Negative Half-line Research Articles

Harmonic Oscillator with a Step and/or a Ramp

We discuss the one-dimensional Schrödinger equation for a harmonic oscillator with a finite step at the origin and/or an external field described by a ramp function. The first half of this paper is a partial review of our recent work. The latter half is devoted to an extension of the problem, i.e., imposing an external linear field on the negative half line. The solvability of the problem via the Hermite polynomials is discussed. We demonstrate that a harmonic oscillator with a step and a ramp can have one eigenstate whose wavefunction is expressed in terms of Hermite polynomials of different orders. Explicit examples are also provided at appropriate places in the text.

Current fluctuations in stochastically resetting particle systems.

We consider a system of noninteracting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) Each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run-and-tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomized. We study the effects of resetting on the distribution P(Q,t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution P_{an}(Q,t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution P_{qu}(Q,t) attains its stationary form for Q<Q_{crit}(t), while it remains time dependent for Q>Q_{crit}(t). We show that Q_{crit}(t) increases linearly with t for large t. On the scale where Q∼Q_{crit}(t), we show that P_{qu}(Q,t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10^{-14000}, we were able to compute numerically this nonanalytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

Effusion of stochastic processes on a line

We consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.

Tracy–Widom at each edge of real covariance and MANOVA estimators

We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy–Widom limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin Schnelli, replacing a continuous Green function flow by a discrete Lindeberg swapping scheme.

Total number of births on the negative half-line of the binary branching Brownian motion in the boundary case

The binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time 0. The particle moves according to a Brownian motion with drift μ=2 and diffusion coefficient σ2=2, until an independent exponential time of parameter 1. At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to ∞. The aim of this note is to provide a precise estimate for the total number of particles that were born on the negative half-line, investigating in particular the tail decay of this random variable.

GaN-Based High-Power-Density AC–DC–AC Converter for Single-Phase Transformerless Online Uninterruptible Power Supply

This article introduces a new transformerless ac-dc-ac converter topology suitable for a high-power-density high-frequency single-phase online uninterruptible power supply (UPS) with a common neutral between the input and output ac ports. The proposed converter comprises an input power factor correction rectification (ac-dc) stage which is followed by an inversion (dc-ac) stage. The rectification stage operates in boost mode during the input positive half line cycle and buck-boost mode during the input negative half line cycle, while the inversion stage operates in buck and buck-boost modes in the output positive and negative half line cycles, respectively. The rectification stage utilizes boundary conduction mode control enabling soft-switching and allowing high-frequency operation. The inversion stage is operated in continuous conduction mode, wherein a digital controller regulates the output voltage of the converter across both resistive and reactive loads. The proposed online UPS utilizes a single (nonsplit) dc-bus between the rectification and inversion stages, resulting in a 50% reduction in dc-bus capacitance requirement compared to conventional split-bus online UPS topologies. Additionally, two battery interface solutions are also investigated which provide a trade-off between passive volume and additional switch-count. To verify the performance and control of the proposed online UPS, a GaN-based electrolytic-free 1-kVA prototype online UPS is designed, built, and tested. The prototype ac-dc-ac converter achieves a peak efficiency of 95.2% and maintains a high efficiency of above 92.3% across the full output power range. The electrolytic-free prototype ac-dc-ac converter for the online UPS achieves a power density of 26.4 W/in <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> .

Remark on a Mittag–Leffler function of Le Roy type

We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag–Leffler function of Le Roy type. It is conjectured that the underlying positive random variable, when it exists, must be logarithmically infinitely divisible.

Some Properties of the Kilbas-Saigo Function

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

The extension of analytic solutions to FDEs to the negative half-line

An analytical framework for the extension of solutions to fractional differential equations (FDEs) to the negative half-line is presented in this paper. The proposed technique is based on the construction of a special characteristic equation corresponding to the original FDE (when the characteristic equation does exist). This characteristic equation enables the construction analytic solutions to FDEs are expressed in the form of infinite fractional power series. Necessary and sufficient conditions for the existence of such an extension are discussed in detail. It is demonstrated that the extension of solutions to FDEs to the negative half-line is not a single-valued operation. Computational experiments are used to illustrate the efficacy of the proposed scheme.

Heavy-tailed distributions, correlations, kurtosis and Taylor's Law of fluctuation scaling.

Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44, 2089-2097; p. 2091) speculated that 'the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ·· ·'. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor's Law (a power-law variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise unexpectedly converges much faster than when they have finite variances.

