15,048 research outputs found
Compressing the hidden variable space of a qubit
In previously exhibited hidden variable models of quantum state preparation
and measurement, the number of continuous hidden variables describing the
actual state of a single realization is never smaller than the quantum state
manifold dimension. We introduce a simple model for a qubit whose hidden
variable space is one-dimensional, i.e., smaller than the two-dimensional Bloch
sphere. The hidden variable probability distributions associated with the
quantum states satisfy reasonable criteria of regularity. Possible
generalizations of this shrinking to a N-dimensional Hilbert space are
discussed.Comment: References updated and added some more discussions of result
Microeconomic Structure determines Macroeconomic Dynamics. Aoki defeats the Representative Agent
Masanao Aoki developed a new methodology for a basic problem of economics:
deducing rigorously the macroeconomic dynamics as emerging from the
interactions of many individual agents. This includes deduction of the fractal
/ intermittent fluctuations of macroeconomic quantities from the granularity of
the mezo-economic collective objects (large individual wealth, highly
productive geographical locations, emergent technologies, emergent economic
sectors) in which the micro-economic agents self-organize.
In particular, we present some theoretical predictions, which also met
extensive validation from empirical data in a wide range of systems: - The
fractal Levy exponent of the stock market index fluctuations equals the Pareto
exponent of the investors wealth distribution. The origin of the macroeconomic
dynamics is therefore found in the granularity induced by the wealth / capital
of the wealthiest investors. - Economic cycles consist of a Schumpeter
'creative destruction' pattern whereby the maxima are cusp-shaped while the
minima are smooth. In between the cusps, the cycle consists of the sum of 2
'crossing exponentials': one decaying and the other increasing.
This unification within the same theoretical framework of short term market
fluctuations and long term economic cycles offers the perspective of a genuine
conceptual synthesis between micro- and macroeconomics. Joining another giant
of contemporary science - Phil Anderson - Aoki emphasized the role of rare,
large fluctuations in the emergence of macroeconomic phenomena out of
microscopic interactions and in particular their non self-averaging, in the
language of statistical physics. In this light, we present a simple stochastic
multi-sector growth model.Comment: 42 pages, 6 figure
The Normal Form Theorem around Poisson Transversals
We prove a normal form theorem for Poisson structures around Poisson
transversals (also called cosymplectic submanifolds), which simultaneously
generalizes Weinstein's symplectic neighborhood theorem from symplectic
geometry and Weinstein's splitting theorem. Our approach turns out to be
essentially canonical, and as a byproduct, we obtain an equivariant version of
the latter theorem.Comment: 15 pages; v2: the title was changed; v3: proof of Lemma 2 was
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