With the above results we can make the following remarks about the graph of F.

The following image shows the graph of the gamma function along the real line:

f ( x ) whose graph crosses through the origin, that is, whose y-intercept is 0, has the following properties:

The usual assumed path of theories is given in the following graph, where each unification step leads one level up:

The graph of the delta function is usually thought of as following the whole x-axis and the positive y-axis.

A tree is an undirected simple graph G that satisfies any of the following equivalent conditions:

This may be denoted by error bars on a graph, or by the following notations:

Suppose we have a connected graph G = ( V, E ), The following statements are equivalent:

In the following example, the formula R ( 3, 3 ) provides a solution to the question which asks the minimum number of vertices a graph must contain in order to ensure that either ( 1 ) at least 3 vertices in the graph are connected or ( 2 ) at least 3 vertices in the graph are unconnected.

A classic form of state diagram for a finite state machine or finite automaton ( FA ) is a directed graph with the following elements ( Q, Σ, Z, δ, q < sub > 0 </ sub >, F ):

However, if a graph contains a " negative cycle ", i. e., a cycle whose edges sum to a negative value, then walks of arbitrarily low weight can be constructed by repeatedly following the cycle, so there may not be a shortest path.

However, the notion of isomorphism may be applied to all other variants of the notion of graph, by adding the requirements to preserve the corresponding additional elements of structure: arc directions, edge weights, etc., with the following exception.

For a graph and a number, the following are two standard isoperimetric parameters for graphs.

In the following example, graph H is a minor of graph G:

For example, consider the following control flow graph:

The following is a simple example of how to do interactive array manipulations and plot a graph with NumPy and Matplotlib.

These conditions are equivalent to the following one: If G is the graph of this vector valued function then for all tangent vectors X, Y of the surface G then s ( X, Y ) = 0 with s the symplectic form.

The oriented or unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L ( G ) by the following theorem:

The differences between the absolute and relative performance measures can be illustrated by the following graph showing marginal and average productivity.

The probability mass function of the random variable X may be depicted by the following bar graph:

In the list of popes given in the Holy See's annual directory, Annuario Pontificio, the following note is attached to the name of Pope Leo VIII ( 963 – 965 ): At this point, as again in the mid-eleventh century, we come across elections in which problems of harmonising historical criteria and those of theology and canon law make it impossible to decide clearly which side possessed the legitimacy whose factual existence guarantees the unbroken lawful succession of the successors of Saint Peter.

On the other hand, for integer order α, the following relationship is valid ( note that the Gamma function becomes infinite for negative integer arguments ):

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

These scholars note parallels with a series of Morisco forgeries, the Sacromonte tablets of Granada, dating from the 1590s ; or otherwise with Morisco reworkings of Christian and Islamic traditions, produced following the expulsion of the Moriscos from Spain.

The following demographic statistics are from the CIA World Factbook ().< ref name =" CIA_20110303 ">< sup >†</ sup > note: estimates for this country explicitly take into account the effects of excess mortality due to AIDS ; this can result in lower life expectancy, higher infant mortality and death rates, lower population and growth rates, and changes in the distribution of population by age and sex than would otherwise be expected < sup >‡</ sup > note: the preliminary 2011 numbers differ significantly from those of 2010, which were strongly influenced by the demographic effect of the January 2010 earthquake ; the latest figures more closely correspond to those of 2009 </ ref >

Let ( A < sub > i </ sub >)< sub > i ∈ I </ sub > be a family of groups and suppose we have a family of homomorphisms f < sub > ij </ sub >: A < sub > j </ sub > → A < sub > i </ sub > for all i ≤ j ( note the order ) with the following properties:

The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following ( note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton – Simon experiment ; it is not applicable to most present-day measurements within the quantum domain ):

* 1969 – The United Kingdom introduces the British fifty-pence coin, which replaces, over the following years, the British ten-shilling note, in anticipation of the decimalization of the British currency in 1971, and the abolition of the shilling as a unit of currency anywhere in the world.

In the following table, the second harmonic ( the lowest normally playable open note, written as middle C ) is presented using scientific pitch notation to identify the octave.

Quantum information differs from classical information in several respects, among which we note the following:

During the programming of a sequence, the user can determine whether a note should be accented, and whether it should employ portamento, a smooth transition to the following note.

In December 1971, following an informative note from Ion Stănescu, the President of the Council of State Security of the Romanian Socialist Republic, to Yuri Andropov, the chief of KGB, three of the leaders of the National Patriotic Front, Alexandru Usatiuc-Bulgar, Gheorghe Ghimpu and Valeriu Graur, as well as a fourth person, Alexandru Soltoianu, the leader of a similar clandestine movement in northern Bukovina ( Bucovina ), were arrested and later sentenced to long prison terms.

