The Order of the Coif is an honor society for United States law school graduates. A student at an American law school who earns a Juris Doctor degree and graduates in the top 10 percent of his or her class is eligible for membership if the student's law school has a chapter of the Order.

Membership[link]

According to the organization's constitution, "The purpose of The Order is to encourage excellence in legal education by fostering a spirit of careful study, recognizing those who as law students attained a high grade of scholarship, and honoring those who as lawyers, judges and teachers attained high distinction for their scholarly or professional accomplishments."^[1]

The exact induction process varies by law school, but students are generally notified of their membership after the final class ranks at their schools are announced. A new member receives a certificate of membership, a badge of membership for wear during academic ceremonies, a Coif key, and in some cases an actual coif or a representation of one.

The basic requirement for membership is ranking in the top 10% of a member school's graduating class. If a member law school graduates fewer than 30 students, it may induct its top three students. A school can decide not to allow an otherwise eligible student to receive the honor, and may impose additional requirements for membership beyond the organization's national requirement of being in the top 10% of the class.^[2]

Each member school may also induct a faculty member and one honorary member each year.^[3] The national organization's executive committee may also elect a limited number of honorary members.^[4] Those chosen for honorary membership are usually United States Supreme Court justices and other preeminent members of the legal profession.^[5]

Notable members[link]

• Jim Bacchus
• Mack Barham
• Elizabeth Warren
• Warren E. Burger
• Warren Christopher
• Ann Coulter
• Roger Groot
• Richard Nixon
• Kathleen M. O'Malley
• Robert G. Pugh
• Diane Wood
• Barbara, Lady Judge

Chapters[link]

As of 2011^[update], 81 of 199 United States law schools accredited by the American Bar Association to award the J.D. degree had Order of the Coif chapters. ^{[6] [7]} In that year, all but five of the top fifty law schools, as ranked by U.S. News, were member schools. ^{[8] [9]} The others, Boston University School of Law, Columbia Law School, Harvard Law School, George Mason University School of Law, and Notre Dame Law School, have never applied for a chapter.^{[citation needed]} Notre Dame and Columbia are ineligible because they do not rank the top 10% of their graduating class by grade point average,^{[citation needed]} which the order's constitution requires.

For a law school to establish a chapter, the school must apply for a charter. If the organization's executive committee determines, after considering the law school's written submissions and its own investigation (which may include an examination of the school by a visitation team), that the applicant merits a chapter, it will submit the application for a vote by the existing chapters. A charter for a new chapter requires approval by 80% of the existing chapters.^[10]

Criteria considered when a law school applies for a chapter of the Order include: (1) American Bar Association and Association of American Law Schools approval; (2) at least ten years of existence as a law school; (3) affiliation with a university; (4) if a part-time J.D. program exists, the part-time program must offer students and faculty affiliated with the part-time program the same scholarship opportunities as all other students and faculty; (5) a stimulating intellectual environment for the study of law; (6) commitment of the university and law school administration to quality legal education; (7) faculty scholarship and institutional support for same; (8) a diverse educational program; (9) a diverse student body with strong academic credentials; (10) a law library that will support and encourage research activity; and (11) appropriate physical facilities.^[11]

A law school can also be removed from the Order if a two-thirds majority of member schools agrees to bring the matter to a vote and a four-fifths majority (excluding the school in question) then votes to remove the school.

The University of Illinois established the first Order of the Coif chapter in 1902. ^[12]

References[link]

• Michael Herz, Coif Comes to Cardozo, Cardozo Life (Spring 1999) (includes information on the order's history).
• Order of the Coif, Constitution (2003) [cited herein as Constitution]. Retrieved January 2, 2010.
• Frank R. Strong, Order of the Coif: English Antecedents and American Adaptation, 63 A.B.A. J. 1725, 1726 (1977).

External links[link]

• The Order of the Coif

Example of Big O notation: f(x) ∈ O(g(x)) as there exists c > 0 (e.g. c = 1) and x₀ (e.g. x₀ = 5) such that f(x) < cg(x) whenever x > x₀.

In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann–Landau notation (after Edmund Landau and Paul Bachmann), or asymptotic notation. In computer science, big O notation is used to classify algorithms by how they respond (e.g., in their processing time or working space requirements) to changes in input size.

Big O notation characterizes functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

Big O notation is also used in many other fields to provide similar estimates.

