Slit Plane Research Articles

Novel band‐notch UWB antenna design with slit ground plane

AbstractA newly designed microstrip feed, octagonal‐shaped ultra‐wideband printed antenna is proposed. The UWB operation within the required 3.1–10.6 GHz band is achieved by simply embedding in the ground plane a small rectangular slit, located below the feed line of an octagonal patch. With the dimensions (length and height) of the slit carefully tuned, a notched band of around 5.1–6.4 GHz can be achieved, without applying the commonly recommended techniques of loading the radiating patch with a slot or an open‐ended slit. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 2229–2233, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23621

Bieberbach’s conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [4]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [5] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [2] about certain hypergeometric functions played a crucial role in de Branges’ proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [72] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

Ethane adsorption in slit-shaped micropores: influence of molecule orientation on adsorption capacity

Adsorption of ethane in a slit shaped micropore system has been studied by Monte Carlo molecular simulation by considering this hydrocarbon as a two interacting sites molecule. Ethane adsorption in pore sizes from 0.41 to 1.66 nm was simulated at 303 K. Microscopic characteristics of the adsorbed phase have been studied for pores of different size, comparing two density profiles: the molecule centre of mass profile and the molecular interaction site profile. Averaged angle distribution of molecule positions with respect to the slit plane across the pore width has been also obtained by simulation. These results were related to ethane molecule packing efficiency, which is also related to the adsorption capacity in terms of the adsorbed phase density. Packing efficiency presents an oscillation shape as the result of the adsorbate disorder inside the pore. Pressure influence on the adsorption has been studied by following pore filling by simulation. When pore condensation takes place and for pressures above condensation, fluid-fluid interactions are determinant in molecule disorder observed between the two adsorbed layers.

Proof of a conjecture on the slit plane problem

Let a i, j ( n) denote the number of walks in n steps from (0,0) to ( i, j), with steps (±1,0) and (0,±1), never touching a point (− k,0) with k⩾0 after the starting point. Bousquet-Mélou and Schaeffer conjectured a closed form for the number a − i, i (2 n) when i⩾1. In this paper, we prove their conjecture, and give a formula for a − i, i (2 n) for i⩽−1.

Evaluation of in-column energy filters for analytical electron microscopes

Performance of three electron energy filters: B-type omega filter (dispersion D=1.2 μm/eV at 200 kV), twin column filter ( D=5.1 μm/eV) and S-filter ( D=9.5 μm/eV ) are compared. Comparison was made for the non-isochromaticity at various total magnifications and beam illumination conditions in imaging and diffraction modes and for the energy resolution in electron energy loss spectroscopy. Although non-isochromaticity is considered to be a property of the imaging filter, it depends on the size and energy spread of the incident beam in practice. The incident beam size and energy width are changed to investigate how the non-isochromaticity and energy resolution changes by those parameters. When the energy width is 1 eV (without using monochromator), the large dispersion filter (S-filter) has a merit under low magnification and large illumination angle. The non-isochromaticity approaches up to the energy width of the incident beam when the magnification becomes high (100,000 times) or the beam illumination angle becomes small (1 mrad). Under the diffraction condition, the B-type omega filter has a difficulty in obtaining 4 eV non-isochromaticity when the diffracted beam with the half angle of 60 mrad is used. However, the other two filters can give smaller non-isochromaticity because of its larger dispersion. The elongation of the beam on the slit plane occurs due to the energy spread of the incident beam. The energy resolution of the filter is determined by the incident beam energy width when it is larger than 0.1 eV for all filters.

A new dual‐band planar inverted‐F antenna for ISM‐band applications

AbstractThis paper proposes a new dual‐band planar inverted‐F antenna (PIFA) for ISM‐band applications. The proposed antenna is designed using the Microwave Studio Software of Computer Simulation Technology (CST). The measured results of the antenna are compared with the simulated results. The effects of antenna parameters such as the slit geometry and ground plane are studied. Both lower and upper frequency bands can be determined independently, which is very useful for the design of dual‐band antennas. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 39: 211–214, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11172

Walks on the slit plane

In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line . We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three variable is algebraic of degree 8. Moreover, for any point (i, j), the length for walks of this type ending at (i, j) is also algebraic, of degree 2 or 4, and involves the famous Catalan numbers. Our method is based on the solution of a functional equation, established via a simple combinatorial argument. It actually works for more general models, in which walks take their steps in a finite subset of ℤ2 satisfying two simple conditions. The corresponding are always algebraic. In the second part of the paper, we derive from our enumerative results a number of probabilistic corollaries. For instance, we can compute exactly the probability that an ordinary random walk starting from (i, j) hits for the first time the half-line at position (k, 0), for any triple (i, j, k). This generalizes a question raised by R. Kenyon, which was the starting point of this paper. Taking uniformly at random all n-step walks on the slit plane, we also compute the probability that they visit a given point (k, 0), and the average number of visits to this point. In other words, we quantify the transience of the walks. Finally, we derive an explicit limit law for the coordinates of their endpoint.

