Strong-coupling expansion for the momentum distribution of the Bose-Hubbard model with benchmarking against exact numerical results

A strong-coupling expansion for the Green's functions, self-energies, and correlation functions of the Bose-Hubbard model is developed. We illustrate the general formalism, which includes all possible (normal-phase) inhomogeneous effects in the formalism, such as disorder or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator–to–superfluid transition along with a generalization of the random-phase-approximation-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions. The accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott-phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling-theory results are benchmarked against numerically exact quantum Monte Carlo simulations in two and three dimensions and against density-matrix renormalization-group calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.

Digitoitujen väri-infrailmakuvien numeerinen monikuvatulkinta metsäninventoinnissa

The use of remote sensing imagery as auxiliary data in forest inventory is based on the correlation between features extracted from the images and the ground truth. The bidirectional reflectance and radial displacement cause variation in image features located in different segments of the image but forest characteristics remaining the same. The variation has so far been diminished by different radiometric corrections. In this study the use of sun azimuth based converted image co-ordinates was examined to supplement auxiliary data extracted from digitised aerial photographs. The method was considered as an alternative for radiometric corrections. Additionally, the usefulness of multi-image interpretation of digitised aerial photographs in regression estimation of forest characteristics was studied. The state owned study area located in Leivonmäki, Central Finland and the study material consisted of five digitised and ortho-rectified colour-infrared (CIR) aerial photographs and field measurements of 388 plots, out of which 194 were relascope (Bitterlich) plots and 194 were concentric circular plots. Both the image data and the field measurements were from the year 1999. When examining the effect of the location of the image point on pixel values and texture features of Finnish forest plots in digitised CIR photographs the clearest differences were found between front-and back-lighted image halves. Inside the image half the differences between different blocks were clearly bigger on the front-lighted half than on the back-lighted half. The strength of the phenomenon varied by forest category. The differences between pixel values extracted from different image blocks were greatest in developed and mature stands and smallest in young stands. The differences between texture features were greatest in developing stands and smallest in young and mature stands. The logarithm of timber volume per hectare and the angular transformation of the proportion of broadleaved trees of the total volume were used as dependent variables in regression models. Five different converted image co-ordinates based trend surfaces were used in models in order to diminish the effect of the bidirectional reflectance. The reference model of total volume, in which the location of the image point had been ignored, resulted in RMSE of 1,268 calculated from test material. The best of the trend surfaces was the complete third order surface, which resulted in RMSE of 1,107. The reference model of the proportion of broadleaved trees resulted in RMSE of 0,4292 and the second order trend surface was the best, resulting in RMSE of 0,4270. The trend surface method is applicable, but it has to be applied by forest category and by variable. The usefulness of multi-image interpretation of digitised aerial photographs was studied by building comparable regression models using either the front-lighted image features, back-lighted image features or both. The two-image model turned out to be slightly better than the one-image models in total volume estimation. The best one-image model resulted in RMSE of 1,098 and the two-image model resulted in RMSE of 1,090. The homologous features did not improve the models of the proportion of broadleaved trees. The overall result gives motivation for further research of multi-image interpretation. The focus may be improving regression estimation and feature selection or examination of stratification used in two-phase sampling inventory techniques. Keywords: forest inventory, digitised aerial photograph, bidirectional reflectance, converted image co­ordinates, regression estimation, multi-image interpretation, pixel value, texture, trend surface

The transient response of certain third-ordernon-linear systems

In this paper a method of solving certain third-order non-linear systems by using themethod of ultraspherical polynomial approximation is proposed. By using the method of variation of parameters the third-order equation is reduced to three partial differential equations. Instead of being averaged over a cycle, the non-linear functions are expanded in ultraspherical polynomials and with only the constant term retained, the equations are solved. The results of the procedure are compared with the numerical solutions obtained on a digital computer. A degenerate third-order system is also considered and results obtained for the above system are compared with numerical results obtained on the digital computer. There is good agreement between the results obtained by the proposed method and the numerical solution obtained on digital computer.

