Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ( such as the process for a six-sided die ); this generalization is known as the Bernoulli scheme.

An abstract algebraic variety is a particular kind of scheme ; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings.

The sheaf of rational functions K < sub > X </ sub > of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry.

After completing a doctorate under the supervision of Alexander Grothendieck, he worked with him at the Institut des Hautes Études Scientifiques ( IHÉS ) near Paris, initially on the generalization within scheme theory of Zariski's main theorem.

One high-dimensional generalization scheme which maximizes the mutual information between the joint distribution and other target variables is found to be useful in feature selection.

This scheme is a generalization of linear interpolation.

A generalization of the Proj construction replaces the ring S with a sheaf of algebras and produces, as the end result, a scheme which might be thought of as a fibration of Proj's of rings.

In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes.

Sir Henry Sumner Maine, a hundred years before Communism was a force to be reckoned with, wrote his brilliant legal generalization, that `` the progress of society is from status to contract ''.

The culmination of their investigations, the Arzelà – Ascoli theorem, was a generalization of the Bolzano – Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions.

Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.

Although Bischoff ’ s 1840 surveys depict how similar the early embryos of man are to other vertebrates, he later demanded that such hasty generalization was inconsistent with his recent findings regarding the dissimilarity between hamster embryos and those of rabbits and dogs.

This generalization suffered from many exceptions, and was subsequently replaced by the suggestion that valences were fixed at certain oxidation states.

He is the third, and possibly most important, pillar of that field ( he soon was the only one to have discovered a relativistic generalization of the theory ).

( as opposed to the general G = T theory ) was amenable to exact solutions in terms of a generalization of the Lambert W function.

Perhaps the most novel feature was the use of a point-based skill system, an extension and generalization of the magic system inherited from Wizard.

That idea, however, was only an unfortunate generalization from a technological limitation.

Although physical modelling was not a new concept in acoustics and synthesis, having been implemented using finite difference approximations of the wave equation by Hiller and Ruiz in 1971, it was not until the development of the Karplus-Strong algorithm, the subsequent refinement and generalization of the algorithm into the extremely efficient digital waveguide synthesis by Julius O. Smith III and others, and the increase in DSP power in the late 1980s that commercial implementations became feasible.

Henri Léon Lebesgue ForMemRS (; June 28, 1875 – July 26, 1941 ) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration — summing the area between an axis and the curve of a function defined for that axis.

However, this generalization was statistical and was not approved of by de Broglie, who said " that the particle must be the seat of an internal periodic movement and that it must move in a wave in order to remain in phase with it was ignored by the actual physicists are wrong to consider a wave propagation without localization of the particle, which was quite contrary to my original ideas.

Formal generalization of the pi – theorem for the case of arbitrary number of quantities was for the first time given by A. Vaschy in 1892, and later and, apparently, independently, by A. Federman, D. Riabouchinsky in 1911 and by Buckingham in 1914.

They say that implicit in the behavior of Perls and Satir was the ability to challenge distortion, generalization and deletion in a client's language.

* Chirplet transform, 1991: Mann was the first to propose and reduce to practice a signal representation based on a family of chirp signals, each associated with a coefficient, in a generalization of the wavelet transform that is now referred to as the chirplet transform.

The current algorithm with the lowest known exponent k is a generalization of the Coppersmith – Winograd algorithm that has an asymptotic complexity of O ( n < sup > 2. 3727 </ sup >) thanks to Vassilevska Williams .< ref > The original algorithm was presented by Don Coppersmith and Shmuel Winograd in 1990, has an asymptotic complexity of O ( n < sup > 2. 376 </ sup >).</ ref > This algorithm, and the Coppersmith-Winograd algorithm on which it is based, are similar to Strassen's algorithm: a way is devised for multiplying two k × k-matrices with fewer than k < sup > 3 </ sup > multiplications, and this technique is applied recursively.

In line with this generalization, sertraline was no better than placebo in inpatients ( see History ) and as effective as the TCA clomipramine for severe depression.

Hintikka's example is: "... a generalization like ' no material bodies are infinitely divisible ' seems to be completely unaffected by questions concerning immaterial entities, independently of what one thinks of the relative frequencies of material and immaterial entities in one's universe of discourse.

In 1927, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation.

An exception to this generalization must be pointed out where groups of individuals are dancing independently of each other, but with the purpose of creating a " group " feeling or experience, such as might accompany various forms of ritual dancing.

According to and, the first mathematicians to prove independently the theorem in the nowadays common abstract measure space setting were Frigyes Riesz in,, and Wacław Sierpiński in: an earlier generalization is due to Nikolai Luzin, who succeeded in slightly relaxing the requirement of finiteness of measure of the domain of convergence of the pointwise converging functions in the ample paper, as recalls.

Schrödinger, during the same period that he discovered his famous equation in 1926, also independently found the relativistic generalization of it known as the Klein-Gordon equation but dismissed it since, without spin, it predicted impossible properties for the hydrogen spectrum.

In 1965 Donald Knuth invented the LR ( k ) parser ( Left to right, Rightmost derivation parser ) a type of shift-reduce parser, as a generalization of existing precendence parsers.

He invented the Turán graph, a generalization of the complete bipartite graph, to prove his theorem.

MSP is a generalization of the simple precedence parser method invented by Niklaus Wirth for PL360.

Nevertheless, the extended semi-detailed balance conditions invented by Boltzmann in this discussion remain the remarkable generalization of the detailed balance.

In music a sound object ( objet sonore: Pierre Schaeffer 1959, 1977, p. 95 ), a generalization of the concept of a musical note, is any sound from any source which in duration is on the time scale of 100 ms to several seconds.

He shifted attention from the study of individual varieties to the relative point of view ( pairs of varieties related by a morphism ), allowing a broad generalization of many classical theorems.

A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate ( i. e. ) the subalgebra generated by those elements is associative.

A generalization of the class of Horn formulae is that of renamable-Horn formulae, which is the set of formulae that can be placed in Horn form by replacing some variables with their respective negation.

The maximum satisfiability problem, an FNP generalization of SAT, asks for the maximum number of clauses which can be satisfied by any assignment.

This generalization can be proved by comparing derivatives.

Adapting the algorithm to use an additional array to store the shape of the region allows generalization to cover " fuzzy " flood filling, where an element can differ by up to a specified threshold from the source symbol.

Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization, the post-Newtonian expansion, both of which were developed by Einstein.

Enumerative graph theory then rose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937 and the generalization of these by De Bruijn in 1959.

The study and the generalization of this problem by Tait, Heawood, Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus.

The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods.

Multiplication of rational numbers ( fractions ) and real numbers is defined by systematic generalization of this basic idea.

A further generalization is given by the Veronese variety, when there is more than one input variable.

The apparent paradox is caused by a hasty generalization, for if the surgeon is the boy's father, the statement cannot be true.

They were a generalization of the concept of ideal numbers developed by Ernst Kummer.

Further, there is a generalization of the Stone – Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone – Weierstrass theorem and described below.

This modern form of Stokes ' theorem is a vast generalization of a classical result first discovered by Lord Kelvin, who communicated it to George Stokes in July 1850.

The immediate predecessor of vectors were quaternions, devised by William Rowan Hamilton in 1843 as a generalization of complex numbers.

This is a generalization of Einstein's equation above, since the momentum of a photon is given by p

An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series.