675 research outputs found
Variable Order Fractional Variational Calculus for Double Integrals
We introduce three types of partial fractional operators of variable order.
An integration by parts formula for partial fractional integrals of variable
order and an extension of Green's theorem are proved. These results allow us to
obtain a fractional Euler-Lagrange necessary optimality condition for variable
order two-dimensional fractional variational problems.Comment: This is a preprint of a paper whose final and definite form will be
published in: 51st IEEE Conference on Decision and Control, December 10-13,
2012, Maui, Hawaii, USA. Article Source/Identifier: PLZ-CDC12.1240.d4462b33.
Submitted 07-March-2012; accepted 17-July-201
Minimum Number of Probes for Brain Dynamics Observability
In this paper, we address the problem of placing sensor probes in the brain
such that the system dynamics' are generically observable. The system dynamics
whose states can encode for instance the fire-rating of the neurons or their
ensemble following a neural-topological (structural) approach, and the sensors
are assumed to be dedicated, i.e., can only measure a state at each time. Even
though the mathematical description of brain dynamics is (yet) to be
discovered, we build on its observed fractal characteristics and assume that
the model of the brain activity satisfies fractional-order dynamics.
Although the sensor placement explored in this paper is particularly
considering the observability of brain dynamics, the proposed methodology
applies to any fractional-order linear system. Thus, the main contribution of
this paper is to show how to place the minimum number of dedicated sensors,
i.e., sensors measuring only a state variable, to ensure generic observability
in discrete-time fractional-order systems for a specified finite interval of
time. Finally, an illustrative example of the main results is provided using
electroencephalogram (EEG) data.Comment: arXiv admin note: text overlap with arXiv:1507.0720
Mathematical Economics
This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus
Long-term correlations and multifractal nature in the intertrade durations of a liquid Chinese stock and its warrant
Intertrade duration of equities is an important financial measure
characterizing the trading activities, which is defined as the waiting time
between successive trades of an equity. Using the ultrahigh-frequency data of a
liquid Chinese stock and its associated warrant, we perform a comparative
investigation of the statistical properties of their intertrade duration time
series. The distributions of the two equities can be better described by the
shifted power-law form than the Weibull and their scaled distributions do not
collapse onto a single curve. Although the intertrade durations of the two
equities have very different magnitude, their intraday patterns exhibit very
similar shapes. Both detrended fluctuation analysis (DFA) and detrending moving
average analysis (DMA) show that the 1-min intertrade duration time series of
the two equities are strongly correlated. In addition, both multifractal
detrended fluctuation analysis (MFDFA) and multifractal detrending moving
average analysis (MFDMA) unveil that the 1-min intertrade durations possess
multifractal nature. However, the difference between the two singularity
spectra of the two equities obtained from the MFDMA is much smaller than that
from the MFDFA.Comment: 10 latex pages, 4 figure
Generalized beta models and population growth: so many routes to chaos
Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models proportional to either geo-max-stable distributions (log-logistic and backward log-logistic) or to other max-stable distributions (Fréchet or max-Weibull). We show that the former arise when in the hyper-logistic Blumberg equation, connected to the Beta (Formula presented.) function, we use fractional exponents (Formula presented.) and (Formula presented.), and the latter when in the hyper-Gompertz-Turner equation, the exponents of the logarithmic factor are real and eventually fractional. The use of a BetaBoop function establishes interesting connections to Probability Theory, Riemann–Liouville’s fractional integrals, higher-order monotonicity and convexity and generalized unimodality, and the logistic map paradigm inspires the investigation of the dynamics of the hyper-logistic and hyper-Gompertz maps.info:eu-repo/semantics/publishedVersio
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