Even though in most cases the completion of the definitive editions of their writings is still years off, enough documentation has already been assembled to warrant drawing a new composite profile of the leadership which performed the heroic dual feats of winning American independence and founding a new nation.

When the grammatical dual form of Ajax is used in the Iliad, it was once believed that it indicated the lesser Ajax fighting side-by-side with Telamonian Ajax, but now it is generally thought that that usage refers to the Greater Ajax and his brother Teucer.

The system is a dual purpose test and interception facility in Alaska, and in 2006 was operational with a few interceptor missiles.

* there were no " half-open " connections ; once one end initiated a tear-down of the connection, the whole connection would be closed ( i. e., ADSP is full-duplex, not dual simplex ).

Changing the balance of serotonin and norepinephrine would help the brain cells send and receive messages, therefore boosting the mood, which also is the main reason why these types of medications are called dual reuptake inhibitors.

Since the reference model given for traffic policing in the network is the GCRA, this algorithm is normally used for shaping as well, and single and dual leaky bucket implementations may be used as appropriate.

It is unknown when dual wielding was first used in fiction, but it was likely from early western novels before spilling over into other works.

Alcoholism is called a " dual disease " since it includes both mental and physical components.

The town is served by junction seven of the M65 and is linked from the A680 and the A56 dual carriageway which briefly merge ; linking to the M66 motorway heading towards Manchester.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

The spaceX * of all linear maps into K ( which is called the algebraic dual space to distinguish it from X ′) also induces a weak topology which is finer than that induced by the continuous dual since X ′ ⊆ X *.

There is a natural map F: X → X ′′ ( the dual of the dual

In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.

Just as kets and bras can be transformed into each other ( making into ) the element from the dual space corresponding with is where A < sup >†</ sup > denotes the Hermitian conjugate ( or adjoint ) of the operator A.

Moreover ( and more embarrassingly, although this is essentially trivial ), mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *- symbol, but an overline ( which the physicists reserve to averages ) to denote conjugate-complex numbers, i. e. for scalar products mathematicians usually write

There is dual suspension:

When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involution, i. e.,

The dual of a uniform hypergraph is regular and vice-versa.

The transpose of the incidence matrix defines a hypergraph called the dual of, where is an m-element set and is an n-element set of subsets of.

And, the hypergraph dual of vertex coloring is edge coloring.

A self-dual polyhedron is a polyhedron whose dual is a congruent figure, though not necessarily the identical figure: for example, the dual of a regular tetrahedron is a regular tetrahedron " facing the opposite direction " ( reflected through the origin ).

This causes a split in Jean's psyche — between Jean Grey and the Phoenix ( what her dual personality calls itself ) with Jean whose limited powers were always in her control, and the dormant side, " The Phoenix ," a purely instinctual creature.

The two sites whose Islamic sanctity is unchallengeably the highest of all are Mecca and Medina in Arabia, so that the Arabic dual form الحرمان or الحرمين refers to these two places.

The Lagrangian is thus a function on the jet bundle J over E ; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T < sup >*</ sup > E < sub > t </ sub >, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian.

This dual appointment was a way of satisfying two divergent shades of opinion now hardening throughout the army, one, like Hephaestion, broadly supportive of Alexander's policy of integration, and the other, that of Philip's older veterans in particular, whose implacable resentment of Persian ways was well represented by Cleitus.

Other Québécois musicians with notable influences from Bolduc include Oscar Thiffault, whose style was descended from Bolduc, André Gagnon, whose composition Les Turluteries is based on Bolduc's use of Turlutage, and Robert Charlebois, who also sang in the dual French Canadian and English derived style and used the everyday slang of Quebec.

In mathematics, the predual of an object D is an object P whose dual space is D.

The minimum vertex cover problem can be formulated as a half-integral linear program whose dual linear program is the maximum matching problem.

In the case of Boa Morte, whose members were relatively poor and almost all elderly — from 50 to 70 years old — the funds raised during members ' lifetimes were always meant to pay for a decent funeral, whose preparations, given the dual religious activities of its members, required both rigor and understanding, besides being a nest-egg for the burial.

Satisfiability and validity are dual because the invalid formulas are precisely those whose negations are satisfiable, and the unsatisfiable formulas are those whose negations are valid.

Instantons in the dual theory are now vortices whose action is given by the exponential of the FI term.

The N4 is the only one of the main inter-urban national routes whose dual carriageway section runs right into the city centre.

The toric variety has a map to the polytope in the dual of N whose fibers are topological tori.

The tribes, whose members have dual status as citizens of the United States and as " citizens " of the tribe, can impose taxes, make laws, and control their lands.

To efficiently operate valves whose tops were such a long way apart, the dual overhead cam arrangement was found to be the most effective.

Over a field, its dual line bundle is the line bundle associated to the hyperplane divisor H, whose global sections are the linear forms.

On occasions when it was necessary for " Fiona MacLeod " to write to someone unaware of the dual identity, Sharp would dictate the text to his sister ( Mary Beatrice Sharp ), whose handwriting would then be passed off as Fiona's manuscript.

Covering problems are minimization problems and usually linear programs, whose dual problems are called packing problems.

This kind of anomaly can occur in odd-dimensional gauge theories with fermions whose gauge groups have odd dual Coxeter numbers.

The dual of a cube is an octahedron, shown here with vertices at the cube face centers.

The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges.

The vertices of the dual, then, are the poles reciprocal to the face planes of the original, and the faces of the dual lie in the polars reciprocal to the vertices of the original.

Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual.

It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph ( sometimes called a Schlegel diagram ) on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph.

An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosahedron located at the midpoints of the edges of either.

The 24 vertices of 24-cell ( red ) and 24 vertices of its dual ( yellow ) represent the 48 root vectors of F < sub > 4 </ sub > in this Coxeter plane projection

The 48 root vectors of F < sub > 4 </ sub > can be found as the vertices of the 24-cell in two dual configurations:

A quotient of any tiling in the ( 2, 3, 7 ) family can be used ( and will have the same automorphism group ); of these, the two regular tilings are the tiling by 24 regular hyperbolic heptagons, each of degree 3 ( meeting at 56 vertices ), and the dual tiling by 56 equilateral triangles, each of degree 7 ( meeting at 24 vertices ).

In particular, in 2 dimensions, if the vertices can take on values in a locally compact abelian group G and the action / energy only depends on the edges ( e. g. the Ising model for Z < sub > 2 </ sub >, the Potts model for Z < sub > n </ sub >, the XY model for U ( 1 ) ), then its dual via the Kramers – Wannier duality to a model where the vertices take on values in the dual group G ′.

The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal.

Again, such authorities agree that the Archimedean polyhedra ( having regular faces and equivalent vertices ) have no inspheres while the Archimedean dual or Catalan polyhedra do have inspheres.

The weak dual of an embedded planar graph is the subgraph of the dual graph whose vertices correspond to the bounded faces of the primal graph.