Formal definition[link]

Let f(x) and g(x) be two functions defined on some subset of the real numbers. One writes

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=O(g(x))\text{ as }x\to\infty\,

if and only if there is a positive constant M such that for all sufficiently large values of x, f(x) is at most M multiplied by g(x) in absolute value. That is, f(x) = O(g(x)) if and only if there exists a positive real number M and a real number x₀ such that

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(x)| \le \; M |g(x)|\text{ for all }x>x_0.

In many contexts, the assumption that we are interested in the growth rate as the variable x goes to infinity is left unstated, and one writes more simply that f(x) = O(g(x)). The notation can also be used to describe the behavior of f near some real number a (often, a = 0): we say

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=O(g(x))\text{ as }x\to a\,

if and only if there exist positive numbers δ and M such that

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(x)| \le \; M |g(x)|\text{ for }|x - a| < \delta.

If g(x) is non-zero for values of x sufficiently close to a, both of these definitions can be unified using the limit superior:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=O(g(x))\text{ as }x \to a\,

if and only if

Failed to parse (Missing texvc executable; please see math/README to configure.): \limsup_{x\to a} \left|\frac{f(x)}{g(x)}\right| < \infty.

Example[link]

In typical usage, the formal definition of O notation is not used directly; rather, the O notation for a function f(x) is derived by the following simplification rules:

• If f(x) is a sum of several terms, the one with the largest growth rate is kept, and all others omitted.
• If f(x) is a product of several factors, any constants (terms in the product that do not depend on x) are omitted.

For example, let Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = 6x^4 - 2x^3 +5 , and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x⁴, −2x³, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x⁴. Now one may apply the second rule: 6x⁴ is a product of 6 and x⁴ in which the first factor does not depend on x. Omitting this factor results in the simplified form x⁴. Thus, we say that f(x) is a big-oh of (x⁴) or mathematically we can write f(x) = O(x⁴). One may confirm this calculation using the formal definition: let f(x) = 6x⁴ − 2x³ + 5 and g(x) = x⁴. Applying the formal definition from above, the statement that f(x) = O(x⁴) is equivalent to its expansion,

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(x)| \le \; M |g(x)|

for some suitable choice of x₀ and M and for all x > x₀. To prove this, let x₀ = 1 and M = 13. Then, for all x > x₀:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align}|6x^4 - 2x^3 + 5| &\le 6x^4 + |2x^3| + 5\\ &\le 6x^4 + 2x^4 + 5x^4\\ &\le 13x^4\\ &\le 13|x^4|\end{align}

so

Failed to parse (Missing texvc executable; please see math/README to configure.): |6x^4 - 2x^3 + 5| \le 13 \,|x^4 |.

Usage[link]

Big O notation has two main areas of application. In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion. In computer science, it is useful in the analysis of algorithms. In both applications, the function g(x) appearing within the O(...) is typically chosen to be as simple as possible, omitting constant factors and lower order terms. There are two formally close, but noticeably different, usages of this notation: infinite asymptotics and infinitesimal asymptotics. This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

Infinite asymptotics[link]

Big O notation is useful when analyzing algorithms for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n² − 2n + 2. As n grows large, the n² term will come to dominate, so that all other terms can be neglected — for instance when n = 500, the term 4n² is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term n³ or n⁴. Even if T(n) = 1,000,000n², if U(n) = n³, the latter will always exceed the former once n grows larger than 1,000,000 (T(1,000,000) = 1,000,000³= U(1,000,000)). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either

Failed to parse (Missing texvc executable; please see math/README to configure.): \ T(n)= O(n^2) \,

or

Failed to parse (Missing texvc executable; please see math/README to configure.): T(n)\in O(n^2) \,

and say that the algorithm has order of n² time complexity. Note that "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is technically accurate (see the "Equals sign" discussion below) while the first is a common abuse of notation.^[1]

Infinitesimal asymptotics[link]

Big O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. For example,

Failed to parse (Missing texvc executable; please see math/README to configure.): e^x=1+x+\frac{x^2}{2}+O(x^3)\qquad\text{as } x\to 0

expresses the fact that the error, the difference Failed to parse (Missing texvc executable; please see math/README to configure.): \ e^x - (1 + x + x^2/2) , is smaller in absolute value than some constant times Failed to parse (Missing texvc executable; please see math/README to configure.): |x^3|

when Failed to parse (Missing texvc executable; please see math/README to configure.): x
is close enough to 0.

