Curves Of Width Research Articles

Real algebraic curves of constant width

Rabinowitz constructed a parametric curve of constant width and expressed it as a plane algebraic curve; however, the algebraic curve also contains isolated points separate from the original curve. We show how to modify his example in order to produce a curve with no isolated points. We then conjecture a method for producing a family of such curves and prove the conjecture in several cases.

Changes of Time-Attenuation Curve Blood Flow Parameters in Patients with and without Carotid Stenosis

From the time-attenuation curves of DSA flow parameters, maximal intensity, maximal slope, and full width at half maximum of selected vascular points are defined. The study explores the reliability of defining the flow parameters by the time-attenuation curves of DSA. Seventy patients with unilateral carotid artery stenosis (group A) and 56 healthy controls (group B) were retrospectively enrolled. Fixed contrast injection protocols and DSA acquisition parameters were used with all patients. The M1, sigmoid sinus, and internal jugular vein on anteroposterior view DSA and the M2, parietal vein, and superior sagittal sinus on lateral view DSA were chosen as ROI targets for measuring flow parameters. The difference of time of maximal intensity between 2 target points was defined as the circulation time between the target points. The maximal intensity difference of 2 selected points from the ICA to the M1, sigmoid sinus, internal jugular vein, M2, parietal vein, and superior sagittal sinus was significantly longer in group A than in group B. The maximum slope of M1, M2, and the superior sagittal sinus was significantly lower in group A than in group B. The full width at half maximum of M1 and M2 was significantly larger in group A than in group B. The maximal slope of M1 demonstrated the best diagnostic performance. The maximal intensity difference of 2 selected points derived from DSA can be used as a definitive alternative flow parameter for intracranial circulation time measurement. Maximal slope and full width at half maximum complement the maximal intensity difference of 2 selected points in defining flow characteristics of healthy subjects and patients with carotid stenosis.

A note on algebraic convex curves of constant width

We show that if a plane convex curve of constant width is sufficiently smooth and has strictly positive curvature at all of its points, then it can be approximated by algebraic convex curves of constant width.

Experimental Study on Mechanical Behavior of Serving Prestressed Concrete Poles

In order to assess accurately safety performance of prestressed concrete poles servicing 30 years, bending loading tests are carried out on 3 samples extracted randomly, these ones are studied on the mechanical behavior and failure mechanism, failure modes are revealed, and important experimental data including cracking load, the ultimate load of normal use, the ultimate load of carrying capacity is obtained. Furthermore, based on measured test data, the curves of load-crack width and moment-deflection are made among the whole force process. The results show that failure modes of specimens are similar to one of “rare-reinforced beams”. Moreover, based on design method of serviceability limit state, the average security surplus coefficient is 1.2 before collapse damage.

Modeling and performance analysis of energy selective electron (ESE) engine with heat leakage and transmission probability

A model of an energy selective electron (ESE) engine with linear heat leakage and Lorentzian transmission probability is established in this paper. The expressions for the main performance parameters of the ESE engine operating as a heat engine or a refrigerator are derived by using the theory of finite time thermodynamics. The optimum performances of the ESE engine are explored and the influences of the heat leakage, the central energy level of the resonance, and the width of the resonance on the performance of the ESE engine are analyzed by using detailed numerical examples. The optimal operation regions of power output and efficiency (or cooling load and coefficient of performance (COP)) are also discussed. Moreover, the performances of the ESE engine with Lorentzian transmission probability are compared with those with rectangular transmission probability. It is shown that the power output versus efficiency (or cooling load versus COP) characteristic curves with and without heat leakage are all closed loop-shaped ones. The efficiency (or COP) of the ESE engine decreases as the heat leakage increases. It is found that as the resonance width increases, the power output and efficiency (or cooling load and COP) increase to a maximum and then decrease due to the finite range of energies which contribute positively to the power generation or refrigeration in the electron system. Especially, when heat leakage is taken into account, the characteristic curves of maximum efficiency (or maximum COP) versus half resonance width are parabolic-like ones, which are quite different from the monotonic decreasing characteristic curves obtained in previous analyses without considering heat leakage. The results obtained in this paper can provide some theoretical guidelines for the design and operation of practical electron energy conversion devices such as solid-state thermionic refrigerators.