Vave propagation and bandgaps in a parametrically modulated composite laminate

Wave propagation and its frequency bandgaps in a parametrically modulated composite laminate are reported in this paper. The modulated properties under considerations are due to periodic microstructure, for example honeycomb core sandwich composite, which can be parameterized and homogenized in a suitable scale. Wave equations are derived by assuming a third-order shear deformation theory. Homogenization of the wave equations is carried out in the scale of wavelength. In-plane wave and flexural-shear wave dispersions are obtained for a range of values of a stiffness modulation coefficient (alpha). A clear pattern of stop-bands is observed for alpha >= 4. To validate the band-gap phenomena, we take recourse to time domain response obtained from finite element simulation. As predicted by the proposed analytical technique, a distinct correlation between the chosen frequency band and the simulated wave arrival time and amplitude reduction is found. This promises practical applications of the proposed analytical technique to designing parametrically modulated composite laminate for wave suppression. (C) 2009 Elsevier B.V. All rights reserved.

Structure and representation of correlation functions and the density matrix for a statistical wave field in optics

A systematic structure analysis of the correlation functions of statistical quantum optics is carried out. From a suitably defined auxiliary two‐point function we are able to identify the excited modes in the wave field. The relative simplicity of the higher order correlation functions emerge as a byproduct and the conditions under which these are made pure are derived. These results depend in a crucial manner on the notion of coherence indices and of unimodular coherence indices. A new class of approximate expressions for the density operator of a statistical wave field is worked out based on discrete characteristic sets. These are even more economical than the diagonal coherent state representations. An appreciation of the subtleties of quantum theory obtains. Certain implications for the physics of light beams are cited.

Higher-order analysis of crack-tip fields in power-law hardening materials

In this paper, we present an exact higher-order asymptotic analysis on the near-crack-tip fields in elastic-plastic materials under plane strain, Mode I. A four- or five-term asymptotic series of the solutions is derived. It is found that when 1.6 < n less-than-or-equal-to 2.8 (here, n is the hardening exponent), the elastic effect enters the third-order stress field; but when 2.8< n less-than-or-equal-to 3.7 this effect turns to enter the fourth-order field, with the fifth-order field independent. Moreover, if n>3.7, the elasticity only affects the fields whose order is higher than 4. In this case, the fourth-order field remains independent. Our investigation also shows that as long as n is larger than 1.6, the third-order field is always not independent, whose amplitude coefficient K3 depends either on K1 or on both K1 and K2 (K1 and K2 arc the amplitude coefficients of the first- and second-order fields, respectively). Firmly, good agreement is found between our results and O'Dowd and Shih's numerical ones[8] by comparison.

I. Numerical solution of exact pair equations. II. Numerical solution of first-order pair equations

Part I

Numerical solutions to the S-limit equations for the helium ground state and excited triplet state and the hydride ion ground state are obtained with the second and fourth difference approximations. The results for the ground states are superior to previously reported values. The coupled equations resulting from the partial wave expansion of the exact helium atom wavefunction were solved giving accurate S-, P-, D-, F-, and G-limits. The G-limit is -2.90351 a.u. compared to the exact value of the energy of -2.90372 a.u.

Part II

The pair functions which determine the exact first-order wavefunction for the ground state of the three-electron atom are found with the matrix finite difference method. The second- and third-order energies for the (1s1s)¹S, (1s2s)³S, and (1s2s)¹S states of the two-electron atom are presented along with contour and perspective plots of the pair functions. The total energy for the three-electron atom with a nuclear charge Z is found to be E(Z) = -1.125•Z² +1.022805•Z-0.408138-0.025515•(1/Z)+O(1/Z²)a.u.

Shape signifiers for control of a low-order compressor model

Rotating stall and surge, two instability mechanisms limiting the performance of aeroengines compressors, are studied on the third-order Moore-Greitzer model. The skewness of the compressor characteristic, a single parameter shape signifier, is shown to determine the key qualitative properties of feedback control.

Two-dimensional simulation of high-order laterally-coupled GaAs-AlGaAs DFB laser diodes

Laterally-coupled distributed feedback (LC-DFB) laser diodes made without an epitaxial re-growth process have the advantage of a simple fabrication process. In this paper, two-dimensional optical field distribution of the fundamental quasi TE (transverse electric) mode is calculated by means of a semivectorial finite-difference method (SV-FDM). The dependence of the effective coupling coefficient (kappa(eff)) on the dutycycle of first-, second- and third-order LC-DFB LDs is investigated using modified coupled wave equations.