Properties[link]

If a function f(n) can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) = 9 \log n + 5 (\log n)^3 + 3n^2 + 2n^3 = O(n^3)\,\!.

In particular, if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial. O(n^c) and O(cⁿ) are very different. If c is greater than one, then the later grows much faster. A function that grows faster than n^c for any c is called superpolynomial. One that grows more slowly than any exponential function of the form Failed to parse (Missing texvc executable; please see math/README to configure.): c^n

is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for integer factorization.

O(log n) is exactly the same as O(log(n^c)). The logarithms differ only by a constant factor (since Failed to parse (Missing texvc executable; please see math/README to configure.): \log(n^c)=c \log n ) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. Exponentials with different bases, on the other hand, are not of the same order. For example, Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n

and Failed to parse (Missing texvc executable; please see math/README to configure.): 3^n
are not of the same order.

Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n², replacing n by cn means the algorithm runs in the order of Failed to parse (Missing texvc executable; please see math/README to configure.): c^2n^2 , and the big O notation ignores the constant Failed to parse (Missing texvc executable; please see math/README to configure.): c^2 . This can be written as Failed to parse (Missing texvc executable; please see math/README to configure.): c^2n^2 \in O(n^2) . If, however, an algorithm runs in the order of Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n , replacing n with cn gives Failed to parse (Missing texvc executable; please see math/README to configure.): 2^{cn} = (2^c)^n . This is not equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): 2^n

in general.

Changing of variable may affect the order of the resulting algorithm. For example, if an algorithm's running time is O(n) when measured in terms of the number n of digits of an input number x, then its running time is O(log x) when measured as a function of the input number x itself, because n = Θ(log x).

Product[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in O(g_1) \text{ and } f_2\in O(g_2)\, \Rightarrow f_1 f_2\in O(g_1 g_2)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): f\cdot O(g) \subset O(f g)

Sum[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in O(g_1) \text{ and } f_2\in O(g_2)\, \Rightarrow f_1 + f_2\in O(|g_1| + |g_2|)\,

This implies Failed to parse (Missing texvc executable; please see math/README to configure.): f_1 \in O(g) \text{ and } f_2 \in O(g) \Rightarrow f_1+f_2 \in O(g)

, which means that Failed to parse (Missing texvc executable; please see math/README to configure.): O(g)

is a convex cone.

If f and g are positive functions, Failed to parse (Missing texvc executable; please see math/README to configure.): f + O(g) \in O(f + g)

Multiplication by a constant[link]

Let k be a constant. Then:

Failed to parse (Missing texvc executable; please see math/README to configure.): \ O(k g) = O(g)

if k is nonzero.

Failed to parse (Missing texvc executable; please see math/README to configure.): f\in O(g) \Rightarrow kf\in O(g).

Multiple variables[link]

Big O (and little o, and Ω...) can also be used with multiple variables. To define Big O formally for multiple variables, suppose Failed to parse (Missing texvc executable; please see math/README to configure.): f(\vec{x})

and Failed to parse (Missing texvc executable; please see math/README to configure.): g(\vec{x})
are two functions defined on some subset of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^n

. We say

Failed to parse (Missing texvc executable; please see math/README to configure.): f(\vec{x})\text{ is }O(g(\vec{x}))\text{ as }\vec{x}\to\infty

if and only if

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists C\,\exists M>0\text{ such that } |f(\vec{x})| \le C |g(\vec{x})|\text{ for all }\vec{x} \text{ with } x_i>M \text{ for all }i.

For example, the statement

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = n^2 + m^3 + O(n+m) \text{ as } n,m\to\infty\,

asserts that there exist constants C and M such that

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall n, m>M\colon |g(n,m)| \le C(n+m),

where g(n,m) is defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = n^2 + m^3 + g(n,m).\,

Note that this definition allows all of the coordinates of Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{x}

to increase to infinity. In particular, the statement

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n,m) = O(n^m) \text{ as } n,m\to\infty\,

(i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \exists C\,\exists M\,\forall n\,\forall m\dots ) is quite different from

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall m\colon f(n,m) = O(n^m) \text{ as } n\to\infty

(i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \forall m\,\exists C\,\exists M\,\forall n\dots ).

