Infinite Height Research Articles

An improved heuristic and an exact algorithm for the 2D strip and bin packing problem

In this paper, we consider a two-dimensional (2D) packing problem. It consists of packing a given set of rectangular items either into a strip with a fixed width and infinite height or into identical bins. This problem has industrial applications such as cutting glass and Very-Large-Scale Integration (VLSI) design. The guillotine constraint is imposed in this paper. For this problem, we propose some strategies to improve the Shelf Heuristic Filling (SHF) introduced by Ben Messaoud et al. (2003). In a second part, we present a branch-and-bound algorithm for the 2D Strip Packing (2SP) problem that uses a new lower bound and the improved heuristic. All our algorithms are tested on generated and literature-based instances and give good results compared to the existing algorithms.

Bidirectional best-fit heuristic for orthogonal rectangular strip packing

In a non-guillotinable rectangular strip packing problem (RF-SPP), the best orthogonal placement of given rectangular pieces on a strip of stock sheet having fixed width and infinite height are searched. The aim is to minimize the height of the strip while including all the pieces in appropriate orientations. In this study, a novel bidirectional best-fit heuristic (BBF) is introduced for solving RF-SPPs. The proposed heuristic as a new feature considers the gaps in both horizontal and vertical directions during the placement process. The performance of BBF is compared to some previous approaches, including one of the best heuristics from the literature. BBF achieves better results than the existing heuristics and delivers a better or matching performance as compared to the most of the previously proposed meta-heuristics for solving RF-SPPs.

Interface roughness effect on density of states and mobility of narrow Si/Si1−xGex quantum wells: path-integral approach

The in-plane transport of a two-dimensional hole gas in narrow Si/Si1−xGex quantum wells (QWs) with the interface roughness effect was investigated using path-integral theory. The random variation of the well width causes the fluctuation potential in the in-plane direction. This fluctuation was integrated into the path integral via the eigenvalue model of the infinite barrier height QW. The hole mobility and the density of states can be derived in an analytic form. The calculated hole mobility was compared with the experimental data and a different theoretical approach based on the wavefunction model. We found that Δ = 0.2 nm and Λ = 4.5 nm give a good description of the experimental data. Our theory predicted the existence of the localized states, which reduces the hole mobility in addition to the prediction from the wavefunction method.

THREE-DIMENSIONAL SIMULATIONS OF MAGNETIZED SUPERBUBBLES: NEW INSIGHTS INTO THE IMPORTANCE OF MHD EFFECTS ON OBSERVED QUANTITIES

We present three-dimensional magnetohydrodynamic (MHD) simulations of superbubbles, to study the importance of MHD effects in the interpretation of images from recent surveys of the Galactic plane. These simulations focus mainly on atmospheres defined by an exponential density distribution and the Dickey & Lockman density distribution. In each case, the magnetic field is parallel to the Galactic plane and we investigate cases with either infinite scale height (constant magnetic field) or a constant ratio of gas pressure to magnetic pressure. The three-dimensional structure of superbubbles in these simulations is discussed with emphasis on the axial ratio of the cavity as a function of magnetic field strength and the age of the bubble. We investigate systematic errors in the age of the bubble and scale height of the surrounding medium that may be introduced by modeling the data with purely hydrodynamic models. Age estimates derived with symmetric hydrodynamic models fitted to an asymmetric magnetized superbubble can differ by up to a factor of 4, depending on the direction of the line of sight. The scale height of the surrounding medium based on the Kompaneets model may be up to 50% lower than the actual scale height. We also present the first ever predictions of Faraday rotation by a magnetized superbubble based on three-dimensional MHD simulations. We emphasize the importance of MHD effects in the interpretation of observations of superbubbles.

An Exact Method for the 2D Guillotine Strip Packing Problem

We consider the two-dimensional strip packing problem with guillotine cuts. The problem consists in packing a set of rectangular items on one strip of widthWand infinite height. The items packed without overlapping must be extracted by a series of cuts that go from one edge to the opposite edge (guillotine constraint). To solve this problem, we use a dichotomic algorithm that uses a lower bound, an upper bound, and a feasibility test algorithm. The lower bound is based on solving a linear program by introducing new valid inequalities. A new heuristic is used to compute the upper bound. Computational results show that the dichotomic algorithm, using the new bounds, gives good results compared to existing methods.

