450 research outputs found
Royal Road to Coupling Classical and Quantum Dynamics
We present a consistent framework of coupled classical and quantum dynamics.
Our result allows us to overcome severe limitations of previous
phenomenological approaches, like evolutions that do not preserve the
positivity of quantum states or that allow to activate quantum nonlocality for
superluminal signaling. A `hybrid' quantum-classical density is introduced and
its evolution equation derived. The implications and applications of our result
are numerous: it incorporates the back-reaction of quantum on classical
variables, it resolves fundamental problems encountered in standard mean field
theories, and remarkably, also the quantum measurement process, i.e. the most
controversial example of quantum-classical interaction is consistently
described within our approach, leading to a theory of dynamical collapse.Comment: 4 pages, RevTe
Shattering-Extremal Set Systems of Small VC-Dimension
We say that a set system shatters a given set
if . The Sauer inequality
states that in general, a set system shatters at least
sets. Here we concentrate on the case of equality. A set system
is called shattering-extremal if it shatters exactly sets. We
characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1
in terms of their inclusion graphs. Also from the perspective of extremality,
we relate set systems of bounded Vapnik-Chervonenkis dimension to their
projections.Comment: 17 page
On algebraic endomorphisms of the Einstein gyrogroup
We describe the structure of all continuous algebraic endomorphisms of the
open unit ball of equipped with the Einstein
velocity addition. We show that any nonzero such transformation originates from
an orthogonal linear transformation on
Exploring projective norm graphs
The projective norm graphs provide tight constructions for
the Tur\'an number of complete bipartite graphs with . In
this paper we determine their automorphism group and explore their small
subgraphs. To this end we give quite precise estimates on the number of
solutions of certain equation systems involving norms over finite fields. The
determination of the largest integer , such that the projective norm graph
contains for all large enough prime powers is
an important open question with far-reaching general consequences. The best
known bounds, , are far apart for . Here we
prove that does contain (many) for any prime power
not divisble by or . This greatly extends recent work of Grosu,
using a completely different approach. Along the way we also count the copies
of any fixed -degenerate subgraph, and find that projective norm graphs are
quasirandom with respect to this parameter. Some of these results also extend
the work of Alon and Shikhelman on generalized Tur\'an numbers. Finally we also
give a new, more elementary proof for the -freeness of
.Comment: 41 pages + 6 pages of Appendi
Post-Markov master equation for the dynamics of open quantum systems
A systematic first-order correction to the standard Markov master equation
for open quantum systems interacting with a bosonic bath is presented. It
extends the Markov Lindblad master equation to the more general case of
non-Markovian evolution. The meaning and applications of our `post'-Markov
master equation are illustrated with several examples, including a damped
two-level atom, the spin-boson model and the quantum Brownian motion model.
Limitations of the Markov approximation, the problem of positivity violation
and initial slips are also discussed.Comment: 7 pages, 2 figures, RevTe
Preserving the measure of compatibility between quantum states
In this paper after defining the abstract concept of compatibility-like
functions on quantum states, we prove that every bijective transformation on
the set of all states which preserves such a function is implemented by an
either unitary or antiunitary operator.Comment: 11 pages, submitted for publicatio
Páronként érintkező hengerek
Littlewood több mint ötven évig nyitott diszkrét geometriai
kérdését válaszoljuk meg: létezik a térben hét páronként
érintkező, végtelen hosszú, azonos átmérőjű körhenger
- …