Matters of notation[link]

Equals sign[link]

The statement "f(x) is O(g(x))" as defined above is usually written as f(x) = O(g(x)). Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, O(x) = O(x²) is true but O(x²) = O(x) is not.^[2] Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n² from the identities n = O(n²) and n² = O(n²)."^[3] For these reasons, it would be more precise to use set notation and write f(x) ∈ O(g(x)), thinking of O(g(x)) as the class of all functions h(x) such that |h(x)| ≤ C|g(x)| for some constant C.^[3] However, the use of the equals sign is customary. Knuth pointed out that "mathematicians customarily use the = sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle."^[4]

Other arithmetic operators[link]

Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) = h(x) + O(f(x))\,

expresses the same as

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) - h(x) \in O(f(x))\,.

[edit] Example

Suppose an algorithm is being developed to operate on a set of n elements. Its developers are interested in finding a function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of O(n²), and after the subroutine runs the algorithm must take an additional Failed to parse (Missing texvc executable; please see math/README to configure.): 55n^3+2n+10

time before it terminates.  Thus the overall time complexity of the algorithm can be expressed as

Failed to parse (Missing texvc executable; please see math/README to configure.): T(n)=O(n^2)+55n^3+2n+10.\

This can perhaps be most easily read by replacing O(n²) with "some function that grows asymptotically no faster than n² ". Again, this usage disregards some of the formal meaning of the "=" and "+" symbols, but it does allow one to use the big O notation as a kind of convenient placeholder.

Declaration of variables[link]

Another feature of the notation, although less exceptional, is that function arguments may need to be inferred from the context when several variables are involved. The following two right-hand side big O notations have dramatically different meanings:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(m) = O(m^n)\,,

Failed to parse (Missing texvc executable; please see math/README to configure.): g(n)\,\, = O(m^n)\,.

The first case states that f(m) exhibits polynomial growth, while the second, assuming m > 1, states that g(n) exhibits exponential growth. To avoid confusion, some authors use the notation

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x) \in O(f(x))\,.

rather than the less explicit

Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)\,,

Multiple usages[link]

In more complicated usage, O(...) can appear in different places in an equation, even several times on each side. For example, the following are true for Failed to parse (Missing texvc executable; please see math/README to configure.): n\to\infty

Failed to parse (Missing texvc executable; please see math/README to configure.): (n+1)^2 = n^2 + O(n)\

Failed to parse (Missing texvc executable; please see math/README to configure.): (n+O(n^{1/2}))(n + O(\log n))^2 = n^3 + O(n^{5/2})\

Failed to parse (Missing texvc executable; please see math/README to configure.): n^{O(1)} = O(e^n).\

The meaning of such statements is as follows: for any functions which satisfy each O(...) on the left side, there are some functions satisfying each O(...) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)=O(1)\, , there is some function Failed to parse (Missing texvc executable; please see math/README to configure.): g(n)=O(e^n)\,

such that Failed to parse (Missing texvc executable; please see math/README to configure.): n^{f(n)}=g(n)

." In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "=" is a formal symbol that unlike the usual use of "=" is not a symmetric relation. Thus for example Failed to parse (Missing texvc executable; please see math/README to configure.): n^{O(1)} = O(e^n)\,

does not imply the false statement Failed to parse (Missing texvc executable; please see math/README to configure.): O(e^n) = n^{O(1)}\, 

.

Orders of common functions[link]

Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a constant and n increases without bound. The slower-growing functions are generally listed first.

The statement Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)=O(n!)\,

is sometimes weakened to Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)=O\left(n^n\right)
to derive simpler formulas for asymptotic complexity.

For any Failed to parse (Missing texvc executable; please see math/README to configure.): k>0

and Failed to parse (Missing texvc executable; please see math/README to configure.): c>0

, Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^c(\log n)^k)

is a subset of Failed to parse (Missing texvc executable; please see math/README to configure.): O(n^{c+\varepsilon })
for any Failed to parse (Missing texvc executable; please see math/README to configure.):  \varepsilon >0

, so may be considered as a polynomial with some bigger order.

Related asymptotic notations[link]

Big O is the most commonly used asymptotic notation for comparing functions, although in many cases Big O may be replaced with Big Theta Θ for asymptotically tighter bounds. Here, we define some related notations in terms of Big O, progressing up to the family of Bachmann–Landau notations to which Big O notation belongs.

Little-o notation[link]

The relation Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) \in o(g(x))

is read as "Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)
is little-o of Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

". Intuitively, it means that Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

grows much faster than Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

, or similarly, the growth of Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

is nothing compared to that of Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)

. It assumes that f and g are both functions of one variable. Formally, f(n) = o(g(n)) as n → ∞ means that for every positive constant ε there exists a constant N such that

Failed to parse (Missing texvc executable; please see math/README to configure.): |f(n)|\leq\epsilon|g(n)|\qquad\text{for all }n\geq N~.