A survey and comparison of heuristics for the 2D oriented on-line strip packing problem

The two dimensional oriented on-line strip packing problem requires items to be packed, one at a time, into a strip of fixed width and infinite height so as to minimise the total height of the packing. The items may neither be rotated nor overlap. In this paper, ten heuristics from the literature are considered for the special case where the items are rectangles. Six modifications to some of these heuristics are proposed, along with two entirely new shelf algorithms. The performances and efficiencies of all the algorithms are compared in terms of the total packing height achieved and computation time required in each case, when applied to 542 benchmark data sets documented in the literature.

Computation of 2-groups of narrow logarithmic divisor classes of number fields

We present an algorithm for computing the 2-group C ℓ ˜ F res of narrow logarithmic divisor classes of degree 0 for number fields F . As an application, we compute in some cases the 2-rank of the wild kernel W K 2 ( F ) and the 2-rank of its subgroup K 2 ∞ ( F ) : = ∩ n ≥ 1 K 2 n ( F ) of infinite height elements in K 2 ( F ) .

Prediction of daily heavy precipitation in a lowland area using GPS and surface weather data

Recent progress in GPS technology has enabled us to estimate the total amount of water vapor in an atmospheric column with infinite height as ’GPS-derived precipitable water vapor’ (GPS-PWV). In the present study, we analyzed the GPS and meteorological data obtained in Saga Plain, in northern Kyushu, Japan. An attempt was made to predict heavy precipitation (≥50 mm d−1) on a daily basis using the analyzed relations between surface temperature and GPS-PWV. Several features were revealed that were associated with the daily heavy precipitation at the study area. An index was made to predict heavy precipitation on a daily basis. Applying the index to observations, we obtained acceptable results of the prediction. It is suggested that GPS is useful for predicting heavy precipitation events.

Correlations of eigenvalues and Riemann zeros

Interest in comparing the statistics of the zeros of the Riemann zeta function with random matrix theory dates back to the 1970s and the work of Montgomery and Dyson. Twelve years ago Rudnick and Sarnak and, independently, Bogomolny and Keating showed that the n-point correlation function of the Riemann zeros, correctly scaled and in the limit of infinite height on the critical line, agrees with the scaling limit of the n-correlation of eigenvalues of random unitary matrices. The former piece of work holds only for a restricted class of test functions, and the latter relies on a heuristic method and the conjectures of Hardy and Littlewood. Neither tells us more than the asymptotic limit for the general n-correlation. In this article we use the ratios conjecture for average values of the Riemann zeta function to produce the lower order terms in a very precise formula for the n-correlation of the Riemann zeros. The same method can be applied rigorously in the random matrix case, yielding a formula which shows identical structure (though with none of the arithmetic details) to the correlations of the Riemann zeros something which cannot be seen from the classical determinantal formula for the random matrix correlation functions.

Gate capacitance-voltage characteristics of submicron-long-gate diamond field-effect transistors with hydrogen surface termination

The radio-frequency characteristics of p-type diamond field-effect transistors with hydrogen surface termination were numerically analyzed using an equivalent-circuit model. From the gate-source capacitance (CGS)-voltage (VGS) results extracted from measured s parameters, the authors found a plateau in CGS within a certain VGS range. This means that a two-dimensional hole gas channel forms parallel to the surface and that the channel is separated by a thin energy-barrier layer with an infinite height from the gate metal. At a high negative VGS, as negative VGS is increased, CGS increases steeply. This results from holes penetrating the energy barrier.

Reactive GRASP for the strip-packing problem

This paper presents a greedy randomized adaptive search procedure (GRASP) for the strip packing problem, which is the problem of placing a set of rectangular pieces into a strip of a given width and infinite height so as to minimize the required height. We investigate several strategies for the constructive and improvement phases and several choices for critical search parameters. We perform extensive computational experiments with well-known instances which have been previously reported, first to select the best alternatives and then to compare the efficiency of our algorithm with other procedures. The results show that the GRASP algorithm outperforms recently reported metaheuristics.

