Banach spaces are a different generalization of Hilbert spaces.

It is a generalization of the plane into a different number of dimensions.

* In quantum field theory and string theory, a generalization of coherent states to the case of infinitely many degrees of freedom is used to define a vacuum state with a different vacuum expectation value from the original vacuum.

* A different type of generalization is the normal-Wishart distribution, essentially the product of a multivariate normal distribution with a Wishart distribution.

Similarly, Plattner ( 1989 ) posited that some generalization across different societies are still possible, meaning that Western and non-Western economics are not entirely different.

This extra structure is necessary because there is no canonical way to compare vectors from different vector spaces, as is necessary for this generalization of the directional derivative.

A generalization of the gyrator is conceivable, in which the forward and backward gyration conductances have different magnitudes, so that the admittance matrix is

Klug agrees that it is a prejudice, because it is a generalization about individuals ; nevertheless, he argues, it is " not rooted in the ideology of ' the Jew '," and is therefore a different phenomenon from antisemitism.

Other generalizations, such as a " votes-only " generalization that makes no reference to voter preferences, may result in a different analysis.

Kleinberg realized that this generalization implies two different classes of important web pages, which he called " hubs " and " authorities ".

It thus operates with a fairly high level of generalization, but does also allow for some redundancy in that it recognizes extensions of different types.

In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties ; two different generalizations are in common use, Cartier divisors and Weil divisors ( named for Pierre Cartier and André Weil ).

At first, this may appear rather different from the Bayes rule approach of the previous section, not a generalization.

Shaping assists in " discrimination ", which is the ability to tell the difference between stimuli that are and are not reinforced, and in " generalization ", which is the application of a response learned in one situation to a different but similar situation.

* A language may use different sets of symbols or different rules for distinct sets of vocabulary items, such as the Japanese hiragana and katakana syllabaries, or the various rules in English for spelling words from Latin and Greek, or the original Germanic vocabulary.

For English, this is partly because the Great Vowel Shift occurred after the orthography was established, and because English has acquired a large number of loanwords at different times, retaining their original spelling at varying levels.

Note that this is subtly different to the ' thanks ' list in the original BBC Model B.

Because the original designer had left the company another employee completely redesigned most of the system, ( adding a display snow remover circuit, true 80 / 64 column text mode support, ( with different size letters for TRS-80 and CP / M mode, so that in TRS-80 mode the full screen was also used, not just a 64x16 portion of the 80x25 screen ) with an improved font set ( adding " gray scale " version of the TRS-80 mozaik graphics and many special PETSCII like characters ), and a more flexible and reliable floppy disk controller and keyboard interface plus many other small improvements ), also an enclosure was developed for the main computer system, ( in the form of a 19-inch rack for the Eurocards ) and for two floppy disk drives and the power supply.

An existing pop song can be re-recorded with a different arrangement to the original.

Two different versions of the original game were released.

This discovery has shed much light on the differences between the two versions ; while it was previously maintained that the Greek Septuagint ( the version used by the earliest Christians ) was only a poor translation, professor Emanuel Tov, senior editor of the Dead Sea Scrolls ' publication, wrote that the Masoretic edition either represents a substantial rewriting of the original Hebrew, or there had previously been two different versions of the text.

Iterated product ciphers carry out encryption in multiple rounds, each which uses a different subkey derived from the original key.

These supplements may be set in different time frames or even different game universes from the original game.

However, the original Counter-Strike is still available and played by many people via Steam, as the two variants are quite different, and players inevitably prefer one variant over the other.

German SABA also released a few compatible carts different from the original carts, translation in Videocart 1 Tic-Tac-Toe to German words, Videocart 3 released with different abbreviations ( German ), Videocart 18 changed graphics and German word list and the SABA 20 that's a Chess game released only by SABA.

It offers different and unique flavors like a cloves aroma, bottled in an original cider bottle.

Aliovalent substitution is where the ion that is substituting the original ion is of a different oxidation state as the ion it is replacing.

The ratio between A and B in the vapor will be different from the ratio in the liquid: the ratio in the liquid will be determined by how the original mixture was prepared, while the ratio in the vapor will be enriched in the more volatile compound, A ( due to Raoult's Law, see above ).

Diasporic cultural development often assumes a different course from that of the population in the original place of settlement.

However the new version of the second Superman film ( known as The Richard Donner Cut ) restores as much of the original director's conception as possible, making it a considerably different picture.

Sometimes, a different actor is used from the original actor on set.

There have been dozens of arcade-based releases across several countries and hundreds of home video game console releases, promoting a music library of original songs produced by Konami's in-house artists and an eclectic set of licensed music from many different genres.

The intersection of any two different cells is empty ; the union of all the cells equals the original set.

In different Indo-European languages, each of these words has a difficult etymology because of taboo deformations — a euphemism was substituted for the original, which no longer occurs in the language.

There were a few changes to the rules in the 2nd Edition ; some classes were slightly different or altered abilities from the original.

Note that this is different from the original Elfengold published by White Wind.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

That is, energy is conserved because the laws of physics do not distinguish between different instants of time ( see Noether's theorem ).

Eberhard, Bell's theorem and the different concepts of locality.

Thus, in a sense, there is a different completeness theorem for each deductive system.

Hayes ( 1973 ) developed an equational language, Golux, in which different procedures could be obtained by altering the behavior of the theorem prover.

One obtains the value f ( r ) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function ( see Fermat's little theorem for an example where R is the integers modulo p ).

The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules.

In this case A is called the hypothesis of the theorem ( note that " hypothesis " here is something very different from a conjecture ) and B the conclusion ( A and B can also be denoted the antecedent and consequent ).

Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem.

In some cases, as the relative importance of different theorems becomes more clear, what was once considered a lemma is now considered a theorem, though the word " lemma " remains in the name.

This problem may be alleviated by using a different fitness function, increasing the rate of mutation, or by using selection techniques that maintain a diverse population of solutions, although the No Free Lunch theorem proves that there is no general solution to this problem.

Bayes ' theorem links these probabilities, which are in general different.

One proof of the impossibility of finding a planar embedding of K < sub > 3, 3 </ sub > uses a case analysis involving the Jordan curve theorem, in which one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding.

On curved surfaces, the formula for C ( r ) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand – Diquet – Puiseux theorem as

Stronger forms of Dirichlet's theorem state that, for any arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges, and that different arithmetic progressions with the same modulus have approximately the same proportions of primes.

It is also known that there exist arbitrarily long finite arithmetic progressions consisting only of primes, but this is a different result, known as the Green – Tao theorem.

* Experimentally finding the centroid of a triangle with different weights at the vertices: a practical application of Ceva's theorem at Dynamic Geometry Sketches, an interactive dynamic geometry sketch using the gravity simulator of Cinderella.

This also is a special case of Ramsey's theorem, which says that for any given integer c, any given integers n < sub > 1 </ sub >,..., n < sub > c </ sub >, there is a number, R ( n < sub > 1 </ sub >,..., n < sub > c </ sub >), such that if the edges of a complete graph of order R ( n < sub > 1 </ sub >,..., n < sub > c </ sub >) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order n < sub > i </ sub > whose edges are all colour i. The special case above has c

* Van der Waerden's theorem: For any given c and n, there is a number V, such that if V consecutive numbers are coloured with c different colours, then it must contain an arithmetic progression of length n whose elements are all the same colour.

A theorem similar to van der Waerden's theorem is Schur's theorem: for any given c there is a number N such that if the numbers 1, 2, ..., N are coloured with c different colours, then there must be a pair of integers x, y