Scaling Theorem Research Articles

Fluid Limits for Shortest Remaining Processing Time Queues

We consider a single-server queue with renewal arrivals and i.i.d. service times in which the server uses the shortest remaining processing time policy. To describe the evolution of this queue, we use a measure-valued process that keeps track of the residual service times of all buffered jobs. We propose a fluid model (or formal law of large numbers approximation) for this system and, under mild assumptions, prove the existence and uniqueness of fluid model solutions. Furthermore, we prove a scaling limit theorem that justifies the fluid model as a first-order approximation of the stochastic model. The state descriptor of the fluid model is a measure-valued function whose dynamics are governed by certain inequalities in conjunction with the standard workload equation. In particular, these dynamics determine the evolution of the left edge (infimum) of the state descriptor's support, which yields conclusions about response times. We characterize the evolution of this left edge as an inverse functional of the initial condition, arrival rate, and service time distribution. This characterization reveals the manner in which the growth rate of the left edge depends on the service time distribution. By considering varying examples, the authors show that the rate can vary from logarithmic to polynomial.

Investigation of 2P Be− Shape Resonances Using a Quadratically Convergent Complex Multiconfigurational Self-Consistent Field Method

We develop, implement, and apply a quadratically convergent complex multiconfigurational self-consistent field method (CMCSCF) that uses the complex scaling theorem of Aguilar, Balslev, and Combes within the framework of the multiconfigurational self-consistent field method (MCSCF) in order to theoretically investigate the resonances originated due to scattering of a low-energy electron off of a neutral or an ionic target (atomic or molecular). The need to scale the electronic coordinates of the Hamiltonian as prescribed in the complex scaling theorem requires the use of a modified second quantization algebra suitable for biorthonormal spin orbital bases. In order to control the convergence to a stationary point in the complex energy hypersurface, a modified step-length control algorithm is incorporated. The position and width of 2P Be- shape resonances are calculated by inspecting the continuum states of Be-. To our knowledge, this is the first time that CMCSCF has been directly used to determine electron-atom/molecule scattering resonances. We demonstrate that both relaxation and nondynamical correlation are important for accurately describing shape resonances. For all of the calculations, the quadratically convergent CMCSCF was found to converge to the correct stationary point with a tolerance of 1.0 x 10(-10) au for the energy gradient within 10 iterations or less.

A virial theorem for vortices in ferromagnetic elements

We derive a virial theorem for a disc-shaped ferromagnetic particle with an axially symmetric magnetic configuration. This is a generalization of Derrick's scaling theorem which is now valid in the presence of surfaces. The relation gives simple results when applied to elementary magnetic states such as a single domain in an infinitely elongated cylinder. We further calculate the vortex state and verify numerically that it satisfies the virial relation. The vortex profile has a simple form in the limit of a very thin particle where also the virial relation simplifies and effectively gives the vortex core radius. Away from the very thin limit, the vortex configuration becomes more complicated with a varying vortex core radius along the thickness of the particle.

Fast Frequency Estimation by Zero Crossings of Differential Spline Wavelet Transform

Zero crossings or extrema of a wavelet transform constitute important signatures for signal analysis with the advantage of great simplicity. In this paper, we introduce a fast frequency-estimation method based on zero-crossing counting in the transform domain of a family of differential spline wavelets. The resolution and order of the vanishing moments of the chosen wavelets have a close relation with the frequency components of a signal. Theoretical results on estimating the highest and the lowest frequency components are derived, which are particularly useful for frequency estimation of harmonic signals. The results are illustrated with the help of several numerical examples. Finally, we discuss the connection of this approach with other frequency estimation methods, with the high-order level-crossing analysis in statistics, and with the scaling theorem in computer vision.

Oligomer‐to‐Polymer Transition in Short Ethylene Glycol Chains Connected to Mobile Hydrophobic Anchors

We studied the structure of short ethylene glycol (EG) chains with N repeating units (EGN, N = 3, 6, 9, 12, and 15) connected to hydrophobic dihexadecyl chains by means of a combination of differential scanning calorimetry (DSC) and small- and wide-angle X-ray scattering (SAXS/WAXS). These synthetic amphiphiles dispersed in water form planar lamellar stacks and hexagonal cylinders confining the EG chains to restricted geometries. Owing to the self-assembly of the anchoring points, the lateral density of EG chains in planar lamella can be quantitatively controlled. Furthermore, the chain-melting phase transition of the anchors enables us to "switch" the intermolecular distance reversibly. SAXS/WAXS results suggest that the shorter EG chains (N = 3, 6, and 9) assume a helical conformation in stacks of planar lamella. When the EG chains are further elongated (N = 12 and 15), the lamellar periodicities cannot be explained by a linear extrapolation of shorter oligomers, but can be interpreted well as polymer brushes following the scaling theorem. Such rich phase behaviors of EGN molecules can be used as a simple model of oligo/poly-saccharide chains on cell surfaces, which act not only as flexible repellers between neighboring cells but also as stable spacers for functional ligands.

