Asymptotic Frequency Research Articles

Magnetic Field Reference Levels for Arbitrary Periodic Waveforms for Prevention of Peripheral Nerve Stimulation.

Guidelines for prevention of peripheral nerve stimulation from exposure to low frequency magnetic fields have been developed by standard-setting bodies. Exposure limits or reference levels (RLs) are typically set in terms of the maximum root-mean-square amplitude of a sinusoidal waveform; however, environmental flux densities are often periodic, non-sinusoidal waveforms. This work presents a procedure for deriving RLs for any generalized periodic waveform using the empirical nerve-stimulation threshold data obtained from human volunteer MRI experiments. For this purpose, the "Law of Electrostimulation" (LOE), which sets forth conditions of a waveform necessary to trigger the action potential required to depolarize cell membranes, is applied to various waveforms. The results of the LOE analysis are waveform-specific, amplitude thresholds of stimulation that are found in terms of the empirically-derived rheobase threshold time-rate-of-change flux density and chronaxie from trapezoidal pulse MRI experiments. The thresholds are converted to amplitude RLs in two asymptotic frequency regimes as per the usual practice in standard setting. The resulting RLs have the same frequency dependence as in existing standards (i.e., inverse-frequency below a transition frequency and flat above). It is shown that the transition frequency is dependent only on the shape of the waveform. Both sinusoidal and non-sinusoidal waveforms have identical peak-to-peak amplitude RLs above their respective transition frequencies. Below these frequencies, all peak-to-peak amplitude RLs have the same functional dependence on frequency when the frequency is normalized to the waveform-specific transition frequency. This results in simple criteria for testing the amplitude of any arbitrary periodic waveform against potential for stimulation. These criteria are compared to guidance given for non-sinusoidal waveforms in the ICNIRP 1 Hz-100 kHz exposure standard.

Critical behavior at dynamical phase transition in the generalized Bose-Anderson model

Critical properties of the dynamical phase transition in the quenched generalized Bose-Anderson impurity model are studied in the mean-field limit of an infinite number of channels. The transition separates the evolution toward ground state and toward the branch of stable excited states. We perform numerically exact simulations of a close vicinity of the critical quench amplitude. The relaxation constant describing the asymptotic evolution toward ground state, as well as asymptotic frequency of persistent phase rotation and number of cloud particles at stable excited state are power functions of the detuning from the critical quench amplitude. The critical evolution (separatrix between the two regimes) shows a non-Lyapunov power-law instability arising after a certain critical time. The observed critical behavior is attributed to the irreversibility of the dynamics of particles leaving the cloud and to memory effects related to the low-energy behavior of the lattice density of states.

A weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrödinger equations

We prove a weak form of the soliton resolution conjecture for — [Formula: see text] uniformly bounded in time — solutions of semilinear fourth-order Schrödinger equations, in dimensions [Formula: see text], and with a mass supercritical-energy subcritical power type nonlinearity, by using a strategy developed by T. Tao in 2007. More precisely, we prove that the solutions are decomposed into a sum of two terms: a free solution and a nonradiative term that approaches asymptotically an object that has similar properties to those of a finite sum of solitons. The asymptotic behavior of the nonradiative term is derived from its asymptotic frequency localization and its asymptotic spatial localization. There are two main differences between the present contribution and T. Tao’s earlier work. The first one appears when we prove the asymptotic frequency localization: we fill a gap of regularity by using the better dispersive properties of the high frequency pieces of the free solution. The second one appears when we prove the asymptotic spatial localization. A key estimate depends on the fundamental solution that does not have an explicit form. We overcome the difficulty by introducing a modified fundamental solution and exploiting the symmetries of the characters of the phases.

High-efficiency surface plasmonic polariton waveguides with enhanced low-frequency performance in microwave frequencies.

In this paper, a planar waveguide based on spoof surface plasmon polaritons (SSPPs) with metals on both sides of the corrugated strip as grounds is firstly proposed in microwave region. Simple and efficient conversion between guided waves and SSPPs is realized by gradient corrugated strip with grounds on both sides. Compared with plasmonic waveguide with flaring ground [Laser Photonics Rev. 8, 146 (2014)], the addition of grounds suppresses the radiation loss effectively and improves the low-frequency performance with tighter field confinement, which leads to a wider operating bandwidth. Moreover, as the asymptotic frequency of SSPPs decreasing, the confinement of SSPPs is further enhanced by a defected ground structure (DGS), which is achieved by the periodic grooves symmetrical to those on the corrugated strip. Therefore, miniaturization of the proposed waveguide can be realized. Measured results validate both high efficiency of momentum and impedance matching and enhanced performance in the region of lower frequencies with the wave vectors close to those in free space. Such results have significant values in plasmonic functional devices and integrated circuits in microwave frequencies.