Infinite rate symbiotic branching on the real line: The tired frogs model

Considérons une population de grenouilles infinitésimales sur la droite réelle. Au début, toutes les grenouilles à droite de l’origine sont endormies tandis que les grenouilles à gauche sont éveillées et bougent comme un flux de chaleur. A l’interface, les grenouilles éveillées qui arrivent essaient de réveiller les grenouilles dormantes. Elles le font avec succès avec une probabilité proportionnelle à leur proportion par rapport à la population totale de grenouilles à cet endroit. Si elles échouent, les grenouilles arrivantes s’endorment aussi. Ce modèle de grenouilles est un modèle spécifique de branchement symbiotique sur la droite réelle où les populations bougent avec des vitesses différentes. Nous construisons le modèle par une procédure d’approximation et nous calculons la structure du processus de sauts. Nous montrons que notre modèle des grenouilles peut être décrit par une équation différentielle partielle stochastique avec sauts.

A Step Up/Down Power-Factor-Correction Converter with Modified Dual Loop Control

A step up/down AC/DC converter with modified dual loop control is proposed. The step up/down AC/DC converter features the bridgeless characteristic which can reduce bridge-diode conduction losses. Based on the step up/down AC/DC converter, a modified dual loop control scheme is proposed to achieve input current shaping and output voltage regulation. Fewer components are needed compared with the traditional bridge and bridgeless step up/down AC/DC converters. In addition, the intermediate capacitor voltage stress can be reduced. Furthermore, the top and bottom switches still have zero-voltage turn-on function during the negative and positive half-line cycle, respectively. Hence, the thermal stresses can also be reduced and balanced. Simulation and experimental results are provided to verify the validity of the proposed step up/down AC/DC converter and its control scheme.

Integrated step‐up non‐isolated inverter with leakage current elimination for grid‐tied photovoltaic system

This study presents a non-isolated step-up inverter without leakage current for low-voltage renewable energy generation such as photovoltaic (PV) cells grid connection. From the structure of the proposed inverter, the negative terminal of the PV array is directly linked with the grid neutral, so the common mode voltage of the parasitic capacitance can be kept constant and the leakage current will not be generated in theory. Moreover, this transformerless inverter can produce an AC output voltage higher than the input DC voltage, which will reduce the sizes of the generation system. A switching modulation strategy carrier-based is analysed in detail when the input inductor current is operated in discontinuous conduction mode. The operation modes of the proposed inverter are also discussed, respectively, in positive and negative half line cycles. Finally, a digital experimental prototype is tested in the laboratory, and the experimental results are consistent with the theoretical analysis.

The Korteweg–de Vries equation on a metric star graph

We prove local well-posedness for the Cauchy problem associated to Korteweg-de Vries equation on a metric star graph with three semi-infinite edges given by one negative half-line and two positives half-lines attached to a common vertex, for two classes of boundary conditions. The results are obtained in the low regularity setting by using the Duhamel Boundary Forcing Operator, in context of half-lines, introduced by Colliander, Kenig (2002),and extended by Holmer (2006) and Cavalcante (2017).

Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation

We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of $M^+$, then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of $M^+$ are fundamental to guarantee the solvability of classical minimization problems given by linear-quadratic control processes.

Splitting and time reversal for Markov additive processes

We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into the behaviour of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener–Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Restricting to the case of no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line. The latter result is obtained using three quite different approaches based on the established path decomposition theory, which further demonstrates its applicability.

Distribution of the smallest visited point in a greedy walk on the line

AbstractWe consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.

On a Brownian motion with a hard membrane

Local perturbations of a Brownian motion are considered. As a limit we obtain a non-Markov process that behaves as a reflecting Brownian motion on the positive half line until its local time at zero reaches some exponential level, then changes a sign and behaves as a reflecting Brownian motion on the negative half line until some stopping time, etc.

Inverse Scattering on the Half-Line Revisited

The inverse scattering problem on the half-line has been studied in the literature in detail. V. Marchenko presented the solution to this problem. In this paper, the invertibility of the steps of the inversion procedure is discussed and a new set of necessary and sufficient conditions on the scattering data is given for the scattering data to be generated by a potential suitable potential. Our proof is new and in contrast with known proof it does not use equations on the negative half-line.

Single-Stage and Boost-Voltage Grid-Connected Inverter for Fuel-Cell Generation System

According to the requirement of fuel cell generation system, this paper presents a new single-stage and boost-voltage grid-connected inverter, as well as the signal modulation diagram in a line cycle. In the proposed inverter, one of the bridge legs is shared with a boost converter. The boost inductor current is designed in discontinuous conduction mode, which can effectively decrease low-frequency current ripple (LFCR) in the output current of the fuel cell. The switching modes of the proposed inverter are analyzed in positive and negative half line cycles, respectively. The LFCR in the output current of the fuel cell is analyzed, from which the design principle of the boost voltage inductor is obtained. The per-unit current values of the shared switches and the loss of the whole system are derived and compared with those of the conventional inverter. Experimental verifications are presented to verify the theoretical analysis.