Checking the influence of numerically induced chaos in the computational study of intramolecular dynamics using trajectory equivalence

Influence of the numerical chaos induced by the numerical round-off errors in the classical trajectory study of intramolecular dynamics has been investigated using HOD as an example. Calculation of the trajectory equivalence based on the scaling theorem for classical dynamics has been found useful to check the onset of the numerically induced chaos. It has also been found that the use of higher precision arithmetic is not very effective to delay this onset.

Scaling of classical rate constants on scaled potential-energy surfaces

The scaling relation for the classical rate constants on the scaled potential-energy surfaces has been derived using the scaling theorem in classical dynamics reported previously. This applies to the classical rate constants, both for unimolecular and for bimolecular reactions, that can be obtained by the classical trajectory method and the transition state theory. Validity of the theory has been tested for the prototype reactions, H2CO→H2+CO and Cl+H2→HCl+H. Exact scaling of the rate constants obtained by the classical trajectory calculations has been demonstrated. The rate-energy relations for the former reaction calculated with the statistical Rice–Ramsperger–Kassel–Marcus theory also displayed excellent scaling in the high-energy limit. The scaling relation does not hold rigorously near the reaction threshold due to the quantum mechanical zero-point energy effect. Regardless, the order of magnitude prediction of the threshold rate constant by scaling was possible even in extreme cases. The present method may allow reliable prediction of the classical rate constant by using potential energy data obtained at moderately high levels of electronic structure calculation.

Permittivity measurements using a coaxial-line conical-tipprobe

A conical-tip probe for the measurement of permittivity is presented which retains many of the advantages of the normal-plane probe including ease of sample preparation and moderate sample volume requirement, but which is mechanically better suited than monopole probes for the measurement of a wide range of biological materials since it does not promote free moisture regions resulting from cell rupture. The probe input admittance is found by modelling the probe as a short partially-sheathed monopole in free space and then a well-known antenna scaling theorem is used to determine the admittance and hence permittivity in the dielectric medium.

Scaling theorems for zero crossings of bandlimited signals

Scale-space filtering is the only known method which provides a hierarchic signal description method by extracting features across a continuum of scales. One of its important characteristics is that it demands the filtering involved does not create generic features as the scale increases. It has been shown that the Gaussian filter is unique in holding this remarkable property. This is in essence the so-called scaling theorem. In this paper, we propose two scaling theorems for band-limited signals. They are applicable to a broader class of signals and a bigger family of filtering kernels than in Babaud et al. (1986), Yuille et al. (1986) and Wu-Xie (1990). An in-depth discussion of our theorems and the previously published ones is also given.

Fractal relations. Modelling fractals and analogy with Feigenbaum's period-doubling diagram