Plasmonic waveguide with folded stubs for highly confined terahertz propagation and concentration.

We proposed a novel planar terahertz (THz) plasmonic waveguide with folded stub arrays to achieve excellent terahertz propagation performance with tight field confinement and compact size based on the concept of spoof surface plasmon polaritons (spoof SPPs). It is found that the waveguide propagation characteristics can be directly manipulated by increasing the length of the folded stubs without increasing its lateral dimension, which exhibits much lower asymptotic frequency of the dispersion relation and even tighter terahertz field confinement than conventional plasmonic waveguides with rectangular stub arrays. Based on this waveguiding scheme, a terahertz concentrator with gradual step-length folded stubs is proposed to achieve high terahertz field enhancement, and an enhancement factor greater than 20 is demonstrated. This work offers a new perspective on very confined terahertz propagation and concentration, which may have promising potential applications in various integrated terahertz plasmonic circuits and devices, terahertz sensing and terahertz nonlinear optics.

Hawking Radiation-Quasinormal Modes Correspondence for Large AdS Black Holes

It is well-known that the nonstrictly thermal character of the Hawking radiation spectrum generates a natural correspondence between Hawking radiation and black hole quasinormal modes. This main issue has been analyzed in the framework of Schwarzschild black holes, Kerr black holes, and nonextremal Reissner-Nordstrom black holes. In this paper, by introducing the effective temperature, we reanalyze the nonstrictly thermal character of large AdS black holes. The results show that the effective mass corresponding to the effective temperature is approximatively the average one in any dimension. And the other effective quantities can also be obtained. Based on the known forms of frequency in quasinormal modes, we reanalyze the asymptotic frequencies of the large AdS black hole in three and five dimensions. Then we get the formulas of the Bekenstein-Hawking entropy and the horizon’s area quantization with functions of the quantum “overtone” number n.

Asymptotic frequency of shapes in supercritical branching trees

Abstract The shapes of branching trees have been linked to disease transmission patterns. In this paper we use the general Crump‒Mode‒Jagers branching process to model an outbreak of an infectious disease under mild assumptions. Introducing a new class of characteristic functions, we are able to derive a formula for the limit of the frequency of the occurrences of a given shape in a general tree. The computational challenges concerning the evaluation of this formula are in part overcome using the jumping chronological contour process. We apply the formula to derive the limit of the frequency of cherries, pitchforks, and double cherries in the constant-rate birth‒death model, and the frequency of cherries under a nonconstant death rate.

Asymptotic Time Averages and Frequency Distributions

Consider an arbitrary nonnegative deterministic process (in a stochastic setting {X(t), t≥0} is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space S=(-∞,∞). Using a sample-path approach, we give necessary and sufficient conditions for the long-run time average of a measurable function of process to be equal to the expectation taken with respect to the same measurable function of its long-run frequency distribution. The results are further extended to allow unrestricted parameter (time) space. Examples are provided to show that our condition is not superfluous and that it is weaker than uniform integrability. The case of discrete-time processes is also considered. The relationship to previously known sufficient conditions, usually given in stochastic settings, will also be discussed. Our approach is applied to regenerative processes and an extension of a well-known result is given. For researchers interested in sample-path analysis, our results will give them the choice to work with the time average of a process or its frequency distribution function and go back and forth between the two under a mild condition.

Distribution of the quasispecies for a Galton–Watson process on the sharp peak landscape

Abstract We study a classical multitype Galton–Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to ∞ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k digits from the master sequence approaches (σe-a - 1)(ak/k!)∑i≥ 1ik/σi, where σ is the selective advantage of the master sequence and a is the product of the length of the chains with the mutation probability. The probability distribution Q(σ, a) on the nonnegative integers given by the above equation is the quasispecies distribution with parameters σ and a.

