We have a language where is a constant symbol and is a unary function and the following axioms:

Area can be defined as a function from a collection M of special kind of plane figures ( termed measurable sets ) to the set of real numbers which satisfies the following properties:

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

/* The following function will print a non-negative number, n, to

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 auxiliary verb function derives from the copular function ; and, depending on one's point of view, one can still interpret the verb as a copula and the following verbal form as being adjectival.

It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g ′( a ) and a function ε ( h ) that tends to zero as h tends to zero, and furthermore

In 3 dimensions, a differential 0-form is simply a function f ( x, y, z ); a differential 1-form is the following expression: a differential 2-form is the formal sum: and a differential 3-form is defined by a single term: ( Here the a-coefficients are real functions ; the " wedge products ", e. g. can be interpreted as some kind of oriented area elements,, etc.

The function F is called universal if the following property holds: for every computable function f of a single variable there is a string w such that for all x, F ( w x ) = f ( x ); here w x represents the concatenation of the two strings w and x.

A real number a is said to be computable if it can be approximated by some computable function in the following manner: given any integer, the function produces an integer k such that:

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

For example, consider the following C ++ function:

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

It has the following universal property: if N is any complete metric space and f is any uniformly continuous function from M to N, then there exists a unique uniformly continuous function f ' from M ' to N, which extends f. The space M is determined up to isometry by this property, and is called the completion of M.

There is a product rule of the following type: if is a scalar valued function and F is a vector field, then

The majority of patients go through a period of spontaneous recovery following brain injury in which they regain a great deal of language function.

Your application must include the following information: ( 1 ) your reasons for requesting an extension, ( 2 ) whether you filed timely income tax returns for the 3 preceding years, and ( 3 ) whether you were required to file an estimated return for the year, and if so whether you did file and have paid the estimated tax payments on or before the due dates.

The parent of a group filing consolidated returns might be treated as the same corporation following a reorganization defined in section 368(a)(1), but as a different corporation for this purpose after a tax-free acquisition by another corporation which had not, for example, elected to file consolidated returns with its own subsidiaries.

Caligula, by French author Albert Camus, is a play in which Caligula returns after deserting the palace for three days and three nights following the death of his beloved sister, Drusilla.

Psychiatrist Dr. Frasier Crane ( Grammer ) returns to his hometown of Seattle, Washington, following the end of his marriage and his life in Boston ( as seen in Cheers ).

Achilles, moved by Priam's actions and following his mother's orders sent by Zeus, returns Hector's body, and promises Priam a truce of twelve days to allow the Trojans to perform funeral rites for Hector.

A male Ivory-bill leaving the nest as the female returns, this and the following photos were taken in Singer Tract, Louisiana, April 1935 by Arthur A. Allen

Initially successful with annualized returns of over 40 % ( after fees ) in its first years, in 1998 it lost $ 4. 6 billion in less than four months following the Russian financial crisis requiring financial intervention by the Federal Reserve, with the fund liquidating and dissolving in early 2000.

The following hand has one card, then progressively returns to the maximum number of cards ( again with a maximum of 9 ).

( In the following, we will say that " returns " if either, or both and are undefined.

* 2007 – Former Prime Minister of Pakistan Nawaz Sharif returns to Pakistan after seven years in exile, following a military coup in October 1999.

Coleman's 8 kickoff returns for 244 yards and a touchdown set the following Super Bowl records: most kickoff returns in a Super Bowl, most kickoff return yards in a Super Bowl, and most combined net yards gained in a Super Bowl.

Ryan returns to the United States, where he and several others attend the funeral of Filitov, who had died of heart disease in the months following his CIA debriefing period.

After the exploration party returns to base, the Greenlanders decide to return home the following spring.

** Newfoundland returns to Crown Colony status following financial collapse.

* March 16 – Louis Antoine de Bougainville returns to Saint-Malo following a three-year circumnavigation of the world with the ships La Boudeuse and Étoile, with the loss of only seven out of 330 men ; among the returning crew is Jeanne Baré, the first woman known to have circumnavigated the globe.

Manon dies of exposure and exhaustion the following morning ; after burying her, Des Grieux returns to France to become a cleric.

* July 27 – The Bank of England is founded through Royal charter by the Whig-dominated Parliament of England following a proposal by the Scottish merchant William Paterson to raise capital by offering safe and steady returns of interest guaranteed by future taxes.

* September – Cosimo de ' Medici returns to Florence one year, to the day, following his exile by the Alberti and Strozzi faction.

He returns the following year.

The following morning at Cartier, Kate returns the stolen necklace to the inspector and purchases a Cartier check for $ 45, 782 with her own savings to create the illusion that she actually sold the necklace.

As a pedagogical exercise, a physics university instructor might ask what the aftermath would be, as nature returns to normal, following a hypothetical miraculous intervention by God, similar to a modern thought experiment.

He is redeemed by his friendship with Brother François, and following the friar's death he returns to Italy to find Bonnyfeather has died and his housekeeper, Faith Paleologus ( now married to Don Luis ), will inherit the man's estate fortune unless Anthony goes to Paris to claim his inheritance.

As an example, the over operator can be accomplished by applying the following formula to each pixel value:

Tolstoy defined art ( and by no coincidence also characterized its value ) as the following: " Art is a human activity consisting in this, that one man consciously, by means of certain external signs, hands on to others feelings he has lived through, and that other people are infected by these feelings and also experience them.

The following routine prints the value of AC1 as a 16 digit binary number, on the RDOS console.

The multiplicity depends on the value of N modulo 4, and is given by the following table:

These rights were subject to the following conditions: only good quality goods were to be supplied by the Faroese and were to be made in numbers proportionate to the rest of the market ; the goods were to be brought at their market value ; and the traders were to deal fairly and honestly with the Faroese.

This code defines a new word ( again, word is the term used for a subroutine ) called using the following commands: duplicates the number on the stack ; places a 6 on top of the stack ; compares the top two numbers on the stack ( 6 and the ed input ), and replaces them with a true-or-false value ; takes a true-or-false value and chooses to execute commands immediately after it or to skip to the ; discards the value on the stack ; and ends the conditional.

Heilbrun argued that men have for centuries completely misinterpreted Gertrude, accepting at face value Hamlet's view of her instead of following the actual text of the play.

Older economies would revert to hard currency and barter when the circulating medium became excessively devalued, generally following a " run " on the store of value.

They were used extensively following the collapse of Lehman Brothers in September 2008, when the market for certain types of assets held by hedge funds collapsed, preventing the funds from selling or obtaining a market value for the assets.

Following the 1869 finalisation of UK telegraph nationalisation into a General Post Office monopoly, the Isle of Man Telegraph Company was nationalised in 1870 under the Telegraph Act 1870 ( an Act of Parliament ) at a cost to the British Government of £ 16, 106 ( paid in 1872 following arbitration proceedings over the value ).

The intermediate value theorem states the following: If f is a real-valued continuous function on the interval b, and u is a number between f ( a ) and f ( b ), then there is a c ∈ b such that f ( c ) = u.

The CISA Review Manual 2006 provides the following definition of risk management: " Risk management is the process of identifying vulnerabilities and threats to the information resources used by an organization in achieving business objectives, and deciding what countermeasures, if any, to take in reducing risk to an acceptable level, based on the value of the information resource to the organization.

IEEE 802. 3 is an exception ; Ethernet originally used Ethernet II framing, with a 2-octet type field containing an EtherType value, and no 802. 2 header, but the original IEEE 802. 3 specification used a framing wherein the 2-octet field following the 6-octet destination and source MAC addresses contained a frame length, with an 802. 2 LLC header following the length field.

Jains are encouraged to reflect on these thoughts with the following four virtues or value systems clearly in force.

He states that there is at least the possibility of another type of nihilist in the wake of Christianity's self-dissolution, one that does not stop after the destruction of all value and meaning and succumb to the following nothingness.

The following C language fragment is syntactically correct, but performs operations that are not semantically defined ( the operation < tt >* p >> 4 </ tt > has no meaning for a value having a complex type and < tt > p -> im </ tt > is not defined because the value of < tt > p </ tt > is the null pointer ):

A review and evalutation of scholarly work regarding the current state of the value of individuals ' privacy of online social networking show the following results: " first, adults seem to be more concerned about potential privacy threats than younger users ; second, policy makers should be alarmed by a large part of users who underestimate risks of their information privacy on OSNs ; third, in the case of using OSNs and its services, traditional one-dimensional privacy approaches fall short ".

In the following table, n is the value of the header byte as a signed integer.

The following chart shows the points awarded based on the point value of cards taken during the hand.