The Technology Research of CO<sub>2</sub> Laser Micropore Drilling on Carbon Steel

. Five kind of carbon steel were chosen to do the test of CO2 laser micropore drilling in the paper, it were obtained that the influence curves of laser power, pulse width, defocusing amount and repetition rate to the upper aperture and the taper of the hole, and the influence of laser parameters and the carbon content to the drilling quality were system analyzed using double factors without repeating test method, the result show that the value of laser power and pulse width has significant influence to the upper aperture, to taper of hole is less. The influence of the repetition rate of laser and defocus amount to the upper aperture was weak, and the influence to the upper aperture was very significant with the change of the value of carbon content and the repetition rate.

Measuring and Analyzing Cross-Sectional Profiles of Rotating Objects Using Light Sectioning

This paper presents the instrumentation and analysis methods for the measurement of cross-sectional profiles. The object is rotated in front of a light-sectioning system, which repeatedly measures the local geometry. A least-mean-square approximation of the local geometry by a circle is performed. The radius of the circle is used as an estimate for the radius of curvature, and the position of the upper horizontal tangent to the circle is used to estimate the position of the object. It is shown that the tangent position is a more accurate measure of position than the center of the circle. The period of revolution is determined from the aperiodic autocorrelation sequence of the eccentricity of the object. The corresponding to the profile of the object is a smooth Jordan curve: This fact is used to take advantage of harmonic filtering to suppress noise and perturbations and to reconstruct the from the measured radii of curvature. The implementation of the curve from algorithm in Matlab is presented. Diameter deviations of the are analyzed by simulating a caliper measurement. The periodicity of the curvature is used as a measure to firmly detect curves of constant width. A least-square approximation of curves of constant width using elliptical Fourier descriptors is presented. All methods are demonstrated with real data from industrial measurements.

A new construction of curves of constant width

We show that a curve of constant width in the Euclidean plane can be continuously constructed from a Reuleaux triangle while preserving the property of having constant width in all intermediate steps of this procedure. Also we discuss some consequences of this constructive approach.

Inflection points and double tangents on anti-convex curves in the real projective plane

A simple closed curve in the real projective plane is called anti-convex if for each point on the curve, there exists a line which is transversal to the curve and meets the curve only at that given point. Our main purpose is to prove an identity for anti-convex curves that relates the number of independent (true) inflection points and the number of independent double tangents on the curve. This formula is a refinement of the classical Möbius theorem. We also show that there are three inflection points on a given anti-convex curve such that the tangent lines at these three inflection points cross the curve only once. Our approach is axiomatic and can be applied in other situations. For example, we prove similar results for curves of constant width as a corollary.

Axial temperature gradient and mobile phase gradient in microcolumn high-performance liquid chromatography

The effect of axial temperature gradient (ATG) along a microcolumn on the separation performance at both isocratic and gradient elution mode was investigated. A thermostat system was designed to form an ATG along the packed column. Polycyclic aromatic hydrocarbons (PAHs) were separated on a 0.53 mm × 150 mm i.d. 5 μm C 18 microcolumn, with water and acetonitrile as mobile phase. The separation results obtained at mobile phase gradient (MPG) and ATG in microcolumn HPLC were compared with the results performed at ambient conditions. Extrapolated curves of peak width at half height ( w h ) versus ln k showed that w h is narrower at the same retention time when ATG was applied in addition to MPG. The column efficiency was enhanced 20–30% and the resolution was slightly reduced because of reduction of selectivity at elevated temperature at ATG condition. The RSD of retention time in ATG mode was less than 2.5%.