The equivalence relation, l, defined previously on Kekulé spaces of benzenoid hydrocarbons (S. El-Basil, J. Chem. Soc., Faraday Trans., 89 (1993) 909; J. Mol. Struct. (Theochem), 288 (1993) 67; J. Math Chem., 14 (1993) 305) is used to map Kekulé spaces of benzenoid systems onto various stages of deterministic fractals including the Cantor set, the Sierpinski triangle, the Sierpinski carpet, the Koch curve and the box fractal as well as certain stages of cellular automata. Furthermore, the Kekulé spaces of quasicrystalline benzenoids (defined by S. El-Basil in J. Chem. Soc., Faraday Trans., 89 (1993) 909) exhibit a period-doubling pattern which manifests itself through a full analogy with Feigenbaum's scaling theorem (M. Feigenbaum, J. Stat. Phys., 19 (1978) 25) including a bifurcation universality constant, a control parameter (of the logistic equation) and cycle-sizes (which can only be integral powers of two). It was found that in certain instances l is also a “ percolation” process where larger clustering (of the Kekulé space) occurs (D. Stauffer, Introduction to Percolation Theory, Taylor and Francis, London, 1985). Cases in which l percolates the Kekulé space correspond to benzenoid systems which are energetically more stable than those in which the resulting clusters are too small to allow percolation of the space by l. A fractal-like scale factor, s, is defined as the limit of the ratio of Clar counts to Kekulé counts in a given homologous series of benzenoid hydrocarbons as the molecule's length approaches infinity. This limit shows that with quasicrystalline benzenoids (branched and/or unbranched), the extent of data reduction using Clar structures is maximum (i.e. s = 0) and hence the use of such structures as a quantum-mechanical basis set in the method of Herndon and Hosoya (W.C. Herndon and H. Hosoya, Tetrahedron, 40 (1984) 3987) is most suited for this class of benzenoid systems. The linear acenes, however, lead to a value of s = 1 and hence, as the size of the molecule gets larger, the computational scheme of Herndon and Hosoya becomes progressively less efficient. This scale factor has certain arithmetical properties which are in harmony with the ultraviolet spectra of the corresponding benzenoid systems.

Absorbed-dose calibrations in high-energy photon beams at the National Physical Laboratory: conversion procedure

The absorbed-dose calibration service from NPL is based on a primary-standard calorimeter that measures absorbed dose to graphite. Secondary-standard dosemeters are calibrated in absorbed dose to water in a 60Co gamma-ray beam and in X-ray beams over a range of generating potentials from 4 MV to 19 MV. Two methods were used to convert the calibrations of working-standard ionization chambers from absorbed dose to graphite into absorbed dose to water. One method involved the use of published interaction data for photons and secondary electrons, and required a knowledge of the chamber construction. The second method involved the calculation of the ratio of absorbed dose in graphite and water phantoms irradiated consecutively in the same photon beam using the photon-fluence scaling theorem. The two methods were in agreement to 0.1%.

Measurements of phantom scatter factors for small field sizes in high energy x rays.

An alternate method to measure the phantom scatter factor in small fields is provided for high energy photon beams. The measurement technique is based on the density scaling theorem described by O'Connor [Phys. Med. Biol. 1, 352-369 (1957)]. The phantom scatter factor (Sp) is measured in balsa and cedar wood to give effective field sizes in 3 x 3 to 0.5 x 0.5 cm2 water. The extrapolated zero area phantom scatter factor from the average data of balsa and cedar is 0.45. This indicates that the variation of output is largely due to a sharp decrease in the phantom scatter factor for small field sizes.

Dosimetry of 125I sources in a low-density material using scaling.

One may apply O'Connor's scaling theorem to dose measurements with brachytherapy sources in order to overcome the difficulties associated with the need for high spatial accuracy. This possibility has been evaluated by measuring the dose distribution around 125I sources in a low-density styrofoam phantom and comparing it with the dose distributions in water and solid water. Some generalization of the scaling theorem is proposed to allow for the minor differences in atomic composition between styrofoam and water, and the distances are scaled according to the ratio of the linear attenuation coefficients, instead of the physical densities, of the two media. The validity of this application of the scaling theorem has also been tested using Monte Carlo calculations. The results indicate that the scaling of the styrofoam measurements to water is a useful approximation in brachytherapy dosimetry.

Exponentially damped Breit-Pauli Hamiltonian for the description of positronium decay and other high energy processes

The conventional explanation for the 1.022 MeV decay of positronium in terms of the annihilation of an electron and its antiparticle is questioned because of the impossibility of proving experimentally that matter actually disappears. Recent work has provided evidence that photons are not destroyed in radiative absorption, for example, but rather attain an undetectable (E=h(nu)=0) state. Questioning the creation-annihilation hypothesis in the case of positronium decay leads to the construction of an exponentially damped Breit-Pauli Hamiltonian which when employed in a Schroedinger equation gives an e+e- ground state with a binding energy of 1.022 MeV. The analogous treatment does not produce any (unobserved) states of similarly low energy for the hydrogen atom, but by virtue of a simple scaling theorem it does lead to a maximal proton-antiproton binding energy of 1.876 GeV, which is exactly mp/me times larger than for e+e-. Arguments are presented to identify the resulting tightly bound e+ e- system with the photon itself, since both are ascribed a zero rest mass. The above Hamiltonian is identified with the unified electroweak interaction, especially since it also proves applicable to the description of neutrino processes. It is pointed out that the analogous (nu)(bar nu) binary systems would be subject to predissociation into free neutrinos and antineutrinos, however, unlike their much more strongly bound e+ e- and p+ p- counterparts. This observation suggests a different explanation for the well-known solar neutrino problem than the flavor-oscillation theory which has been proposed earlier.

