8,325 research outputs found
Galois Unitaries, Mutually Unbiased Bases, and MUB-balanced states
A Galois unitary is a generalization of the notion of anti-unitary operators.
They act only on those vectors in Hilbert space whose entries belong to some
chosen number field. For Mutually Unbiased Bases the relevant number field is a
cyclotomic field. By including Galois unitaries we are able to remove a
mismatch between the finite projective group acting on the bases on the one
hand, and the set of those permutations of the bases that can be implemented as
transformations in Hilbert space on the other hand. In particular we show that
there exist transformations that cycle through all the bases in every dimension
which is an odd power of an odd prime. (For even primes unitary MUB-cyclers
exist.) These transformations have eigenvectors, which are MUB-balanced states
(i.e. rotationally symmetric states in the original terminology of Wootters and
Sussman) if and only if d = 3 modulo 4. We conjecture that this construction
yields all such states in odd prime power dimension.Comment: 32 pages, 2 figures, AMS Latex. Version 2: minor improvements plus a
few additional reference
Lattice-like operations and isotone projection sets
By using some lattice-like operations which constitute extensions of ones
introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new
perspective is gained on the subject of isotonicity of the metric projection
onto the closed convex sets. The results of this paper are wide range
generalizations of some results of the authors obtained for self-dual cones.
The aim of the subsequent investigations is to put into evidence some closed
convex sets for which the metric projection is isotonic with respect the order
relation which give rise to the above mentioned lattice-like operations. The
topic is related to variational inequalities where the isotonicity of the
metric projection is an important technical tool. For Euclidean sublattices
this approach was considered by G. Isac and respectively by H. Nishimura and E.
A. Ok.Comment: Proofs of Theorem 1 and Corollary 4 have been corrected. arXiv admin
note: substantial text overlap with arXiv:1210.232
Bilinearity rank of the cone of positive polynomials and related cones
For a proper cone K ⊂ Rn and its dual cone K the complementary slackness condition xT s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K^* }. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) ∈ C(K). This proves to be very useful when
optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for the cone, and describe a linear algebraic technique to bound this quantity. We examine several well-known cones, in particular
the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1, and compute their bilinearity ranks. We show that there are exactly four linearly independent bilinear identities which hold for all (x,s) ∈ C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential
polynomials
Bilinearity rank of the cone of positive polynomials and related cones
For a proper cone K ⊂ Rn and its dual cone K the complementary slackness condition xT s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K^* }. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) ∈ C(K). This proves to be very useful when
optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for the cone, and describe a linear algebraic technique to bound this quantity. We examine several well-known cones, in particular
the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1, and compute their bilinearity ranks. We show that there are exactly four linearly independent bilinear identities which hold for all (x,s) ∈ C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential
polynomials
GIT Constructions of Moduli Spaces of Stable Curves and Maps
This largely expository paper first gives an introduction to Hilbert
stability and its use in Gieseker's GIT construction of . Then
I review recent work in this area--variants for unpointed curves that arise in
Hassett's log minimal model program, starting with Schubert's moduli space of
pseudostable curves, and constructions for weighted pointed stable curves and
for pointed stable maps due to Swinarski and to Baldwin and Swinarski
respectively. The focus is on the steps at which new ideas are needed. Finally,
I list open problems in the area, particularly some arising in the log minimal
model program that seem inaccessible to current techniques.Comment: 46 pages, 3 figures, written for Surveys in Differential Geometr
Quantum Causality, Stochastics, Trajectories and Information
A history of the discovery of quantum mechanics and paradoxes of its
interpretation is reconsidered from the modern point of view of quantum
stochastics and information. It is argued that in the orthodox quantum
mechanics there is no place for quantum phenomenology such as events. The
development of quantum measurement theory, initiated by von Neumann, and Bell's
conceptual critics of hidden variable theories indicated a possibility for
resolution of this crisis. This can be done by divorcing the algebra of the
dynamical generators and an extended algebra of the potential (quantum) and the
actual (classical) observables. The latter, called beables, form the center of
the algebra of all observables, as the only visible (macroscopic) observables
must be compatible with any hidden (microscopic) observable.
It is shown that within this approach quantum causality can be rehabilitated
within event enhanced quantum mechanics (eventum mechanics) in the form of a
superselection rule for compatibility of the consistent histories with the
statistically predictable future. The application of this rule in the form of
the nondemolition principle leads to the statistical inference of the von
Neumann projection postulate, and also to the more general quantum information
dynamics for instantaneous events, spontaneous localizations (i.e. quantum
jumps), and state diffusions (i.e. continuous trajectories). This gives a
dynamical solution, in the form of a Dirac boundary value problem and reduced
filtering equations, of the notorious decoherence and measurement problems
which was tackled unsuccessfully by many famous mathematicians and physicists
starting with von Neumann, Schroedinger and Bohr.Comment: 67 pages, 120 references. In celebration of the 100th anniversary of
the discovery of quant
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