Curves of constant width in the non-euclidean geometry

Curves of constant width in the non-euclidean geometry

Analysis of wavelength-dependent photoisomerization quantum yields in bilirubins by fitting two exciton absorption bands

The absorption spectra of bilirubins were deconvoluted by two Gaussian curves of equal width representing the exciton bands of the non-degenerate molecular system. The two bands were used to study the wavelength dependence of the (4Z, 15Z) → (4Z, 15E) configurational photoisomerization quantum yield of the bichromophoric bilirubin-IXα (BR-IX), the intrinsically asymmetric bile pigment associated with jaundice and the symmetrically substituted bilirubins (bilirubin-IIIα and mesobilirubin-XIIIα), when they are irradiated in aqueous solution bound to human serum albumin (HSA). The same study was performed for BR-IX in ammoniacal methanol solution (NH4OH/MeOH). The quantum yields of the configurational photoprocesses were fitted with a combination function of the two Gaussian bands normalized to the total absorption, using the proportionality coefficients and a scaling factor as parameters. The decrease of the (4Z, 15Z) → (4Z, 15E) quantum yield with increasing wavelength, which occurs for wavelengths longer than the most probable Franck–Condon transition of the molecule, did not result in a unique function of the exciton absorptions. In particular we found two ranges corresponding to different exciton interactions with different proportionality coefficients and scaling factors. The wavelength-dependent photoisomerization of bilirubins was described as an abrupt change in quantum yield as soon as the resulting excitation was strongly localized in each chromophore. The change was correlated to a variation of the interaction between the two chromophores when the short-wavelength exciton absorption became vanishingly small. With the help of the circular dichroism (CD) spectrum of BR-IX in HSA, a small band was resolved in the bilirubin absorption spectrum, delivering part of the energy required for the (4Z, 15Z) → (4Z, 15E) photoisomerization of the molecule.

A comparison of treatment strategies for hypoplastic left heart syndrome using decision analysis

The possibility of electrical earthquake precursors (EEPs) has long been appreciated, but to date there still exists neither a solid theory to describe their generation and expected characteristics, nor proven techniques to identify and discriminate true precursors from noise. Experimental studies have produced a prolific variety of signal shape, complexity and duration, but no explanation for the apparently indefinite diversity. Statistical analyses on the basis of such poorly constrained data were inconclusive, leading to scepticism and intense debate. The most objective means of EEP identification would be to construct generic models of their source(s) and compare the model predictions with field observations. We attempt to show the merits of this approach with two studies. The first study expands on the phenomena of spontaneous electric field generation during crack propagation (microfracturing), demonstrated by laboratory experiments. Large-scale microfracturing may appear at the terminal stage of earthquake preparation. We apply a generic, qualitative model, based on a kinetic theory of crack interaction and propagation. The model suggests that EEP signals from such a type of source may have a limited class of permissible waveforms, with characteristic bay- or bell-shaped curves of variable width and duration. We provide two examples consistent with this model: the VAN claims of precursors on 15/1/1983 and 18/1/1983. The magnetic field that may accompany an anomalous electric signal is the subject of the second study. This has been a grossly overlooked quantity, although valuable for identification and discrimination, because it is considerably less sensitive to distortion than the electric field, less sensitive to inhomogeneities along the propagation path, insensitive to the local geoelectric structure and sometimes, telltale of the source (for instance, external magnetic fields can only be generated by (sub)horizontal current configurations). We investigate the 18/4/1995 and 19/4/1995 electric and magnetic signals observed at Ioannina, Greece, used for the prediction of the 13/5/1995 M6.6 Kozani event by the VAN group. The electric and magnetic waveforms are inconsistent with the crack propagation model. By their observed characteristics, the magnetic signals preclude any (sub)vertical electrokinetic current. Using analytic formulations, we investigate whether they might have been generated by an electrokinetic source across a lateral interface, either at the focal area or locally, at Ioannina. We conclude that the magnetic field properties are also inconsistent with such a type of source. Conversely, we cannot rule out their local industrial origin. The examples presented herein indicate that the successful identification and discrimination of EEP and noise may be possible by working out plausible theories of the source.

Identification and discrimination of transient electrical earthquake precursors: fact, fiction and some possibilities