X-Ray Diffractometer for Studies on Molecular-Beam-Epitaxy Growth of III–V Semiconductors

An X-ray diffractometer connected with a molecular-beam epitaxy (MBE) system has been constructed for in situ studies on the growing surfaces of III–V compound semiconductors. This diffractometer is based on the (4+2) type, which has four axes for orienting the sample and two axes for positioning the detector. In addition, it is equipped with an axis for rotating the receiving slit about the normal of the slit plane to align the resolution of the receiving slit properly for the surface X-ray diffraction measurement. For the alignment of the sample and the entire setup with respect to the X-ray beam, an XYZ-stage and an adjustable base plate are available. X-rays enter and leave the chamber through two Be windows welded directly to the MBE chamber. A graphite sheet which can be heated up to 250°C is placed along the inside of the Be windows to protect the Be windows from being coated with evaporated materials. Preliminary data are presented to demonstrate the feasibility of static and dynamic measurements of growing surfaces using this instrument.

Walks on the Slit Plane: Other Approaches

Let S be a finite subset of Z2. A walk on the slit plane with steps in S is a sequence (0,0)=w0,w1,…,wn of points of Z2 such that wi+1−wi belongs to S for all i, and none of the points wi, i≥1, lie on the half-line H={(k,0):k≤0}. In a recent paper [Walks on the Slit Plane, preprint 2000, arXiv: match.CO/0012230] G. Schaeffer and the author computed the length generating function S(t) of walks on the slit plane for several sets S. All the generating functions thus obtained turned out to be algebraic: for instance, on the ordinary square lattice, St=1+1+4t1/21+1−4t1/2/21−4t3/4. The combinatorial reasons for this algebraicity remain obscure. In this paper, we present two new approaches for solving slit plane models. One of them simplifies and extends the functional equation approach of the original paper. The other one is inspired by an argument of Lawler; it is more combinatorial, and explains the algebraicity of the product of three series related to the model. It can also be seen as an extension of the classical cycle lemma. Both methods work for any set of steps S. We exhibit a large family of sets S for which the generating function of walks on the slit plane is algebraic, and another family for which it is neither algebraic, nor even D-finite. These examples give a hint at where the border between algebraicity and transcendence lies, and call for a complete classification of the sets S.

A Bijection for Some Paths on the Slit Plane

This paper deals with a study of the class P of lattice paths, made of north, east, south, and west unit steps, which being at (−1,0) and end at (0,0), avoiding the non-negative x-axis. P is bijectively proved to be enumerated by odd index Catalan numbers according to the number of steps.

Fractal structure and fractal dimension determination at nanometer scale

Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of frsctal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in real metallic fractures is correlated to the intrinsic characteristics of the materials in addition to the effect of boundaries. The relationship of fractal dimensions with the mechanical properties of materials at macrometer scale also exists at nanometer scale.

Temperature and velocity measurement of a 3-D thermal flow field using thermo-sensitive liquid crystals

It is an important challenge to analyze a three-dimensional thermal flow field in engineering, science, and agriculture. For such an analysis, it is essential to measure physical quantities such as temperature and velocity over the entire thermal flow field. This paper presents a measurement system based on color image processing for temperature and velocity vector distributions in a three-dimensional thermal flow field. Flow visualization is accomplished by the use of thermo-sensitive liquid crystal tracers. An algorithm for the color-to-temperature transformation using a multi-layer feed-forward neural network is applied to three-dimensional natural convection in a rotating cylindrical cell. Two-dimensional temperature distributions in a slit plane are obtained by using the algorithm. A three-dimensional temperature distribution is consequently constructed by interpolating the two-dimensional distributions using the B-spline function. In addition, the Spatio-Temporal correlation method is applied to the natural convection to obtain a three-dimensional velocity vector distribution.

An additional axis for the surface X-ray diffractometer.

A new surface X-ray diffractometer based on a kappa-type diffractometer will be installed in BL14B1, SPring-8. This diffractometer has an additional axis on its detector arm for rotating the receiving slit about the normal of the slit plane, in addition to two axes for positioning the detector. This additional axis is founded on the consideration of the correction factor which has been derived so as to be valid for the z-axis mode measurement using any incoming and outgoing angles of the X-ray beam. The rotational slit allows accurate measurement of the surface structure factor up to large perpendicular momentum transfer.