^[3]

Note the difference between the earlier formal definition for the big-O notation, and the present definition of little-o: while the former has to be true for at least one constant M the latter must hold for every positive constant ε, however small.^[1] In this way little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O g is also little-o of g (for instance g itself is not, unless it is identically zero near ∞).

If g(x) is nonzero, or at least becomes nonzero beyond a certain point, the relation f(x) = o(g(x)) is equivalent to

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{x \to \infty}\frac{f(x)}{g(x)}=0.

For example,

• Failed to parse (Missing texvc executable; please see math/README to configure.): 2x \in o(x^2) \,\!

• Failed to parse (Missing texvc executable; please see math/README to configure.): 2x^2 \not \in o(x^2)

• Failed to parse (Missing texvc executable; please see math/README to configure.): 1/x \in o(1)

Little-o notation is common in mathematics but rarer in computer science. In computer science the variable (and function value) is most often a natural number. In mathematics, the variable and function values are often real numbers. The following properties can be useful:

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(f) + o(f) \subseteq o(f)

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(f)o(g) \subseteq o(fg)

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(o(f)) \subseteq o(f)

• Failed to parse (Missing texvc executable; please see math/README to configure.): o(f) \subset O(f)

(and thus the above properties apply with most combinations of o and O).

As with big O notation, the statement "Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

is Failed to parse (Missing texvc executable; please see math/README to configure.): o(g(x))

" is usually written as Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = o(g(x)) , which is a slight abuse of notation.

Family of Bachmann–Landau notations[link]

is bounded above by Failed to parse (Missing texvc executable; please see math/README to configure.): g
(up to constant factor) asymptotically

for some k

 or 
 Failed to parse (Missing texvc executable; please see math/README to configure.):   \exists k>0 \; \exists n_0 \; \forall n>n_0 \; f(n) \leq g(n)\cdot k

is bounded below by Failed to parse (Missing texvc executable; please see math/README to configure.): g
(up to constant factor) asymptotically

for some positive k

is bounded both above and below by Failed to parse (Missing texvc executable; please see math/README to configure.): g
asymptotically

for some positive k1, k2

is dominated by Failed to parse (Missing texvc executable; please see math/README to configure.): g
asymptotically

for every Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon

dominates Failed to parse (Missing texvc executable; please see math/README to configure.): g
asymptotically

for every k

is equal to Failed to parse (Missing texvc executable; please see math/README to configure.): g
asymptotically

Bachmann–Landau notation was designed around several mnemonics, as shown in the As Failed to parse (Missing texvc executable; please see math/README to configure.): n \to \infty , eventually... column above and in the bullets below. To conceptually access these mnemonics, "omicron" can be read "o-micron" and "omega" can be read "o-mega". Also, the lower-case versus capitalization of the Greek letters in Bachmann–Landau notation is mnemonic.

• The o-micron mnemonic: The o-micron reading of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in O(g(n))

and of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in o(g(n))
 can be thought of as "O-smaller than" and "o-smaller than", respectively.  This micro/smaller mnemonic refers to: for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f
grows at a rate that may henceforth be less than Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)
or Failed to parse (Missing texvc executable; please see math/README to configure.): g \in o(f)

.

• The o-mega mnemonic: The o-mega reading of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \Omega(g(n))

and of Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \omega(g(n))
can be thought of as "O-larger than".  This mega/larger mnemonic refers to: for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f
grows at a rate that may henceforth be greater than Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in \Omega(f)
or Failed to parse (Missing texvc executable; please see math/README to configure.): g \in \omega(f)

.

• The upper-case mnemonic: This mnemonic reminds us when to use the upper-case Greek letters in Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in O(g(n))

and Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \Omega(g(n))

for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f

grows at a rate that may henceforth be equal to Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)

.

• The lower-case mnemonic: This mnemonic reminds us when to use the lower-case Greek letters in Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in o(g(n))

and Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) \in \omega(g(n))

for sufficiently large input parameter(s), Failed to parse (Missing texvc executable; please see math/README to configure.): f

grows at a rate that is henceforth inequal to Failed to parse (Missing texvc executable; please see math/README to configure.): cg
regarding Failed to parse (Missing texvc executable; please see math/README to configure.): g \in O(f)

. Aside from Big O notation, the Big Theta Θ and Big Omega Ω notations are the two most often used in computer science; the Small Omega ω notation is rarely used in computer science.

