In addition, the term is used in a broader sense to cover a world-view on a wide variety of topics, including chemical evolution as an alternative term for abiogenesis or for nucleosynthesis of chemical elements, galaxy formation and evolution, stellar evolution, spiritual evolution, technological evolution and universal evolution, which seeks to explain every aspect of the world in which we live.

For example, the group R is the universal cover of the circle group S < sup > 1 </ sup >.

In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying universal cover, one guarantees a group structure ( compatible with its other structures ).

The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center.

In Ferdydurke ( his first novel, published in autumn 1937, the date on the cover 1938 ) discusses form as a universal category which was understood both in the philosophical, sociological, and aesthetic sense.

The concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function.

The universal cover of SO ( 3 ) is a Lie group called Spin ( 3 ).

" popularity is based on universal rock & roll virtues ," said Rolling Stone, which put the band on the cover of its January 7, 1993 issue.

is the universal cover of SL ( 2, R ), which fibers over H < sup > 2 </ sup >.

* If π < sub > 1 </ sub >( M ) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H < sup > 2 </ sup >× R or the universal cover of SL ( 2, R ).

Note that this means that algebraic group is narrower than Lie group, when working over the field of real numbers: there are examples such as the universal cover of the 2 × 2 special linear group that are Lie groups, but have no faithful linear representation.

A connected covering space is a universal cover if it is simply connected.

The name universal cover comes from the following important property: if the mapping is a universal cover of the space and the mapping is any cover of the space where the covering space is connected, then there exists a covering map such that.

< BLOCKQUOTE > The universal cover of the space covers all connected covers of the space .</ BLOCKQUOTE >

If the space has a universal cover then that universal cover is essentially unique: if the mappings and are two universal covers of the space then there exists a homeomorphism such that.

Every individual is connected with the rest of the world, and the universe is fashioned for universal harmony.

Some modern hi-fi equipment can be digitally connected using fibre optic TOSLINK cables, universal serial bus ( USB ) ports ( including one to play digital audio files ), or Wi-Fi support.

Baptism is the mark of membership in the covenant of grace and in the universal church, although regeneration is not inseparably connected with baptism.

It is believed to be a field of universal knowledge and is somehow connected to the terrestrial artifact in the Global Dynamics labs.

which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected ( and nonempty ) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ.

The unit of this adjunction is the universal cocone from F to colim F. If J is connected ( and nonempty ) then the counit is an isomorphism, so that colim is a left inverse of Δ.

However, it can also refer to the concept of such a language being determined by international consensus, including even a standardized natural language ( e. g., International English ), and has also been connected to the project of constructing a universal language.

Other noteworthy unknots are those that consist of rigid line segments connected by universal joints at their endpoints ( linkages ), that yet cannot be reconfigured into a convex polygon, thus acquiring the name stuck unknots.

This allows a VGA monitor to be connected with a passive plug adapter and offers a universal solution for the DVI interface.

In South Wales again, where in 1811 the railways were connected with canals, collieries, iron and copper works, and had a total length of nearly, the plateway was almost universal.

For n > 2, Spin ( n ) is simply connected, and thus the universal covering group for SO ( n ).

Menger showed, in the 1926 construction, that the sponge is a universal curve, in that any possible one-dimensional curve is homeomorphic to a subset of the Menger sponge, where here a curve means any compact metric space of Lebesgue covering dimension one ; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways.

The rig is connected to the board by a free-rotating universal joint and consists of a mast, 2-sided boom and sail.

The differential is connected to the engine via a swinging drive shaft and a universal joint.

The space has a universal cover if it is connected, locally path-connected and semi-locally simply connected.

* if is a Lie group ( as in the two examples above ), then so is its universal cover, and the mapping is a homomorphism of Lie groups.

These noncommutative algebras, and the non-associative Lie algebras, were studied within universal algebra before the subject was divided into particular mathematical structure types.

The universal enveloping algebra of the free Lie algebra generated by X and Y is isomorphic to the algebra of all non-commuting polynomials in X and Y.

Some authors require that the center of a simple Lie group be finite ( or trivial ); the universal cover of SL ( 2, R ) is an example of a simple Lie group with infinite center.

Secondly the Lie algebra only determines uniquely the simply connected ( universal ) cover G * of the component containing the identity of a Lie group G. It may well happen that G * isn't actually a simple group, for example having a non-trivial center.

Like in the non graded case, this Hopf algebra can be described purely algebraically as the universal enveloping algebra of the Lie superalgebra.

On the universal cover, there is no obstruction to integrating the infinitesimal symmetries, and so the group of conformal transformations is the infinite dimensional Lie group

Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.

Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

Algae constitute a paraphyletic and polyphyletic group, as they do not include all the descendants of the last universal ancestor nor do they all descend from a common algal ancestor, although their plastids seem to have a single origin.

However, there is no universal standard for the Goðar amongst organizations, and the title is usually only significant to the particular group with whom they work.

Namely φ is universal for homomorphisms from G to an abelian group H: for any abelian group H and homomorphism of groups f: G → H there exists a unique homomorphism F: G < sup > ab </ sup > → H such that.

Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space.

One group of opponents of this point of view, including many European nations, maintain that all civilized nations have certain norms of conduct expected of them, including the prohibition of genocide, slavery and the slave trade, wars of aggression, torture, and piracy, and that violation of these universal norms represents a crime, not only against the individual victims, but against humanity as a whole.

The homomorphism η is characterized by the following universal property: given any profinite group H and any group homomorphism f: G → H, there exists a unique continuous group homomorphism g: G < sup >^</ sup > → H with f

Some religions claim to be universal, believing their laws and cosmology to be binding for everyone, while others are intended to be practiced only by a closely defined or localized group.

Some scholars classify religions as either universal religions that seek worldwide acceptance and actively look for new converts, or ethnic religions that are identified with a particular ethnic group and do not seek converts.

:: Corollary: There is no universal solvable word problem group.

⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.