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 5−12{\frac{\sqrt{5}-1}{2}}}

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 ${\frac{\sqrt{5}-1}{2}}$, 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 $1/f$ 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 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 $\dot{x}=y-F(x)$, $\dot {y}=-g(xt)$, under weaker conditions on the odd functions $F(x)$ and $g(x)$ 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 $F$ as $|x| \to\infty$.Comment: Latex 2e, 27 pages, 9 figure

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 1/f1/f spectrum in Physics and Biology

We show that the generic $1/f$ 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

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