A remark on renormalization group theoretical perturbation in a class of ordinary differential equations

We revisit the renormalization group (RG) theoretical perturbation theory on oscillator type second order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all order of the RG series. The result covers non-autonomous as well as autonomous cases and refines earlier studies including the classic examples as Van der Pol, Mathieu, Duffing and Rayleigh equations.

Active Integrated Antenna

This chapter discusses the active integrated antenna (AIA), which is the integration of an antenna with an active circuitry that acts as a radiating element as well as to perform additional functions simultaneously. The designs of AIA can be generalized into three different classifications, the amplifier type, the oscillator type, and the frequency conversion type. An AIA is classified as the amplifier type when its active device functions as an amplifier. Correspondingly, an AIA is classified as the oscillator type when its active device offers the function of an oscillator while the frequency conversion type of AIA integrates an active device with a passive antenna element for the purpose of frequency translation.

Solution of fractional oscillator equations using ultraspherical wavelets

Fractional oscillator type equations are well-known model equations to describe several phenomenon in mathematical physics, engineering and biology. In this paper, a new method incorporated by the ultraspherical wavelet operational matrix of general order integration and block-pulse functions are adopted to investigate the solution of fractional oscillator type equations. To facilitate this, the ultraspherical wavelets are first presented and the corresponding operational matrix of fractional-order integration is derived by virtue of block pulse functions. The properties of ultraspherical wavelets and block pulse functions are used to transform the underlying problem to a system of algebraic equations which can be easily solved by any of the usual numerical methods. The efficiency and accuracy of the proposed method is demonstrated by presenting several benchmark test problems. Moreover, special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

Non-Hermitian inverted harmonic oscillator-type Hamiltonians generated from supersymmetry with reflections

By modifying and generalizing known supersymmetric models, we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of [Formula: see text]. Also, by changing the derivative [Formula: see text] by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted harmonic oscillator [Formula: see text], we generate the third set of Hamiltonians. The fourth set of Hamiltonians emerges by allowing a parameter of the supersymmetric two-body Calogero-type model to take imaginary values. The eigensolutions of definite parity for each set of Hamiltonians are given.

Study on Singular Spectrum Analysis as a New Technical Oscillator for Trading Rules Design

The connection between Singular Spectrum Analysis (SSA) decomposition and short-term market movements is investigated. Since SSA is a non-parametric approach, suitable to decompose general time-series into meaningful components, such as trends, oscillations and noise, it is proposed as a new oscillator-type Technical Indicator, replacing popular ones. New Technical Trading Rules (TTRs) are designed and applied to some major global stock indexes to illustrate the benefits in terms of revealing market movements. The performance is evaluated according to different risk-adjustment metrics and the empirical results reveal that the SSA-TTRs may outperform some popular technical oscillators and also the Buy & Hold strategy.