** the classical Lie groups, namely the simple groups related to the projective special linear, unitary, symplectic, or orthogonal transformations over a finite field ;

There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.

A finite projective plane of order q, with the lines as blocks, is an S ( 2, q + 1, q < sup > 2 </ sup >+ q + 1 ), since it has q < sup > 2 </ sup >+ q + 1 points, each line passes through q + 1 points, and each pair of distinct points lies on exactly one line.

Galois also constructed the projective special linear group of a plane over a prime finite field, PSL ( 2, p ), and remarked that they were simple for p not 2 or 3.

The above is for the classical unitary group ( over the complex numbers ) – for unitary groups over finite fields, one similarly obtains special unitary and projective unitary groups, but in general.

The four finite volume manifolds with this geometry are: S < sup > 2 </ sup > × S < sup > 1 </ sup >, the mapping torus of the antipode map of S < sup > 2 </ sup >, the connected sum of two copies of 3 dimensional projective space, and the product of S < sup > 1 </ sup > with two-dimensional projective space.

Another field that emerged from axiomatic studies of projective geometry is finite geometry.

There are many projective geometries, which may be divided into discrete and continuous: a discrete geometry comprises a set of points, which may or may not be finite in number, while a continuous geometry has infinitely many points with no gaps in between.

In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity

The number of points on the projective line and projective space are so easy to calculate because they can be written as disjoint unions of a finite number of copies of affine spaces.

* After blowing up V and extending the base field, one may assume that the variety V has a morphism onto the projective line P < sup > 1 </ sup >, with a finite number of singular fibers with very mild ( quadratic ) singularities.

Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings ( see also the history section in the Algebraic Geometry article ).

** Clement Lam's proof of the non-existence of a finite projective plane of order 10.

While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity.

Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field ; the affine and projective planes so constructed are called Galois geometries.

Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field ( that is, the projectivization of a vector space over a finite field ).

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

in a Hilbert space can be extended to subspaces of any finite dimensions.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets ( for spaces only requiring finite additivity, such as the ba space ).

In 1931 Lemaître went further and suggested that the evident expansion of the universe, if projected back in time, meant that the further in the past the smaller the universe was, until at some finite time in the past all the mass of the Universe was concentrated into a single point, a " primeval atom " where and when the fabric of time and space came into existence.

The universe described by the Einstein model is static ; space is finite and unbounded ( analogous to the surface of a sphere, which has a finite area but no edges ).

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

* Any finite topological space, including the empty set, is compact.

Slightly more generally, any space with a finite topology ( only finitely many open sets ) is compact ; this includes in particular the trivial topology.

# X has a sub-base such that every cover of the space by members of the sub-base has a finite subcover ( Alexander's sub-base theorem )

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense.

Essentially, the rendering process tries to depict a continuous function from image space to colors by using a finite number of pixels.

Still, in the absence of naked singularities, the universe is deterministic — it's possible to predict the entire evolution of the universe ( possibly excluding some finite regions of space hidden inside event horizons of singularities ), knowing only its condition at a certain moment of time ( more precisely, everywhere on a spacelike 3-dimensional hypersurface, called the Cauchy surface ).

The related concept of " standard " numbers, which can only be defined within a finite time and space, is used to motivate axiomatic internal set theory, and provide a workable formulation for illimited and infinitesimal number.

The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space.

The differential equations determining the evolution function Φ < sup > t </ sup > are often ordinary differential equations: in this case the phase space M is a finite dimensional manifold.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

μ is a finite measure on the sigma-algebra, so that the triplet ( X, Σ, μ ) is a probability space.

Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1-4 are consistent with either infinite or finite space ( as in elliptic geometry ), and all five axioms are consistent with a variety of topologies ( e. g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry ).

* Pro-finite groups are defined as inverse limits of ( discrete ) finite groups.

Lie groups are often defined to be finite dimensional, but there are many groups that resemble Lie groups, except for being infinite dimensional.

Typically, when the sample space is finite, any subset of the sample space is an event ( i. e. all elements of the power set of the sample space are defined as events ).

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Power law distributions have a well defined mean only if the exponent exceeds 1 and have a finite variance only when the exponent exceeds two.

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

Although symmetric groups can be defined on infinite sets as well, this article discusses only the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory.

A universal computer is defined as a device with a Turing complete instruction set, infinite memory, and an infinite lifespan ; all general purpose programming languages and modern machine instruction sets are Turing complete, apart from having finite memory.

Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1.

The finite difference of higher orders can be defined in recursive manner as

* A generalized finite difference is usually defined as

One advantage of this restriction is that the structures studied in universal algebra can be defined in any category which has finite products.

The numerical value of an infinite continued fraction will be irrational ; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions.

This group can be defined as the increasing union of the finite simple groups with respect to standard embeddings.

For finite random sequences, Kolmogorov defined the " randomness " as the entropy, i. e. Kolmogorov complexity, of a string of length K of zeros and ones as the closeness of its entropy to K, i. e. if the complexity of the string is close to K it is very random and if the complexity is far below K, it is not so random.

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

If L ′ is of finite height, or at least verifies the ascending chain condition ( all ascending sequences are ultimately stationary ), then such an x ′ may be obtained as the stationary limit of the ascending sequence x ′< sub > n </ sub > defined by induction as follows: x ′< sub > 0 </ sub >=⊥ ( the least element of L ′) and x ′< sub > n + 1 </ sub >= f ′( x ′< sub > n </ sub >).

In the laboratory, uncertainty is eliminated and calculating the expected returns should be a simple mathematical exercise, because participants are endowed with assets that are defined to have a finite lifespan and a known probability distribution of dividends.

Let E be the set of real numbers that can be defined by a finite number of words.

But N has been defined by a finite number of words.

All real numbers which can be defined by a finite number of words form a subset of the real numbers.

If the real numbers can be well-ordered, then there must be a first real number ( according to this order ) which cannot be defined by a finite number of words.