Use in computer science[link]

Informally, especially in computer science, the Big O notation often is permitted to be somewhat abused to describe an asymptotic tight bound where using Big Theta Θ notation might be more factually appropriate in a given context. For example, when considering a function Failed to parse (Missing texvc executable; please see math/README to configure.): T(n) = 73n^3 + 22n^2 + 58 , all of the following are generally acceptable, but tightnesses of bound (i.e., bullets 2 and 3 below) are usually strongly preferred over laxness of bound (i.e., number 1 below).

1. T(n) = O(n¹⁰⁰), which is identical to T(n) ∈ O(n¹⁰⁰)
2. T(n) = O(n³), which is identical to T(n) ∈ O(n³)
3. T(n) = Θ(n³), which is identical to T(n) ∈ Θ(n³).

The equivalent English statements are respectively:

1. T(n) grows asymptotically no faster than n¹⁰⁰
2. T(n) grows asymptotically no faster than n³
3. T(n) grows asymptotically as fast as n³.

So while all three statements are true, progressively more information is contained in each. In some fields, however, the Big O notation (number 2 in the lists above) would be used more commonly than the Big Theta notation (bullets number 3 in the lists above) because functions that grow more slowly are more desirable. For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): T(n)

represents the running time of a newly developed algorithm for input size Failed to parse (Missing texvc executable; please see math/README to configure.): n

, the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound.

Extensions to the Bachmann–Landau notations[link]

Another notation sometimes used in computer science is Õ (read soft-O): f(n) = Õ(g(n)) is shorthand for f(n) = O(g(n) log^k g(n)) for some k. Essentially, it is Big O notation, ignoring logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since log^k n is always o(n^ε) for any constant k and any ε > 0). The L notation, defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): L_n[\alpha,c]=O\left(e^{(c+o(1))(\ln n)^\alpha(\ln\ln n)^{1-\alpha}}\right),

is convenient for functions that are between polynomial and exponential.

Generalizations and related usages[link]

The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any topological group is also possible. The "limiting process" x→x_o can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g. The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,

Failed to parse (Missing texvc executable; please see math/README to configure.): f\sim g \iff (f-g) \in o(g)

which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to Failed to parse (Missing texvc executable; please see math/README to configure.): \lim f/g = 1

if f and g are positive real valued functions.)  For example, 2x is Θ(x), but 2x − x is not o(x).

Graph theory[link]

It is often useful to bound the running time of graph algorithms. Unlike most other computational problems, for a graph G = (V, E) there are two relevant parameters describing the size of the input: the number |V| of vertices in the graph and the number |E| of edges in the graph. Inside asymptotic notation (and only there), it is common to use the symbols V and E, when someone really means |V| and |E|. This convention simplifies asymptotic functions and make them easily readable. The symbols V and E are never used inside asymptotic notation with their literal meaning, since the number of vertices and edges must be non-negative, so this abuse of notation does not risk ambiguity. For example Failed to parse (Missing texvc executable; please see math/README to configure.): O(E + V \log V)\,

means Failed to parse (Missing texvc executable; please see math/README to configure.): O((V,E) \mapsto |E| + |V|\cdot\log|V|)\,
for a suitable metric of graphs.  Another common convention—referring to the values |V| and |E| by the names n and m, respectively—sidesteps this ambiguity.

History[link]

The notation was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory"), the first volume of which (not yet containing big O notation) was published in 1892.^[5] The notation was popularized in the work of number theorist Edmund Landau; hence it is sometimes called a Landau symbol. It was popularized in computer science by Donald Knuth, who re-introduced the related Omega and Theta notations.^[6] He also noted that the (then obscure) Omega notation had been introduced by Hardy and Littlewood^[7] under a slightly different meaning, and proposed the current definition. Hardy's symbols were (in terms of the modern O notation)

Failed to parse (Missing texvc executable; please see math/README to configure.): f\lesssim g \iff f \in O(g)

  and   Failed to parse (Missing texvc executable; please see math/README to configure.):  f\ll g \iff f\in o(g); 

other similar symbols were sometimes used, such as Failed to parse (Missing texvc executable; please see math/README to configure.): \preceq

and Failed to parse (Missing texvc executable; please see math/README to configure.): \prec\!\!\!\!\!\!\!\!\prec

. The big-O, standing for "order of", was originally a capital omicron; today the identical-looking Latin capital letter O is used, but never the digit zero.