Application of the Fano theorem in inhomogeneous media using a convolution algorithm

For photon fields, separate dose distribution kernels have been generated for charged particles produced in the first interaction, for single and multiple scattered photons, including bremsstrahlung and annihilation. These kernels are applied in absorbed dose calculations for radiotherapy treatment planning using a convolution technique. The vast amount of kernel data required for 3D calculations can be accurately generated out of a small subset, due to rotational symmetry. Due to the discrete sampling of both the irradiated object and the dose distribution kernels, application of the Fano theorem and O'Connor's scaling theorem is not possible without difficulties. The scaling process has been thoroughly investigated and a density dependent correction factor applied to the central kernel value has been derived. The calculated absorbed dose distributions were found to agree well with Monte Carlo calculated data using the EGS4 program.

High-energy x-ray beam calibrations using scaling.

When calibrating high-energy x-ray beams, it is convenient to measure in a plastic phantom and convert the result to absorbed dose in water at a specified depth, field size, and distance from the source. By combining the scaling theorem with the observation that TAR is independent of the distance to the source, it is demonstrated that the measurement in plastic can be made at the same source-detector distance but scaling depth and field size. Experiments to support this were made in acrylic and water with 6- and 25-MV x rays and at different distances from the source. The agreement between the calibrations in acrylic and water was comparable to that using the current national protocol, which recommends the scaling of depth only while using the same field size and source-detector distance in the two media and correcting for the difference in scatter dose. It was confirmed that the scaling factor may be the ratio between the electron densities of acrylic and water, or the ratio between the linear attenuation coefficients. A method to determine this factor directly is also described.

On the scaling theorem for interacting Fleming-Viot processes

We prove an extended version of the scaling theorem for interacting Fleming—Viot processes, in the sense of weak convergence for stochastic processes.

Interacting Fleming-Viot processes

We construct a class of interacting Ohta-Kimura stepwise-mutation models and study their macroscopic behavior, i.e., we prove their weak convergence when the population size n increases to infinity, the evolution is speeded up by a factor n 2, the increment effects of a change in each population's distribution of characteristics under study is scaled down by a factor n -1 2 and the level of interaction between populations is decreased by a factor n -1. The measure-valued limits—interacting Fleming-Viot processes—are characterized as unique solutions to certain martingale problems using the method of duality. Finally, we prove a scaling theorem for these processes.

10‐MV x‐ray primary and scatter dose calculation using convolutions

Three-dimensional (3-D) forward and backward primary dose spread functions in water were developed for 10-MV x rays. Three-dimensional forward and backward primary dose spread functions in a heterogeneous medium were constructed from the ones in water using the density scaling theorem. Each of the forward and backward primary dose components were calculated using a method of convolving the primary water collision kerma distribution with the forward or backward primary dose spread function. Scatter dose was separated into forward and backward components. Each scatter dose component was calculated using a differential scatter method, a kind of convolution method, where the primary water collision kerma distribution was convolved with a differential scatter-maximum ratio or differential backscatter factor equation. From the dose calculation and measurement results obtained for water phantoms containing a cork or aluminum slab, it was found that the 3-D forward and backward primary dose functions were effective especially in regions where there was a loss of longitudinal and/or lateral electronic equilibrium.

Monte Carlo法を用いた体内散乱線の解析 : 密度尺度理論についての考察

The density scaling theorem, which is based on 3D CT data, is applied to recent calculating algorithms for inhomogeneous correction in radiotherapy treatment planning. Using the Monte Carlo method, we examined the accuracy of the density scaling theorem, when it was used to convert absorbed doses (primary doses and scattered doses) in several kinds of media into doses in water which is basis of the dose calculations. Absorbed doses in media, which effective atomic number is the same as that of water, coincided with doses in water by means of the density scaling theorem with maximum error of 1%. Absorbed doses in compact bone model, which effective atomic number is much larger compared with that of water, deviated from doses in water despite the conversion by the density scaling theorem. However, subjoining the correction for ratio of mass attenuation coefficients of each medium with the density scaling theorem, doses in compact bone model coincided satisfactorily with that in water.