Broadband quasi-phase-matched second-harmonic generation in a nonlinear photonic crystal

We analyze the properties of quasi-phase-matched second-harmonic generation (SHG) from a defect waveguide in photonic-crystal (PhC) slabs embedded in a periodically χ(2) inverted material with a collinear beam configuration. We show that by controlling the material dispersion with the structural dispersion of a defect waveguide in PhC slabs of infinite height, it is possible to realize a much wider frequency range for quasi-phase-matched SHG than without the PhC structure. Also, taking the fabrication of actual devices into consideration, we examine the case for PhC slabs of finite height and show that, although guided modes are index confined in the vertical direction (while they are confined by the PhC structure in the horizontal direction), the effect that the PhC structure has on the broadening of the frequency range remains and that the broadened range can be comparable to that for the PhC slabs of infinite height if appropriate structural parameters are taken.

Band structure of honeycomb photonic crystal slabs

Two-dimensional (2D) honeycomb photonic crystals with cylinders and connecting walls have the potential to have a large full band gap. In experiments, 2D photonic crystals do not have an infinite height, and therefore, we investigate the effects of the thickness of the walls, the height of the slabs, and the type of the substrates on the photonic bands and gap maps of 2D honeycomb photonic crystal slabs. The band structures are calculated by the plane wave expansion method and the supercell approach. We find that the slab thickness is a key parameter affecting the band gap size, while on the other hand the wall thickness hardly affects the gap size. For symmetric photonic crystal slabs with lower dielectric claddings, the height of the slabs needs to be sufficiently large to maintain a band gap. For asymmetric claddings, the projected band diagrams are similar to that of symmetric slabs as long as the dielectric constants of the claddings do not differ greatly.

Field Computation for Two-Dimensional Array Transducers with Limited Diffraction Array Beams

A method is developed for calculating fields produced with a two-dimensional (2D) array transducer. This method decomposes an arbitrary 2D aperture weighting function into a set of limited diffraction array beams. Using the analytical expressions of limited diffraction beams, arbitrary continuous wave (cw) or pulse wave (pw) fields of 2D arrays can be obtained with a simple superposition of these beams. In addition, this method can be simplified and applied to a 1D array transducer of a finite or infinite elevation height. For beams produced with axially symmetric aperture weighting functions, this method can be reduced to the Fourier-Bessel method studied previously where an annular array transducer can be used. The advantage of the method is that it is accurate and computationally efficient, especially in regions that are not far from the surface of the transducer (near field), where it is important for medical imaging. Both computer simulations and a synthetic array experiment are carried out to verify the method. Results (Bessel beam, focused Gaussian beam, X wave and asymmetric array beams) show that the method is accurate as compared to that using the Rayleigh-Sommerfeld diffraction formula and agrees well with the experiment.

AdS and stabilized extra dimensions in multi-dimensional gravitational models with nonlinear scalar curvature terms R−1 and R4

We study multi-dimensional gravitational models with scalar curvature nonlinearities of types R−1 and R4. It is assumed that the corresponding higher dimensional spacetime manifolds undergo a spontaneous compactification to manifolds with a warped product structure. Special attention has been paid to the stability of the extra-dimensional factor spaces. It is shown that for certain parameter regions the systems allow for a freezing stabilization of these spaces. In particular, we find for the R−1 model that configurations with stabilized extra dimensions do not provide a late-time acceleration (they are AdS), whereas the solution branch which allows for accelerated expansion (the dS branch) is incompatible with stabilized factor spaces. In the case of the R4 model, we obtain that the stability region in parameter space depends on the total dimension D = dim(M) of the higher dimensional spacetime M. For D > 8 the stability region consists of a single (absolutely stable) sector which is shielded from a conformal singularity (and an antigravity sector beyond it) by a potential barrier of infinite height and width. This sector is smoothly connected with the stability region of a curvature-linear model. For D < 8 an additional (metastable) sector exists which is separated from the conformal singularity by a potential barrier of finite height and width so that systems in this sector are prone to collapse into the conformal singularity. This second sector is not smoothly connected with the first (absolutely stable) one. Several limiting cases and the possibility of inflation are discussed for the R4 model.