Low-pass plasmonic filter and its miniaturization based on spoof surface plasmon polaritons

In this work, a low-pass plasmonic filter based on spoof surface plasmon polaritons (SSPPs) is reported by etching the two grounds of coplanar waveguide with periodic grooves. The cutoff frequency of the filter coincides with the asymptotic frequency of the SSPPs that are supported by the periodic grooves, which can be tuned by the depth of the grooves at will. In addition, a T-shaped groove is designed to increase the equivalent depth of the groove so as to realize the miniaturization of the proposed filter by decreasing the equivalent cutoff frequency of the SSPPs. Measured S-parameters of the two proposed low-pass plasmonic filters agree well with the simulated ones, which validates our ideas and designs. This simple low-pass plasmonic filter can find potential applications in plasmonic circuits and systems.

A weak first digit law for a class of sequences

A non-ergodic approach is employed to establish Benford's law for the leading digit d = 1 for the sequence of powers of two. For the sequence of powers f ng, 1 < 10=9, this method is extended to obtain a weak rst digit law by establishing Benford like lower and upper bounds to the asymptotic relative frequency of terms with a given leading digit.

NORMALITY OF THE THUE–MORSE SEQUENCE ALONG PIATETSKI-SHAPIRO SEQUENCES

We prove that for $1<c<4/3$ the subsequence of the Thue--Morse sequence $\mathbf t$ indexed by $\lfloor n^c\rfloor$ defines a normal sequence, that is, each finite sequence $(\varepsilon_0,\ldots,\varepsilon_{T-1})\in \{0,1\}^T$ occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf t\left(\lfloor n^c\rfloor\right)$ with asymptotic frequency $2^{-T}$.

Large-time dynamics of the asymptotic Lohe model with a small time-delay

We study the asymptotic behavior of an ensemble of identical Lohe oscillators on the unit sphere in the presence of small time delay interaction effects. When there is no time delay, the ensemble of identical Lohe oscillators collapses asymptotically to a one-cluster ensemble on the sphere; its asymptotic dynamics are governed by linear motion on the unit sphere with a constant natural velocity. We show that the presence of a small time delay can induce rich dynamical features such as asymptotic changes in the velocity and asymptotic low-dimensional dynamics in high-dimensional cases. For d = 1, the Lohe dynamics is equivalent to the Kuramoto dynamics via polar coordinates. In this case, the modified asymptotic frequency is uniquely determined by an implicit relation based on the natural frequency, coupling strength, and time delay. For d = 3, we show that the dynamics of identical Lohe oscillators converges to the Kuramoto dynamics for properly chosen initial configurations. We also provide several numerical simulations to confirm our analytical results.

A controller design method for 3 phase 4 wire grid connected VSI with LCL filter

Closed loop control of a grid connected VSI requires line current control and dc bus voltage control. The closed loop system comprising PR current controller and grid connected VSI with LCL filter is a higher order system. Closed loop control gain expressions are therefore difficult to obtain directly for such systems. In this work a simplified approach has been adopted to find current and voltage controller gain expressions for a 3 phase 4 wire grid connected VSI with LCL filter. The closed loop system considered here utilises PR current controller in natural reference frame and PI controller for dc bus voltage control. Asymptotic frequency response plot and gain bandwidth requirements of the system have been used for current control and voltage controller design. A simplified lower order model, derived for closed loop current control, is used for the dc bus voltage controller design. The adopted design method has been verified through experiments by comparison of the time domain response.

Vibration insight of a nonlocal viscoelastic coupled multi-nanorod system

Longitudinal vibration of viscoelastic multi-nanorod system (VMNS) is studied. Based on the D' Alembert's principles, nonlocal and viscoelastic constitutive relations, the system of m partial differential equations are derived which described the motion of the presented nano-system. ClampedeClamped and ClampedeFree boundary conditions and two different chain systems, namely “Clamped-Chain” and “Free-Chain” are illustrated. The method of separations of variables and trigonometric method are utilized for solutions. The analytical expressions for critical viscoelastic parameters and asymptotic frequencies are presented. The predicted results are validated with results obtained by direct numerical simulations and results from literature. The effects of nonlocal parameter, number of nanorods, viscoelastic material constant and parameter of viscoelastic layer on the complex eigenvalue are discussed in details.