The possibility of electrical earthquake precursors (EEPs) has long been appreciated, but to date there still exists neither a solid theory to describe their generation and expected characteristics, nor proven techniques to identify and discriminate true precursors from noise. Experimental studies have produced a prolific variety of signal shape, complexity and duration, but no explanation for the apparently indefinite diversity. Statistical analyses on the basis of such poorly constrained data were inconclusive, leading to scepticism and intense debate. The most objective means of EEP identification would be to construct generic models of their source(s) and compare the model predictions with field observations. We attempt to show the merits of this approach with two studies. The first study expands on the phenomena of spontaneous electric field generation during crack propagation (microfracturing), demonstrated by laboratory experiments. Large-scale microfracturing may appear at the terminal stage of earthquake preparation. We apply a generic, qualitative model, based on a kinetic theory of crack interaction and propagation. The model suggests that EEP signals from such a type of source may have a limited class of permissible waveforms, with characteristic bay- or bell-shaped curves of variable width and duration. We provide two examples consistent with this model: the VAN claims of precursors on 15/1/1983 and 18/1/1983. The magnetic field that may accompany an anomalous electric signal is the subject of the second study. This has been a grossly overlooked quantity, although valuable for identification and discrimination, because it is considerably less sensitive to distortion than the electric field, less sensitive to inhomogeneities along the propagation path, insensitive to the local geoelectric structure and sometimes, telltale of the source (for instance, external magnetic fields can only be generated by (sub)horizontal current configurations). We investigate the 18/4/1995 and 19/4/1995 electric and magnetic signals observed at Ioannina, Greece, used for the prediction of the 13/5/1995 M6.6 Kozani event by the VAN group. The electric and magnetic waveforms are inconsistent with the crack propagation model. By their observed characteristics, the magnetic signals preclude any (sub)vertical electrokinetic current. Using analytic formulations, we investigate whether they might have been generated by an electrokinetic source across a lateral interface, either at the focal area or locally, at Ioannina. We conclude that the magnetic field properties are also inconsistent with such a type of source. Conversely, we cannot rule out their local industrial origin. The examples presented herein indicate that the successful identification and discrimination of EEP and noise may be possible by working out plausible theories of the source.

Temperature-dependent bitumen softening studied by NMR

The study of the temperature-dependent bitumen softening by thenuclear magnetic resonance (NMR) techniques of proton NMR line shape andspin-lattice relaxation is presented. The increased molecular translationaland rotational mobility on bitumen softening results in a dynamictransition of the NMR line shape from a static rigid-lattice spectrum to amotionally narrowed spectrum that is accompanied by a drastic decrease ofthe spectral width by a factor of 300. In the type of bitumen investigated,which is of high industrial importance for the road construction industry,the majority of the line width change (and thus the softening) happens inthe temperature interval between + 40 °C and -20 °C (~310-250 K). This is exactly the range of the summer-winter atmospherictemperature variations in most geographical regions on Earth. Thetemperature-dependent curves of the full width at half height of the protonNMR spectrum (that we propose to call the bitumen `NMR melting curves')represent an accurate analytical tool for the study of the bitumensoftening process. Due to its simplicity, high accuracy and ease ofimplementation in industrial laboratories, the method shows the potentialto become a standard analytical tool for the bitumen basic research. It mayalso serve as an on-line bitumen quality control method in industrialprocesses, like the asphalt covering of roads and highways.

Curves of constant width

Curves of constant width have the same breadth regardless of how they are rotated. Highly noncircular curves with this property may be constructed geometrically. Such curves make good rollers, manhole covers, and allow one to drill nearly square holes.

Analytical solution of 1D Ising-like systems modified by weak long range interaction

It is well-known that 1D systems with only nearest neighbour interaction exhibit no phase transition. It is shown that the presence of a small long range interaction treated by the mean field approximation in addition to strong nearest neighbour interaction gives rise to hysteresis curves of large width. This situation is believed to exist in spin crossover systems where by the deformation of the spin changing molecules, an elastic coupling leads to a long range interaction, and strong bonding between the molecules in a chain compound leads to large values for nearest neighbour interaction constants. For this interaction scheme an analytical solution has been derived and the interplay between these two types of interaction is discussed on the basis of experimental data of the chain compound [Fe(Htrz)2(trz)](BF4)2 which exhibits a very large hysteresis of 50 K above RT at 370 K. The width and shape of the hysteresis loop depend on the balance between long and short range interaction. For short range interaction energies much larger than the transition temperature κBTt the hysteresis width is determined by the long range interaction alone.