STUDY ON TURBURENCE STRUCTURES IN PARTLY VEGETATED OPEN-CHANNEL FLOWS WITH A PARTICLE-IMAGE VELOCIMETRY (PIV)

The importance of vegetated zone in rivers has recently been recognized not only for river management but also for water flow environment. From this point of view, some researchers have investigated vegetated open-channel flows to clarify turbulent structures such as mean velocity profiles and turbulence intensity distributions.In this study, images of very small 75μm diameter particles uniformly scattered in vegetated open-channel flows were taken using a CCD camera and 2W Argon-Ion laser slit illumination. Instantaneous velocities at many points in the laser slit plane were obtained from these continuous two images of particles. This new method is called the “Particle-Image Velocimetry (PIV)”, and seems to be very powerful to investigate coherent vortices in space and time, because simultaneous velocities at every point can be obtained easily. It was found that the density of vegetation effects on structures such as instantaneous vectors and vorticities.

Heisenberg microscope and quantum variation measurement

We demonstrate taking the Heisenberg microscope as an example that the standard quantum limit for registration of force can be overcome in a quantum corrdinate measurement. We propose to vary the distance between the diffraction slit and detector plane during measurement so that one can measure a linear combination of coordinate and back action momentum transfered to the probe mass in accordance with the uncertainty principle. Changing the distance in a special way enables one to exclude the influence of the back action fluctuations, thus implementing the idea of a quantum variation measurement.

Velocity Measurements in Compound Open-Channel Flows by making use of Particle-Image Velocimetry

Many turbulence measurements have been ever conducted by point measurement techniques such as hot-film anemometers (CTA) and laser Doppler anemometers (LDA). Compared with these techniques, coherent vortex phenomena are easily recognized by flow visualization techniques.In this study, images of very small 100μm diameter particles uniformly scattered in compound open-channel flows were taken using a CCD camera and 2W Argon-ion laser slit illumination. These images were stored through frame memory board from an optic disc controlled by a personal computer. Instantaneous velocities at many points in the laser slit plane were measured from these continuous four images of one particle. This new method is called the “Particle -Image Velocimetry (PIV)”, and seems to be very powerful to investigate coherent vortices in space and time because simultaneous velocities at every point can be obtained.

Effect of specimen geometry on fatigue crack growth in plane strain. I. Constant-amplitude response: Shercliff, H.R. and Fleck, N.A. Fatigue Fract. Eng. Mater. Struct. 1990 13, (3), 287–296

Fatigue cracks were grown in centre-notched sheet made from BS4360 50B structural steel. Despite the fact that loading was fully compressive, cracks initiated and grew in regions of residual tensile stress at the notch roots. It was observed that crack growth rates decreased with increasing crack length until arrest occurred. Near tip strain gauges were used to monitor crack closure; closure readings agreed well with those deduced from growth rates. The residual stress ahead of the slit tip and normal to the slit plane, σres, was estimated from the crack growth rate response. It was found that σres scaled with distance, x, from the slit tip as σresαx−0.56.

Fatigue analysis and prediction in fillet welded joints in the low thickness range: Ferreira, J.A. and Branco, C.M. Fatigue Fract. Eng. Mater. Struct. 1990 13, (3), 201–212

Fatigue cracks were grown in centre-notched sheet made from BS4360 50B structural steel. Despite the fact that loading was fully compressive, cracks initiated and grew in regions of residual tensile stress at the notch roots. It was observed that crack growth rates decreased with increasing crack length until arrest occurred. Near tip strain gauges were used to monitor crack closure; closure readings agreed well with those deduced from growth rates. The residual stress ahead of the slit tip and normal to the slit plane, σres, was estimated from the crack growth rate response. It was found that σres scaled with distance, x, from the slit tip as σresαx−0.56.

Fatigue crack growth under compressive loading

Fatigue cracks were grown in centre-notched sheet made from BS4360 50B structural steel. Despite the fact that loading was fully compressive, cracks initiated and grew in regions of residual tensile stress at the notch roots. It was observed that crack growth rates decreased with increasing crack length until arrest occurred. Near tip strain gauges were used to monitor crack closure; closure readings agreed well with those deduced from growth rates. The residual stress ahead of the slit tip and normal to the slit plane, σ res, was estimated from the crack growth rate response. It was found that σ res scaled with distance, x, from the slit tip as σ resαx −0.56 .

Raman Spectrometry with High Sensitivity

Raman spectroscopy, for well-understood fundamental reasons, is a relatively insensitive method of analysis, i.e., it is problematic to study dilute solutions, gases at low pressures, adsorbed species, etc. We describe instrumentation incorporating exit slit plane “multiplex” detection and demonstrate that the instrument improves in sensitivity over its conventional counterpart by a significant margin (in this case >X1000). The value that this increased optical efficiency provides is demonstrated and new applications of Raman spectroscopy are discussed. We include a discussion of the factors controlling the efficiency of illuminating and viewing opaque or highly scattering samples sensitive to high brightness lasers and demonstrate that most commercial spectrometers are set up in such a way that they perform in these cases much less adequately than need be.