Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem.

Their work was a first effort to create an abstract specification and invariant basis for designing different alternative implementations using different hardware components.

The upshot is that the first fundamental form () is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E, F, and G. Indeed, by the chain rule,

Its cohomology class is, up to a real constant factor, the first Chern class of the canonical bundle, and is therefore a topological invariant of X ( for X compact ) in the sense that it depends only on the topology of X and the homotopy class of the complex structure.

The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking.

) so the first invariant is:

Yau also made a contribution in the case that the first Chern number c < sub > 1 </ sub > > 0, and conjectured its relation to the stability in the sense of geometric invariant theory in algebraic geometry.

The first invariant says that:

The first advantage is the primacy of Rasch's specific requirements, which ( when met ) provides fundamental person-free measurement ( where persons and items can be mapped onto the same invariant scale ).

For example, the genus of a smooth projective curve C which is an interesting invariant of the curve, is an integer, which can be read off the dimension of the first Betti cohomology group of C. So, the motive of the curve should contain the genus information.

Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that K is in fact an intrinsic invariant of the surface.

A normalized first Casimir invariant operator can likewise be defined

The graphs with Colin de Verdière graph invariant at most μ, for any fixed constant μ, form a minor-closed family, and the first few of these are well-known: the graphs with μ ≤ 1 are the linear forests ( disjoint unions of paths ), the graphs with μ ≤ 2 are the outerplanar graphs, and the graphs with μ ≤ 3 are the planar graphs.

Because the von Mises yield criterion is independent of the first stress invariant,, it is applicable for the analysis of plastic deformation for ductile materials such as metals, as the onset of yield for these materials does not depend on the hydrostatic component of the stress tensor.

In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert ( 1890 ) in connection with the syzygy ( relation ) problem of invariant theory.

According to Joseph Greenberg's linguistic universal No. 12, " If a language has dominant order VSO in declarative sentences, it always puts interrogative words or phrases first in interrogative word questions ; if it has dominant order SOV in declarative sentences, there is never such an invariant rule.

Scarf published his first scientific article `` Group invariant integration and the fundamental theorem of algebra ” in the Proceedings of the National Academy of Sciences, in May, 1952.

The first flownet pictured here ( modified from Craig, 1997 ) illustrates and quantifies the flow which occurs under the dam ( flow is assumed to be invariant along the axis of the dam — valid near the middle of the dam ); from the pool behind the dam ( on the right ) to the tailwater downstream from the dam ( on the left ).

Much of the development of invariant theory in the first half of the twentieth century concerned explicit computations with invariants, and at any rate, followed the logic of algebra rather than geometry.

Geometric invariant theory was founded and developed by Mumford in a monograph, first published in 1965, that applied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometry questions.

In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II < sub > 1 </ sub > subfactor.

The first step in the Cartan method is to express the pullback relation ( 1 ) in as invariant a way as possible through the use of a " prolongation ".

Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

The quantum number n first appeared in the Bohr model where it determines the radius of each circular electron orbit.

Initially, it was relatively unsuccessful, staying at the charts for only one week, but Haley soon scored a major worldwide hit with a cover version of Big Joe Turner's " Shake, Rattle and Roll ", which went on to sell a million copies and became the first ever rock ' n ' roll song to enter British singles charts in December 1954 and became a Gold Record.

This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power one temporarily labels the term X with an index i ( running from 1 to n ), then each subset of k indices gives after expansion a contribution X < sup > k </ sup >, and the coefficient of that monomial in the result will be the number of such subsets.

Bernoulli numbers feature prominently in the closed form expression of the sum of the m-th powers of the first n positive integers.

The first and second Bernoulli numbers now follow by setting n = 0 ( resulting in B < sub > 1 </ sub >=−, " first Bernoulli numbers ") respectively n = 1 ( resulting in B < sub > 1 </ sub >=+, " second Bernoulli numbers ").

from which it is clear that and for k > n. To see that these coefficients count k-combinations from S, one can first consider a collection of n distinct variables X < sub > s </ sub > labeled by the elements s of S, and expand the product over all elements of S:

This means that the shortest program to output the first n bits of Ω must be of size at least n-O ( 1 ).

A real number is called computable if there is an algorithm which, given n, returns the first n digits of the number.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

As mentioned above, the first n bits of Gregory Chaitin's constant Omega are random or incompressible in the sense that we cannot compute them by a halting algorithm with fewer than n-O ( 1 ) bits.

After finite time the first n bits of the output will never change any more ( it does not matter that this time itself is not computable by a halting program ).

Athlon was the first x86 processor with a 128 kB split level 1 cache ; a 2-way associative, later 16-way, cache separated into 2 × 64 kB for data and instructions ( Harvard architecture ).

The first American-made pocket-sized calculator, the Bowmar 901B ( popularly referred to as The Bowmar Brain ), measuring 5. 2 × 3. 0 × 1. 5 in ( 131 × 77 × 37 mm ), came out in the Autumn of 1971, with four functions and an eight-digit red LED display, for $ 240, while in August 1972 the four-function Sinclair Executive became the first slimline pocket calculator measuring 5. 4 × 2. 2 × 0. 35 in ( 138 × 56 × 9 mm ) and weighing 2. 5 oz ( 70g ).

This is a functor which is contravariant in the first and covariant in the second argument, i. e. it is a functor Ab < sup > op </ sup > × Ab → Ab ( where Ab denotes the category of abelian groups with group homomorphisms ).

This defines a functor to Set which is contravariant in the first argument and covariant in the second, i. e. it is a functor C < sup > op </ sup > × C → Set.

To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 10 < sup > 5 </ sup > to give 1. 528535047 × 10 < sup > 5 </ sup >, or 152853. 5047.

The first aspect of the paradox, " the argument by scale ", is a function of the raw numbers involved: there are an estimated 200 – 400 billion ( 2 – 4 × 10 < sup > 11 </ sup >) stars in the Milky Way and 70 sextillion ( 7 × 10 < sup > 22 </ sup >) in the visible universe.

A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of n. The simple formula for the factorial, n < nowiki >!</ nowiki > = 1 × 2 × … × n, cannot be used directly for fractional values of n since it is only valid when n is a natural number ( i. e., a positive integer ).