36 research outputs found
Duality and Scaling in Quantum Mechanics
The nonadiabatic geometric phase in a time dependent quantum evolution is
shown to provide an intrinsic concept of time having dual properties relative
to the external time. A nontrivial extension of the ordinary quantum mechanics
is thus obtained with interesting scaling laws. A fractal like structure in
time is thus revealed.Comment: 10 pages, Latex, email: [email protected], To appear in Physics
Letters
Dynamical real numbers and living systems
Recently uncovered second derivative discontinuous solutions of the simplest
linear ordinary differential equation define not only an nonstandard extension
of the framework of the ordinary calculus, but also provide a dynamical
representation of the ordinary real number system. Every real number can be
visualized as a living cell -like structure, endowed with a definite
evolutionary arrow. We discuss the relevance of this extended calculus in the
study of living systems. We also present an intelligent version of the Newton's
first law of motion.Comment: AMS-latex 2e, 14 page
A new class of scale free solutions to linear ordinary differential equations and the universality of the Golden Mean }
A new class of finitely differentiable scale free solutions to the simplest
class of ordinary differential equations is presented. Consequently, the real
number set gets replaced by an extended physical set, each element of which is
endowed with an equivalence class of infinitesimally separated neighbours in
the form of random fluctuations. We show how a sense of time and evolution is
intrinsically defined by the infinite continued fraction of the golden mean
irrational number , which plays a key role in this
extended SL(2,R) formalism of Calculus. Time may thereby undergo random
inversions generating well defined random scales, thus allowing a dynamical
system to evolve self similarly over the set of multiple scales. The late time
stochastic fluctuations of a dynamical system enjoys the generic
spectrum. A universal form of the related probability density is also derived.
We prove that the golden mean number is intrinsically random, letting all
measurements in the physical universe fundamentally uncertain. The present
analysis offers an explanation of the universal occurrence of the golden mean
in diverse natural and biological processes.Comment: Latex2e,18 pages, Chaos,Solitons & Fractals (2002), to appea
Time inversion, Self-similar evolution, and Issue of time
We investigate the question, "how does time flow?" and show that time may
change by inversions as well. We discuss its implications to a simple class of
linear systems. Instead of introducing any unphysical behaviour, inversions can
lead to a new multi- time scale evolutionary path for the linear system
exhibiting late time stochastic fluctuations. We explain how stochastic
behaviour is injected into the linear system as a combined effect of an
uncertainty in the definition of inversion and the irrationality of the golden
mean number. We also give an ansatz for the nonlinear stochastic behaviour of
(fractal) time which facilitates us to estimate the late and short time limits
of a two-time correlation function relevant for the stochastic fluctuations in
linear systems. These fluctuations are shown to enjoy generic 1/f spectrum. The
implicit functional definition of the fractal time is shown to satisfy the
differential equation dx=dt. We also discuss the relevance of intrinsic time in
the present formalism, study of which is motivated by the issue of time in
quantum gravity.Comment: Latex 2e, 17 pages, no figur
On a new proof of the Prime Number Theorem
A new elementary proof of the prime number theorem presented recently in the
framework of a scale invariant extension of the ordinary analysis is
re-examined and clarified further. Both the formalism and proof are presented
in a much more simplified manner. Basic properties of some key concepts such as
infinitesimals, the associated nonarchimedean absolute values, invariance of
measure and cardinality of a compact subset of the real line under an IFS are
discussed more thoroughly. Some interesting applications of the formalism in
analytic number theory are also presented. The error term as dictated by the
Riemann hypothesis also follows naturally thus leading to an indirect proof of
the hypothesis.Comment: Completed originally on October 2010, this is a revised and extended
version on the basis of a referee report received sometime in December 2010,
currently still under referee evaluatio
On a Variation of the Definition of Limit: Some Analytic Consequences
The basic formalism of a novel scale invarinat nonlinear analysis is
presented. A few analytic number theoretic results are derived independent of
standard approaches.Comment: 13 pages, Latex2e. An omission in the proof of Proposition 2 is
corrected and Corollary 3 is adde
Fractals in Linear Ordinary Differential Equations
We prove the existence of fractal solutions to a class of linear ordinary
differential equations.This reveals the possibility of chaos in the very short
time limit of the evolution even of a linear one dimensional dynamical system.Comment: 6 pages,Latex, submitted to Phys.Lett.A, email: [email protected]
The Golden mean, scale free extension of Real number system, fuzzy sets and spectrum in Physics and Biology
We show that the generic spectrum problem acquires a natural
explanation in a class of scale free solutions to the ordinary differential
equations. We prove the existence and uniqueness of this class of solutions and
show how this leads to a nonstandard, fuzzy extension of the ordinary framework
of calculus, and hence, that of the classical dynamics and quantum mechanics.
The exceptional role of the golden mean irrational number is also explained.Comment: AMS_Latex 2e, 14 page
On the determination of exact number of limit cycles in Lienard Systems
We present a simpler proof of the existence of an exact number of one or more
limit cycles to the Lienard system , , under
weaker conditions on the odd functions and as compared to those
available in literature. We also give improved estimates of amplitudes of the
limit cycle of the Van Der Pol equation for various values of the nonlinearity
parameter. Moreover, the amplitude is shown to be independent of the asymptotic
nature of as .Comment: Latex 2e, 27 pages, 9 figure
Duality Structure, Asymptotic analysis and Emergent Fractal sets
A new, extended nonlinear framework of the ordinary real analysis
incorporating a novel concept of {\em duality structure} and its applications
into various nonlinear dynamical problems is presented. The duality structure
is an asymptotic property that should affect the late time asymptotic behaviour
of a nonlinear dynamical system in a nontrivial way leading naturally to
signatures generic to a complex system. We argue that the present formalism
would offer a natural framework to understand the abundance of complex systems
in natural, biological, financial and related problems. We show that the power
law attenuation of a dispersive, lossy wave equation, conventionally deduced
from fractional calculus techniques, could actually arise from the present
asymptotic duality structure. Differentiability on a Cantor type fractal set is
also formulated.Comment: This is published version of the original preprint "Duality
Structure, Nonarchimedean Extension of the Real Number System and Emergent
Fractals", No. of pages:32. Also removed most of the unfortunate Latex/Macro
definition errors of the published pape