The computable dimension of trees of infinite height

AbstractWe prove that no computable tree of infinite height is computably categorical, and indeed that all such trees have computable dimension ω. Moreover, this dimension is effectively ω, in the sense that given any effective listing of computable presentations of the same tree, we can effectively find another computable presentation of it which is not computably isomorphic to any of the presentations on the list.

The Computable Dimension of I-Trees of Infinite Height

We study computable trees with distinguished initial subtree (briefly, I-trees). It is proved that all I-trees of infinite height are computably categorical, and moreover, they all have effectively infinite computable dimension.

Casimir energy inside a cavity with triangular cross section

For a certain class of triangles (with angles proportional to π/N, N⩾3) we formulate an image method by making use of the group GN generated by reflections with respect to the three lines which form the triangle under consideration. We formulate the regularization procedure by classification of subgroups of GN and corresponding fixed points in the triangle. We then also calculate Casimir energy for a cavity of infinite height with triangular cross section for scalar massless fields. More detailed calculation is given for odd N.

Surface superconductivity in 3 dimensions

We study the Ginzburg-Landau system for a superconductor occupying a 3 3 -dimensional bounded domain, and improve the estimate of the upper critical field H C 3 H_{C_{3}} obtained by K. Lu and X. Pan in J. Diff. Eqns., 168 (2000), 386-452. We also analyze the behavior of the order parameters. We show that, under an applied magnetic field lying below and not far from H C 3 H_{C_{3}} , order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field, and exponentially decay both in the normal and tangential directions away from the sheath in the L 2 L^{2} sense. As the applied field decreases further but keeps in between and away from H C 2 H_{C_{2}} and H C 3 H_{C_{3}} , the superconducting sheath expands but does not cover the entire surface, and superconductivity at the surface portion orthogonal to the applied field is always very weak. This phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by X. Pan in Comm. Math. Phys., 228 (2002), 327-370, where under an axial applied field lying in-between H C 2 H_{C_{2}} and H C 3 H_{C_{3}} the entire surface is in the superconducting state.

Field and size dependence of exciton–LO-phonon interaction in a semiconductor quantum dot

We theoretically study the exciton--LO-phonon interaction in a semiconductor quantum dot in the presence of an electric field. The confinement potential of the dot in the direction of the field is assumed to be a quantum well with various barrier heights while that in the direction transverse to the field is modeled by a parabolic form. The adiabatic approximation is employed to treat the effect of LO phonon. A trial wave function with a parameter describing the electron-hole correlation is used in the calculation of the ground-state energy of the exciton by the variational method. The magnitude of the exciton--LO-phonon interaction energy $|{E}_{\mathrm{e}\mathrm{x}\ensuremath{-}\mathrm{p}\mathrm{h}}|$ and the corresponding Huang-Rhys factor S are evaluated as functions of field strength and the size of the quantum dot for different barrier heights. With the field strength, the separation between electron and hole is enhanced and thus $|{E}_{\mathrm{e}\mathrm{x}\ensuremath{-}\mathrm{p}\mathrm{h}}|$ increases. Moreover, lowering of the barrier height further increases the separation, resulting in greater values of $|{E}_{\mathrm{e}\mathrm{x}\ensuremath{-}\mathrm{p}\mathrm{h}}|.$ With a fixed strength of field, the increase of size of the quantum dot leads to both delocalization of wave packets for electrons and holes and an increase of their separation simultaneously. Consequently the size dependence of $|{E}_{\mathrm{e}\mathrm{x}\ensuremath{-}\mathrm{p}\mathrm{h}}|$ is complicated. For comparison, an additional calculation is performed for a quantum dot with the shape of a cylindrical disk and barrier of infinite height in all directions and the contrast feature is discussed.