One-Dimensional Spoof Surface Plasmon Structures for Planar Terahertz Photonic Integration

The waveguiding properties of spoof surface plasmon modes propagating along three kinds of one-dimensional groove arrays composed of V-shaped, rectangular, and trapezoid grooves, respectively, are numerically investigated. The results show that the trapezoid groove array demonstrates a tighter confinement compared to the other two groove arrays with the same groove opening. The propagation loss of the three structures at the same frequency are analyzed, from which it is concluded that the propagation loss of spoof plasmon waveguides is determined not only by the difference between the operation frequency and asymptotic frequency but also by the loss associated with the TEM mode inside individual grooves. The bending loss through small bending radii of the three structures (~0.5λ) is also calculated showing that although the straight trapezoid groove array exhibits highest propagation loss, the transmission loss through a 90° bend of it is lower resulting from the tighter confinement than that of the other two groove array bends having the same radius. This property together with the ease of fabrication makes the trapezoid structure a better candidate for planar terahertz photonic integration on the subwavelength scale.

Harmonic functions in union of chambers

We characterize the set of harmonic functions with Dirichlet boundary conditions in unboundeddomains which are union of two different chambers. We analyse the asymptotic behavior of the solutions in connection withthe changes in the domain's geometry; we classify all (possibly sign-changing) infinite energy solutions having given asymptotic frequency atthe infinite ends of the domain; finally we sketch the case of several different chambers.

Stabilizing Switching Signals for Switched Systems

This article deals with stability of continuous-time switched systems under constrained switching. Given a family of systems, possibly containing unstable dynamics, we characterize a new class of switching signals under which the switched system generated by it and the family of systems is globally asymptotically stable. Our characterization of such stabilizing switching signals involves the asymptotic frequency of switching, the asymptotic fraction of activation of the constituent systems, and the asymptotic densities of admissible transitions among them. Our techniques employ multiple Lyapunov-like functions, and extend preceding results both in scope and applicability.

A copolymer near a selective interface: Variational characterization of the free energy

In this paper we consider a random copolymer near a selective interface separating two solvents. The configurations of the copolymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The excursion length distribution is assumed to have a tail that is logarithmically equivalent to a power law with exponent � � 1. The monomers carry i.i.d. real-valued types whose distribution is assumed to have zero mean, unit variance, and a finite moment generating function. The interaction Hamiltonian rewards matches and penalizes mismatches of the monomer types and the solvents, and depends on two parameters: the interaction strength � � 0 and the interaction bias h � 0. We are interested in the behavior of the copolymer in the limit as its length tends to infinity. The quenched free energy per monomer (�;h) 㜡 g que (�;h) has a phase transition along a quenched critical curve � 㜡 h que (�) separating a localized phase, where the copolymer stays close to the interface, from a delocalized phase, where the copolymer wanders away from the interface. We derive variational formulas for both these quantities. We compare these variational formulas with their analogues for the annealed free energy per monomer (�;h) 㜡 g ann (�;h) and the annealed critical curve � 㜡 h ann (�), both of which are explicitly computable. This comparison leads to: (1) A proof that h ann (�=� ) 1 and � > 0. (2) A proof that g que (�;h) < g ann (�;h) for all � � 1 and (�;h) in the annealed localized phase. (3) An estimate of the total number of times the copolymer visits the interface in the interior of the quenched delocalized phase. (4) An identification of the asymptotic frequency at which the copolymer visits the interface in the quenched localized phase. The copolymer model has been studied extensively in the literature. The goal of the present paper is to open up a window with a variational view and to settle a number of open problems.

Self-Similarity in Population Dynamics: Surname Distributions and Genealogical Trees

The frequency distribution of surnames turns out to be a relevant issue not only in historical demography but also in population biology, and especially in genetics, since surnames tend to behave like neutral genes and propagate like Y chromosomes. The stochastic dynamics leading to the observed scale-invariant distributions has been studied as a Yule process, as a branching phenomenon and also by field-theoretical renormalization group techniques. In the absence of mutations the theoretical models are in good agreement with empirical evidence, but when mutations are present a discrepancy between the theoretical and the experimental exponents is observed. Hints for the possible origin of the mismatch are discussed, with some emphasis on the difference between the asymptotic frequency distribution of a full population and the frequency distributions observed in its samples. A precise connection is established between surname distributions and the statistical properties of genealogical trees. Ancestors tables, being obviously self-similar, may be investigated theoretically by renormalization group techniques, but they can also be studied empirically by exploiting the large online genealogical databases concerning European nobility.