In the UNL approach, information conveyed by natural language is represented, sentence by sentence, as a hypergraph composed of a set of directed binary labeled links ( referred to as relations ) between nodes or hypernodes ( the Universal Words, or simply UW ), which stand for concepts.

In mathematics, a hypergraph is a generalization of a graph in which an edge can connect any number of vertices.

Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges.

However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality ; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. ( In other words, it is a collection of sets of size k .) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on.

The set of automorphisms of a hypergraph H (= ( X, E )) is a group under composition, called the automorphism group of the hypergraph and written Aut ( H ).

A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G < nowiki >'</ nowiki > of G is a host of the corresponding H < nowiki >'</ nowiki >.

A hypergraph is bipartite if and only if its vertices can be partitioned into two classes U and V in such a way that each hyperedge contains at least one vertex from both classes.

A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and ( x < sub > 1 </ sub >, e < sub > 1 </ sub >) are connected with an edge if and only if vertex x < sub > 1 </ sub > is contained in edge e < sub > 1 </ sub > in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above.

A hypergraph is said to be vertex-transitive ( or vertex-symmetric ) if all of its vertices are symmetric.

Similarly, a hypergraph is edge-transitive if all edges are symmetric.

A hypergraph is said to be uniform if all of its hyperedges have the same number of vertices as each other.

Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of E. A Sperner family is also sometimes called an independent system or, if viewed from the hypergraph perspective, a clutter.

Most classes of CSPs that are known to be tractable are those where the hypergraph of constraints has bounded treewidth ( and there are no restrictions on the set of constraint relations ), or where the constraints have arbitrary form but there exist essentially non-unary polymorphisms of the set of constraint relations.

In particular, there is a bipartite " incidence graph " or " Levi graph " corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.

Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphs, partial hypergraphs and section hypergraphs.

A partition theorem due to E. Dauber states that, for an edge-transitive hypergraph, there exists a partition

As a special case of this correspondence between bipartite graphs and hypergraphs, any multigraph ( a graph in which there may be two or more edges between the same two vertices ) may be interpreted as a hypergraph in which some hyperedges have equal sets of endpoints, and represented by a bipartite graph that does not have multiple adjacencies and in which the vertices on one side of the bipartition all have degree two.

As there is a hypergraph for every Levi graph, and vice-versa, the incidence matrix of an incidence structure describes a hypergraph.

A simple hypergraph is a hypergraph in which at most one hyperedge connects any pair of vertices and there are no hyperedges of size at most one.

For every Levi graph, there is an equivalent hypergraph, and vice versa.

When molten antimony is slowly cooled, metallic antimony crystallizes in a trigonal cell, isomorphic with that of the gray allotrope of arsenic.

* Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.

* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.

* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.

a − λ1 is not invertible ( because the spectrum of a is not empty ) hence a = λ1: this algebra A is naturally isomorphic to C ( the complex case of the Gelfand-Mazur theorem ).

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η < sub > X </ sub > is an isomorphism for every object X in C.

* The cofinality of the real numbers with their usual ordering is ℵ < sub > 0 </ sub >, since N is cofinal in R. The usual ordering of R is not order isomorphic to c, the cardinality of the real numbers, which has cofinality strictly greater than ℵ < sub > 0 </ sub >.

If two cofinal subsets of B have minimal cardinality ( i. e. their cardinality is the cofinality of B ), then they are order isomorphic to each other.

A bijective *- homomorphism π is called a C *- isomorphism, in which case A and B are said to be isomorphic.

A finite-dimensional C *- algebra, A, is canonically isomorphic to a finite direct sum

Each C *- algebra, Ae, is isomorphic ( in a noncanonical way ) to the full matrix algebra M < sub > dim ( e )</ sub >( C ).

Every finite simple group is isomorphic to one of the following groups:

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

The quaternion ring forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

In fact, every real n-dimensional vector space V is isomorphic to R < sup > n </ sup >.

So the fundamental group of the circle is isomorphic to, the additive group of integers.

The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker ( φ ) denotes the kernel of φ.

I consider it to be my job to expose the public to what is being written today ''.

Of the longer pieces of the volume none is so memorable as `` Nameless And Immortal '', which at once took rank among the finest poems ever written in the Swedish language.

This is brought out in the next to last chapter of the book, `` A Hero's Funeral '', written in the form of an impassioned prose poem.

Admirably written, it is a perfect introduction to Swedish history for readers of other countries.

Incest is still a durable theme, but if it wants to get written about it will have to find ways to surprise the emotions, and there is no better way to do this than that of concealment and symbolic representation.

His denials of extensive reading notwithstanding, it is no doubt safe to assume that he has spent time schooling himself in Southern history and that he has gained some acquaintance with the chief literary authors who have lived in the South or have written about the South.

Even when he is called upon for impromptu remarks, he has notes written on the back of handy envelopes.

Carl says it is the greatest poem ever written to the guitar because he has never heard of any other poem to that subtle instrument.

Of all the Whig tracts written in support of the Succession, The Crisis is perhaps the most significant.

The volume is a piece of passionate special pleading, written with the heat -- and often with the wisdom, it must be said -- of a Liberal damning the shortsightedness of politicians from 1782 to 1832.

12 `` where is it written ''??

This magnificent but greatly underestimated book, which bodies forth the very form and pressure of its time as no other comparable creation, has suffered severely from having been written about an historical event -- the Spanish Civil War -- that is still capable of fanning the smoldering fires of old political feuds.

In his recent book, Hurray For Anything ( 1957 ), one of the most important short poems -- and it is the title poem for one of the long jazz arrangements -- is written for recital with jazz.

Finally, there is the undeniable fact that some of the finest American fiction is being written by Jews, but it is not Jewish fiction ; ;

He mentions the beats only once '', when he refers to their having revived through mere power and abandonment and the unwillingness to, commit death in life some idea of a decent equivalent between verbal expression and actual experience,, but the entire narrative, is written in the tiresome vocabulary `` of '' that lost `` and '' dying cause, `` and in the '' `` sprung syntax that is supposed to supplant, our mother, tongue.

Since little is known about autism, and almost nothing has been written for the layman, we'd like to share one experienced mother's comments.

Morrison points out that since our country is more urbanized than the Soviet Union or Red China, it is the most vulnerable of the great powers -- Europe of course must be written off out of hand.

In connection with any claim decided by the Commission pursuant to this Title in which an award is made, the Commission may, upon the written request of the claimant or any attorney heretofore or hereafter employed by such claimant, determine and apportion the just and reasonable attorney's fees for services rendered with respect to such claim, but the total amount of the fees so determined in any case shall not exceed 10 per centum of the total amount paid pursuant to the award.

This is the point on which so many people have written off the aircraft in favor of the missile.