The following relations are known between PSPACE and the complexity classes NL, P, NP, PH, EXPTIME and EXPSPACE ( note that is not ):

The Note is introduced by the following words: " A preliminary note of explanation is being given to the Council Fathers from higher authority, regarding the Modi bearing on Chapter III of the Schema de Ecclesia ; the doctrine set forth in Chapter III ought to be explained and understood in accordance with the meaning and intent of this explanatory note.

An accidental sign raises or lowers the following note from its normal pitch ignoring sharps or flats in the key signature, usually by a semitone, although microtonal music may use " fractional " accidental signs, and one occasionally sees double sharps or flats, which raise or lower the indicated note by a whole tone.

# An accidental carries through the measure affecting both the note it immediately precedes and any following notes on the same line or space in the measure.

Although a barline is nowadays understood to cancel the effect of an accidental ( except for a tied note ), often publishers will use a courtesy accidental ( also referred to as a cautionary accidental or a reminder accidental ) as a reminder of the correct pitch if the same note occurs in the following measure.

The following Esperanto libraries and collections of works in the Esperanto language are worthy of note:

On 27 December 1877, while Sullivan was on holiday on the French Riviera, Gilbert sent him a plot sketch accompanied by the following note:

Maximizing the log likelihood function with respect to x < sub > 0 </ sub > and γ produces the following system of equations:

The following identity (" Hadamard Lemma ") involving nested commutators, underlying the Campbell – Baker – Hausdorff expansion of log ( expA expB ), is also useful:

CVS can also run external, user-specified log processing scripts following each commit.

The following year a log courthouse, jail, and " stray pen " were built on the public square facing Main Street.

Gideon Pond, a missionary, who had been following and recording the Dakota language from Cloud Man's band, relocated later that year, establishing Oak Grove Mission, his log cabin.

In 1795, following the signing of the Treaty of Greenville which provided assurances to Ohio settlers that they would be protected from Indian " intrusions ", Samuel and Rachel Dument Pierson established the first log home in the area near Deer Park.

) Although he returned to Delaware after this initial visit, he returned the following year with his son Thomas and built a log cabin on the property.

For most recordings the information given in the daily log included the following:

All of the following operations work in O ( log n ) time on a binomial heap with n elements:

A heavy duty commercial type of halftrack, originally for highway and farm use, carrying twice their own weight with as much as ten times that on trailers or sleds behind, notable in their use motorizing log or freight hauling on ice roads, with full contour following flexible traction but otherwise had the conventional features and operation of the common truck, while engine and driveline components were purchased and casting made elsewhere the rest was built on site of local materials 1917-1952, approx.

This recorded information can be used for the diver's personal log of their activities or as important information in medical review or legal cases following diving accidents.

The following line specifies that the Apache error log is to be cleared at one minute past midnight ( 00: 01 ) of every day of the month, of every day of the week, assuming that the default shell for the cron user is Bourne shell compliant:

The route was realigned in 1931 effective with the following year's route log and was replaced by US 54 from El Paso to Tularosa, NM, US 70 from Alamogordo to Farwell, US 60 from Clovis to Amarillo, and US 84 between Clovis and Farwell in 1935.

* An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then ( log γ )/( log α ) is either rational or transcendental.

In 1942 it was said to be ' well cared for ' in the church log: the following year it was almost destroyed in a raid by a group of German fighter bombers.

According to the ship's log and the journal of Increase Mather, the following was recorded ;

The farmers find Jerry and Tom below after following the sound of them singing " Do You Really Want to Hurt Me ," and Dan comes from the ground and hits them with a log.

For this reason, DDH is considered a stronger assumption than discrete log, in the following sense: there are groups for which detecting DDH tuples is easy, but computing discrete logs is believed to be hard.

According to the ship's log and the journal of Increase Mather, whose father Richard Mather and family were on the James, the following was recorded ;

* Accumulate and log a specified amount of training and experience, including the following:

* Accumulate and log a specified amount of training and experience ; the following are part of the airplane single-engine land class rating requirements:

** Receive and log training, as well as obtain a logbook endorsement from your CFII on the following areas of operation: preflight preparation, preflight procedures, air traffic control clearances and procedures, flight by reference to instruments, navigation systems, instrument approach procedures, emergency operations, and postflight procedures.

Under FAR 61. 57, to be eligible to fly in Instrument Meteorological Conditions ( IMC ) an IFR-rated pilot must accomplish and log at least the following IFR procedures under actual or simulated IMC every 6 months: