For example, the generalized continuum hypothesis ( GCH ) is not only independent of ZF, but also independent of ZFC.

For waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way:

For example, the peculiar development of the works by modernist composer Arnold Schoenberg ( here an " ontogeny ") is generalized in many histories into a " phylogeny " – a historical development (" evolution ") of Western Music toward atonal styles of which Schoenberg is a representative.

For any generalized parabola, with its equation given in the standard form

For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals.

For example, Herculean when referring to Hercules himself, but often herculean when referring to the figurative, generalized extension sense ; and quixotic and diesel engine only.

For example, the generalized transformation of embedding would take the kernel " Dave said X " and the kernel " Dan likes smoking " and combine them into " Dave said Dan likes smoking.

For this reason, the name L4 has been generalized and no longer only refers to Liedtke's original implementation.

* For a compilation of Sioux and Ojibwe ( Chippewa ) Indian Treaties and a generalized map of ceded territories by year, see

For a deterministic finite automaton ( DFA ), nondeterministic finite automaton ( NFA ), generalized nondeterministic finite automaton ( GNFA ), or Moore machine, the input is denoted on each edge.

For each generalized velocity, there is one corresponding conjugate momentum, defined as:

For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.

For example, if the Lagrangian L does not depend on q < sub > i </ sub > itself, then the generalized momentum:

For example, the conservation of the generalized momentum,

For generalized harmonic numbers we have:

For a more mathematical description of the GLM model, see generalized linear models.

For example, consider the generalized hypergraph consisting of two edges and, and zero vertices, so that and.

For the linearly parameterized Hamiltonian, simply stands for the generalized force operator.

For instance, they often make use of " generalized cultural comparisons ," taking various artefacts and monuments from one society, and highlighting similarities with those of another to support a conclusion that both had a common source — typically an ancient lost civilisation like Atlantis, Mu, or an extraterrestrial influence.

For a non-motorized mode of transport, such as walking or cycling, the generalized travel cost may include additional factors such as safety or gradient.

For n functions of several variables, a generalized Wronskian is the determinant of an n by n matrix with entries D < sub > i </ sub >( f < sub > j </ sub >) ( with 0 ≤ i < n ), where each D < sub > i </ sub > is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish.

For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent.

For a given pair of real symmetric positive-definite matrices, and a given non-zero vector, the generalized Rayleigh quotient is defined as:

For although I had crossed a corner of the hall on my way to the toilet I still could not tell for sure how far to the rear the darkness extended.

Hobbes said :" The Latines called Accounts of mony Rationes ... and thence it seems to proceed that they extended the word Ratio, to the faculty of Reckoning in all other things .... When a man reasoneth hee does nothing else but conceive a summe totall ... For Reason ... is nothing but Reckoning ... of the consequences of generall names agreed upon, for the marking and signifying of our thoughts ...."

For π-electron systems, this was the Hückel method proposed by Erich Hückel, and for all valence electron systems, the extended Hückel method proposed by Roald Hoffmann.

For example, frogs in cold climates can survive for extended periods of time with most of their body water in a frozen state, while desert frogs in Australia can become inactive and dehydrate in dry periods, losing up to 75 % of their fluids, yet return to life by rapidly rehydrating in wet periods.

For example, Ralph Waldo Emerson ’ s contempt for Jane Austen's works often extended to the author herself, with Emerson describing her as “ without genius, wit, or knowledge of the world .” In turn, Emerson himself was called a “ hoary-headed toothless baboon ” by Thomas Carlyle.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

For instance, in 1917 the Cuyamel Fruit Company extended its rail lines into disputed Guatemalan territory.

For non-executive heads of state there is often a degree of censorship by the politically responsible government ( such as the head of government ), discreetly approving agenda and speeches, especially where the constitution ( or customary law ) assumes all political responsibility by granting the crown inviolability ( in fact also imposing political emasculation ) as in the Kingdom of Belgium from its very beginning ; in a monarchy this may even be extended to some degree to other members of the dynasty, especially the heir to the throne.

For some events they can be made in extended forms without a frame at all, but only a long parallel rank of laser beams yet are still called laser harps.

Coke's preeminence extended across the ocean: " For the American revolutionary leaders, ' law ' meant Sir Edward Coke's custom and right reason.

For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola.

For the first time, the Donation made the pope a temporal ruler over a strip of territory that extended diagonally across Italy from the Tyrrhenian to the Adriatic.

For questions not directly addressed in the primary sources, the application of sharia is extended through consensus of the religious scholars ( ulama ) thought to embody the consensus of the Muslim Community ( ijma ).

For the 1978 – 79 season, the NFL extended its schedule from 14 regular season games to 16, and increased the playoffs from an 8-team tournament to 10, creating two extra playoff games.

For many Togolese, there was little optimism for the future and a prevailing sense of déjà vu as Eyadéma extended his record as Africa's longest-serving ruler.

For example, the old style of the King's head was more rectangular, had a crown with identifiable card suits on it ( club, diamond, heart ), his mustache and beard always hid his mouth, and his beard frequently extended to a curved point when the King was shown in profile ( see The Wondrous Wizard of Id, 1970, Fawcett Publications ).

For legal purposes, the definition of a seal may be extended to include rubber stamps, or writing specified words (" seal " or " L. S.

For works authored by individuals on or after January 1, 1978 ( including new works ), the copyright term was extended to equal the life of the author plus 70 years.

For works authored by joint authors, the copyright term was extended to the life of the last surviving author plus 70 years.

For example, extended access lists can filter packets based on layer 4 port numbers.

For example, the Tayronas of Colombia's Sierra Nevada de Santa Marta used to chew the plant before engaging in extended meditation and prayer.

For the purposes of proving the Penrose – Hawking singularity theorems, a spacetime with a singularity is defined to be one that contains geodesics that cannot be extended in a smooth manner.

For instance, Schrödinger originally viewed the wavefunction associated with the electron as corresponding to the charge density of an object smeared out over an extended, possibly infinite, volume of space.

For example, football has a higher chance of being understood than foxtrot in isolation, but foxtrot is superior in extended communication.