20,635 research outputs found
A domain-theoretic investigation of posets of sub-sigma-algebras (extended abstract)
Given a measurable space (X, M) there is a (Galois) connection between
sub-sigma-algebras of M and equivalence relations on X. On the other hand
equivalence relations on X are closely related to congruences on stochastic
relations. In recent work, Doberkat has examined lattice properties of posets
of congruences on a stochastic relation and motivated a domain-theoretic
investigation of these ordered sets. Here we show that the posets of
sub-sigma-algebras of a measurable space do not enjoy desired domain-theoretic
properties and that our counterexamples can be applied to the set of smooth
equivalence relations on an analytic space, thus giving a rather unsatisfactory
answer to Doberkat's question
The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function (Extended Abstract)
In this article we treat a notion of continuity for a multi-valued function F
and we compute the descriptive set-theoretic complexity of the set of all x for
which F is continuous at x. We give conditions under which the latter set is
either a G_\delta set or the countable union of G_\delta sets. Also we provide
a counterexample which shows that the latter result is optimum under the same
conditions. Moreover we prove that those conditions are necessary in order to
obtain that the set of points of continuity of F is Borel i.e., we show that if
we drop some of the previous conditions then there is a multi-valued function F
whose graph is a Borel set and the set of points of continuity of F is not a
Borel set. Finally we give some analogue results regarding a stronger notion of
continuity for a multi-valued function. This article is motivated by a question
of M. Ziegler in "Real Computation with Least Discrete Advice: A Complexity
Theory of Nonuniform Computability with Applications to Linear Algebra",
(submitted)
Thermodynamic phase transition of a black hole in rainbow gravity
In this letter, using the rainbow functions that were proposed by Magueijo
and Smolin, we investigate the thermodynamics and the phase transition of
rainbow Schwarzschild black hole. First, we calculate the rainbow gravity
corrected Hawking temperature. From this modification, we then derive the local
temperature, free energy, and other thermodynamic quantities in an isothermal
cavity. Finally, we analyze the critical behavior, thermodynamic stability, and
phase transition of the rainbow Schwarzschild black hole. The results show that
the rainbow gravity can stop the Hawking radiation in the final stages of black
holes' evolution and lead to the remnants of black holes. Furthermore, one can
observe that the rainbow Schwarzschild black hole has one first-order phase
transition, two second-order phase transitions, and three Hawking-Page-type
phase transitions in the framework of rainbow gravity theory.Comment: 7 pages, 3 figures, accepted for publication in Physical Letter B.
arXiv admin note: substantial text overlap with arXiv:1608.0682
Rainbow gravity corrections to the entropic force
The entropic force attracts a lot of interest for its multifunctional
properties. For instance, Einstein's field equation, Newton's law of
gravitation and the Friedmann equation can be derived from the entropic force.
In this paper, utilizing a new kind of rainbow gravity model that was proposed
by Magueijo and Smolin, we explore the quantum gravity corrections to the
entropic force. First, we derive the modified thermodynamics of a rainbow black
hole via its surface gravity. Then, according to Verlinde's theory, the quantum
corrections to the entropic force are obtained. The result shows that the
modified entropic force is related not only to the properties of the black hole
but also the Planck length , and the rainbow parameter .
Furthermore, based on the rainbow gravity corrected entropic force, the
modified Einstein's field equation and the modified Friedmann equation are also
derived.Comment: 10 page
Dynamics of Moving Average Rules in a Continuous-time Financial Market Model
Within a continuous-time framework, this paper proposes a stochastic heterogeneous agent model (HAM) of financial markets with time delays to unify various moving average rules used indiscrete-time HAMs. The time delay represents a memory length of a moving average rule indiscrete-time HAMs.Intuitive conditions for the stability of the fundamental price of the deterministic model in terms of agents' behavior parameters and memory length are obtained. It is found that an increase in memory length not only can destabilize the market price, resulting in oscillatory market price characterized by a Hopf bifurcation, but also can stabilize another wise unstable market price, leading to stability switching as the memory length increases. Numerical simulations show that the stochastic model is able to characterize long deviations of the market price from its fundamental price and excess volatility and generate most of the stylized factso bserved in financial markets.asset price; financial market behavior; heterogeneous beliefs; stochastic delay differential equations; stability; bifurcations; stylized facts
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