On resonance widths of leaky Lamb modes

For an elastic plate immersed in a lossless fluid, it is known that resonances of leaky Lamb modes contribute to the transmission coefficient in a manner dependent on the resonance widths. Based on numerical solutions in the complex wave-number field, an examination of such resonance widths is presented here. For plates of different materials and for water and air loadings, families of curves of Lamb mode angular resonance width versus frequency are seen to yield a consistency of pattern. For air loading, the widths are 31/2 orders smaller than for water loading. Frequency resonance widths are found to be two orders greater than angular resonance widths. Nulls in the resonance width curves provide added insight into the cause of published experimental nonobservance of radiation effects for individual Lamb modes.

Dynamics of orientationally disordered hydrogen bonds and of water molecules in a molecular cage

The crystalline complex between the ring-shaped β-cyclodextrin (β-CD) and 11 water molecules (β-CD · 11H2O) shows extensive orientational disorder of β-CD hydroxyl groups and water molecules and in addition positional disorder of six water molecules which are enclosed in the β-CD molecular cavity. A quasielastic incoherent neutron scattering (QINS) study was performed employing three different energy resolutions. The quasielastic lineshape can be approximated by a sum of two Lorentzian curves of constant width and intensity ratio and intensities decreasing with the scattering angle φ and extrapolating to zero for φ = 0 consistently for all three energy resolutions. This is indicative of local diffusion of hydrogen atoms in the sample. A previous high resolution (0.6Å) neutron crystallographic study provides the knowledge of the number and jump distances of the disordered hydrogen atoms. This gives a sound basis for model calculations which aim to describe the experimental spectra in terms of theoretical scattering functions. A very simple two-site jump model with two different jump distances already describes the data well. It can be interpreted as the observation of reorientational jumps of hydroxyl groups and water molecules over H ··· H distances around 1.5Å and of diffusive motions of water molecules enclosed in the β-CD molecular cavity over H ··· H (and O ··· O) distances around 3 Å. At room temperature both types of motion occur with a similar distribution of jump rates extending at least from about 2 × 1010 to 2 × 1011 s-1. The temperature dependence of the elastic peak intensity shows that the diffusive motions can be followed down to at least T = 240K.

Theory of spin polarized photoelectron diffraction

We discuss several aspects of the theory of spin-polarized photoelectron diffraction (SPPD). This method makes use of multiplet splittings of core-level binding energies to produce photoelectron peaks with high spin polarization (for example, the two principal peaks associated with Mn 3s emission from Mn 2+). We consider three possible mechanisms for spin-dependent photoelectron scattering and diffraction: exchange scattering by valence electrons (3d 5 for Mn 2+), spin-orbit scattering (which is not expected to yield large effects if the sample does not have a net magnetization), and spin-dependent inelastic scattering (which cannot yet be dealt with in a fully quantitative way, but is estimated to be less important than the other two). The fact that SPPD involves internal sources of polarized electrons references to their respective emitters implies that it can be employed to study magnetic order in both anti-ferromagnets and ferromagnets and at temperatures above their respective Néel or Curie points. The effects of exchange scattering on Mn 3s emission from Mn 2+ in KMnF 3 have been incorporated into a single-scattering cluster model of the diffraction process via either the Dirac-Hara or Kohn-Sham local density approximations. This model is applied to several cases: a single Mn 2+ scatterer, small clusters of Mn 2+ scatterers, and full clusters appropriate to the (110) surface of KMnF 3, with all atoms included. These calculations demonstrate that SPPD should be a short-range probe of magnetic order, a result consistent with conclusions reached in several prior studies of photoelectron diffraction without spin resolution. They also illustrate the perturbative nature of these effects, which are only about ca. 5–15% of the total intensity; this in turn leads to several possible simplifications in the theory. We have in addition phenomenologically modelled the decreases of short-range order with increasing temperature by using a Gaussian modulation of spins; this model produces curves of SPPD spin asymmetry versus Gaussian width which qualitatively agree with those observed experimentally (Phys. Rev. Lett. 55 (1986) 1227). Direct comparisons of the observed angular dependence of these spin asymmetries with theory for (110) KMnF 3 are also encouraging in that trends from one direction to another are in general predicted correctly. However, the maximum degree of asymmetry predicted by theory (13%) is somewhat below that of experiment (17%). Possible reasons for the discrepancies seen between theory and experiment are discussed, together with likely future directions of study.