See also[link]

• Asymptotic expansion: Approximation of functions generalizing Taylor's formula
• Asymptotically optimal: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem
• Limit superior and limit inferior: An explanation of some of the limit notation used in this article
• Nachbin's theorem: A precise method of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated
• Big O in probability notation: O_p,o_p
• Computational complexity theory: A sub-field strongly related to this article

References[link]

Further reading[link]

• Paul Bachmann. Die Analytische Zahlentheorie. Zahlentheorie. pt. 2 Leipzig: B. G. Teubner, 1894.
• Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen. 2 vols. Leipzig: B. G. Teubner, 1909.
• G. H. Hardy. Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond, 1910.
• Donald Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison–Wesley, 1997. ISBN 0-201-89683-4. Section 1.2.11: Asymptotic Representations, pp. 107–123.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw–Hill, 2001. ISBN 0-262-03293-7. Section 3.1: Asymptotic notation, pp. 41–50.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X.  Pages 226–228 of section 7.1: Measuring complexity.
• Jeremy Avigad, Kevin Donnelly. Formalizing O notation in Isabelle/HOL
• Paul E. Black, "big-O notation", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 11 March 2005. Retrieved December 16, 2006.
• Paul E. Black, "little-o notation", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.
• Paul E. Black, "Ω", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.
• Paul E. Black, "ω", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 29 November 2004. Retrieved December 16, 2006.
• Paul E. Black, "Θ", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. Retrieved December 16, 2006.

External links[link]

The Wikibook Data Structures has a page on the topic of Big-O Notation

• Introduction to Asymptotic Notations
• Landau Symbols
• O-Notation Visualizer: Interactive Graphs of Common O-Notations

Leslie Cohen Berlowitz is President and the William T. Golden Chair of the American Academy of Arts and Sciences. Berlowitz is the first woman to serve as Chief Executive Officer of the Academy and was named its 45th President in 2010. At the Academy she is responsible for policy, planning, and resource development. She oversees five research divisions – science and technology policy; global security; social policy and American institutions; the humanities and society; and education – as well as Academy publications, including the quarterly journal Daedalus (journal). She is a scholar in American Studies. Before joining the Academy in 1996, Berlowitz was Vice President at New York University, where she was responsible for academic policy, planning, and major new institutional initiatives. She has served as an advisor to the National Endowment for the Humanities, the Corporation of Yaddo, the National Humanities Alliance, and the Robert Wood Johnson Foundation. Elected a Fellow of the American Academy in 2004,^[1] she was named an honorary Doctor of Humane Letters by Northeastern University in May of 2011.^[2]

American Academy Presidency[link]

Berlowitz led the Academy’s Strategic Plan "2001 and Beyond" and the development of the Initiative for the Humanities and its Humanities Indicators.^[3][4] She has created a network of more than 50 University Affiliates^[5] to work with the Academy on issues vital to the higher education community and also established two residential fellowship programs for young scholars: the Visiting Scholars Program^[6] and the Hellman Fellowship in Science and Technology Policy.^[7] She has organized task forces on science and energy policy, one of which prepared the Academy’s 2008 report, ARISE: Advancing Research In Science and Engineering: Investing in Early-Career Scientists and High-Risk, High-Reward Research.^[8]

On February 17, 2011, Berlowitz announced the creation of the Academy’s Commission on the Humanities and Social Sciences^[9][10] in response to a request^[11] from Members of Congress. The Commission will recommend specific steps that government, schools and universities, cultural institutions, businesses, and philanthropies can take to bolster teaching and research in the humanities and social sciences, fields that are critical to culture, education, and to America's economic competitiveness.

Publications[link]

Berlowitz’s publications include:

Books

• Reflecting on the Humanities, Daedalus, (Cambridge: MIT Press, 2009) with Patricia Meyer Spacks
• Restoring Trust in American Business (Cambridge: MIT Press, 2005) with Jay W. Lorsch and Andy Zelleke
• America in Theory (New York: Oxford University Press, 1988) with Denis Donoghue and Louis Menand
• Greenwich Village: Culture and Counterculture (New Jersey: Rutgers University Press, 1990) with Richard Eric Beard

Articles

• "Humanities: The Case for Data," Liberal Education Winter 2010 and The Bulletin of the American Academy of Arts & Sciences Spring 2010
• With Neal Lane, "Where to Spend Our Research Dollars: A Progressive Stimulus Package Should Include Grant Reform," Science Progress Thursday, January 22, 2009
• "Advice to the Next President," Letters to the Next President: Strengthening America's Foundation in Higher Education, Stephen Joel Trachtenberg and Gerald B. Kauvar, eds., The Korn/Ferry Institute 2008
• With Joel E. Cohen, "Academia in the Crosshairs," The Boston Globe September 30, 2007

Education[link]

Berlowitz graduated from the Fieldston School and earned her undergraduate and graduate degrees from New York University and Columbia University.

