtube

Bisection Method: Example :Learn via an example, the bisection method of finding roots of a nonlinear equation of the form f(x)=0.

Petrus Method Example Solve :Solving the Rubik's Cube using the Lars Petrus method. In this video I am using a 3 look last layer but I am currently in the process of learning pll. Algorithms that I used: Orienting the UF and BR edges: R'U'R Permuting UF and UB corners: LU'R'UL'U'R Orienting Corners 1st algorithm: RUR'URU2R' Orienting Corners 2nd algorithm: R'U'RU'R'U2R 3 edge cycle: R2UFB'R2F'BUR2