In reference to our discussion on Jordan algebras we can think of Ω as being the open cone of positive elements of a simple Euclidean Jordan algebra J .
Maximal Invariants Over Symmetric Cones
If A is a unital operator space or approximately unital operator algebra, then an element x ∈ Ball(A) is in the positive cone of A ∩ A∗ iff k1 − zxk ≤ 1 for al l z ∈ C with |1 − z | ≤ 1.
Metric characterizations II
Extreme points are the fundamental elements of the cone: in fact, by Carath´eodory Theorem (see Theorem 2.2), if the dimension of the vector space is ﬁnite and under some suitable conditions, every element of the cone can be written as a ﬁnite sum of extreme points.
Nonnegative polynomials and their Carath\'eodory number
Since every element of the cone Pn,d of nonnegative polynomials is a ﬁnite sum of extreme points, it is interesting to calculate the height of a generic F ∈ Pn,d and also to calculate the maximum height of a nonnegative polynomial: this value is the so-called Carath´eodory number of Pn,d .
Nonnegative polynomials and their Carath\'eodory number
So, a point x in a cone is extreme if and only if it cannot be non-trivially decomposed as the sum of two other elements of the cone.
Nonnegative polynomials and their Carath\'eodory number
So we found a non-trivial decomposition of F as a sum of other elements of the cone.
Nonnegative polynomials and their Carath\'eodory number
We take a somewhat different point of view and deﬁne light-cone distribution amplitudes as meson-to-vacuum transition matrix elements of nonlocal gaugeinvariant light-cone operators.
Handbook of Higher Twist Distribution Amplitudes of Vector Mesons in QCD
Calculation of exclusive amplitudes involving a large momentum-transfer reduces to evaluation of meson-to-vacuum transition matrix elements of nonlocal operators, which can be expanded in powers of the deviation from the light-cone.
Handbook of Higher Twist Distribution Amplitudes of Vector Mesons in QCD
Since U(n) is the only example we will need we won’t give the general deﬁnition of positive Weyl chamber, but merely state that for every connected compact Lie group K, there is an analogous group T and polyhedral cone t∗ +, such that each orbit in k∗ intersects t∗ + in a unique element.
The symplectic and algebraic geometry of Horn's problem
The matrix elements of the nonlocal operators sandwiched between a hadronic state and the vacuum deﬁnes the hadronic wave functions (about the light cone QCD sum rules see [12, 13] and references therein).
Meson-Baryon Couplings and the F/D ratio in Light Cone QCD
Cattani, Cox, and Dickenstein construct a different element, which they call ∆σ, referring to the choice σ of a cone in the fan, which is a constant times the Jacobian modulo the ideal I = (f1, f2, f3).
The Resultant of an Unmixed Bivariate System
Note that if we pick a different basis of the weight lattice of T2, the moment cone Cp,q will change by an action of an element of SL(2, Z).
Melrose--Uhlmann projectors, the metaplectic representation and symplectic cuts
Very troubling, for instance, is the fact that the topology generated for the boundary of Minkowski space is not that of its conformal embedding into the Einstein static spacetime: In the GKP topology, each cone-element (a null line) is an open set in the boundary.
Boundaries on Spacetimes: An Outline
In this paper we ﬁnd new elements of the dual cone ˜P 2 N, and thus deﬁne new necessary conditions for N -representability of a trial 2-particle density matrix D2 .
Searching for new conditions for fermion N-representability
The space of harmonic polynomials thus becomes a graded, locally ﬁnite representation of G; it can equivalently be thought of as the ring of regular functions on the cone of nilpotent elements in g.
The Cherednik kernel and generalized exponents
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