Personal[link]

Berlowitz has a daughter and son-in-law, Sarah Tuttleton Arron and Joe Arron and two grandchildren.

References[link]

External links[link]

• Official Website – American Academy of Arts and Sciences
• American Academy Commission on the Humanities and Social Sciences
• Humanities Resource Center Online – Humanities Indicators

This article is about Ken Friedman (artist).

For Ken Friedman (screenwriter) see here.

Ken Friedman, (born September 19, 1949 in New London, Connecticut) is a seminal figure in Fluxus, an international laboratory for experimental art, architecture, design, literature, and music. He had his first solo exhibition in New York in 1966. He has also been involved with mail art, and he has written extensively about Fluxus and Intermedia. Friedman has edited several Fluxus publications and has been widely published in peer reviewed academic journals as well as in popular and small press publications. He has worked closely with other Fluxus artists and composers such as George Maciunas, Dick Higgins, and Nam June Paik, as well as collaborating with John Cage and Joseph Beuys. He was the general manager of Dick Higgins Something Else Press in the early 1970s.

Friedman received his Master of Arts degree in Interdisciplinary Studies in education, psychology and social science from San Francisco State University in 1971. He received his doctorate in 1976 from the United States International University. In 2007, Loughborough University in the UK honored Friedman with the degree of Doctor of Science, honoris causa, for outstanding contributions to design research.

In October 2007, he was appointed Dean of the Faculty of Design at Swinburne University of Technology, Melbourne, Australia. From 1994 to 2009, Friedman was professor in the Department of Culture, Communication, and Language at the Norwegian School of Management in Oslo, as well as at the Design Research Center at The Danish Design School in Copenhagen from 2003 to 2009. In the 1970s he was Director of the Institute for Advanced Studies in Contemporary Art, San Diego and during the mid 1980s he was President of the Art Economist Corporation in New York.

Ken Friedman has done research in the philosophy of science, the philosophy of design, and doctoral education in design. He also works with national design policy projects in Estonia, Latvia, Lithuania, and Wales. He continues to develop international research networks and conferences for the design research community. He is an editor of the journal Artifact, and he serves on the Editorial Board of Design Studies. He has been the book reviews editor of Design Research News and the Communications Secretary of the Design Research Society. In 2000 He co-chaired the La Clusaz Conference on Doctoral Education in Design. He co-chaired the 2006 conference of the European Academy of Management in Oslo, and the 2006 conference of the Design Research Society in Lisbon, and the 2009 conference on Design Thinking in Torquay, Australia.

Recent exhibitions of his work include Ken Friedman: 99 Events at Stendhal Gallery in New York, Ken Friedman Art(net)worker Extraordinaire at the University of Iowa Museum of Art in 2000, and Twelve Structures at the Centre of Attention, London in 2004. His work is represented in major museums and galleries around the world, including the Museum of Modern Art and the Guggenheim Museum in New York, the Tate Modern in London, the Hood Museum of Art at Dartmouth College, and Stadtsgalerie Stuttgart. The University of Iowa Alternative Traditions in the Contemporary Arts is the official repository of Friedman’s papers and research notes. Archiv Sohm at Stadtsgalerie Stuttgart and the Mandeville Department of Special Collections at the University of California also hold extensive collections of his work from the 1960s and 1970s.

External links[link]

• Ken Friedman: 99 Events at Stendhal Gallery
• Friedman Biography & Bibliography
• Ken Friedman Art(net)worker Extraordinaire
• Twelve Structures
• 2004 London Fluxus Concert by Ken Friedman
• Forty Years of Fluxus by Ken Friedman
• Interview Ken Friedman by Ruud Janssen
• Professor Ken Friedman
• Ken Friedman at The Danish Design School
• Creating Master Minds. Professor Ken Friedman. Interview published by Design Victoria

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