Statistical distribution of wave-surface elevation for second-order random directional ocean waves in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a statistical distribution of the wave-surface elevation is derived by using the characteristic function expansion method. It is found that the distribution, after normalization of the wave-surface elevation, depends only on two parameters. One parameter describes the small mean bias of the surface produced by the second-order wave-wave interactions. Another one is approximately proportional to the skewness of the distribution. Both of these two parameters can be determined by the water depth and the wave-number spectrum of ocean waves. As an illustrative example, we consider a fully developed wind-generated sea and the parameters are calculated for various wind speeds and water depths by using Donelan and Pierson spectrum. It is also found that, for deep water, the dimensionless distribution reduces to the third-order Gram-Charlier series obtained by Longuet-Higgins [J. Fluid Mech. 17 (1963) 459]. The newly proposed distribution is compared with the data of Bitner [Appl. Ocean Res. 2 (1980) 63], Gaussian distribution and the fourth-order Gram-Charlier series, and found our distribution gives a more reasonable fit to the data. (C) 2002 Elsevier Science B.V. All rights reserved.

准噶尔盆地西北缘三叠系层序地层及有利勘探目标选择

Based on outcrop, borehole, seismic and regional geological data, the sequence stratigraphy, sedimentary facies of the Triassic in the western margin of the Zhugaer basin was studied, and favorable exploration target was forecasted. The major achievements include: (1) the Triassic in the western margin of the Zhugaer basin can be divided into 1 second-order sequence and 5 third-order sequences, which are, in ascending order, TSQ1, TSQ2, TSQ3, TSQ4, and TSQ5. TSQ1 is equivolent to Baikouquan formation, TSQ2 is equivolent to lower Kelamayi formation, TSQ3 is equivolent to upper Kelamai formation, TSQ4 is equivolent to lower and middle Baijiantan formation, and TSQ5 is equivolent to upper Baijiantan formation. Each sequence is divided into transgressive and regressive system tracts. Thus the sequence correlation framework is established. (2) The factors controlling development of sequences are analyzed, and it is believed that tectonic is the major controlling factor. Model of sequence development is summarized. (3)Through study on sedimentary facies, 6 types of facies are recognized: alluvial fan, fan delta, braided river, braided delta, delta and lake. Their microfacies are also recognized. In this study, it is proposed that the upper and lower Kelamayi formation（TSQ2、 TSQ3）is deposited by braided river instead of alluvial fan. This conclusion is of important theoretical and practical significance.(4) The sedimentary facies map of each sequence is compiled, and the sedimentary facies developed in each sequence is determined. In TSQ1, the sedimentary facies developed is alluvial fan and fan delta. In TSQ2, the sedimentary facies developed is mainly alluvial fan and fan delta in the north, and braided river and braided delta in the south. In TSQ3, the sedimentary facies developed is mainly braided river and braided delta. In TSQ4, the sedimentary facies developed is mainly braided delta in the north, and meandering delta in the south. In TSQ5, the sedimentary facies developed is mainly braided river and braided delta. (5) In the framework of sequence stratigrahpy, favorable areas for concealed traps are forecasted, and different types of traps are developed in different system tracts. (6) Favorable areas for future exploration are predicted.

柴西南区下干柴沟组下段层序地层格架内沉积相展布及有利储集砂体分布研究

Lower member of the lower Ganchaigou Formation in the southwestern of Qaidam Basin is one of the main targeted exploration zones. With the advancement of exploration, the targets are gradually switching into the lithologic reservoirs and it is urgent to gain the more precise research results in distribution of sedimentary facies and sandstones. Guided by the theory of sequence stratigraphy and sedimentology as well as on the basis of many logging data, drillings, seismic data and chemical tests, the paper comprehensively analyzes the sedimentary facies and sandstones in the lower member of lower Ganchaigou Formation in the southern of Chaixi. According to the identification marks of the key interface in sequence stratigraphy, the key interfaces in lower member of lower Ganchaigou Formation in the southwestern of Qaidam Basin are identified as two third-order sequences SQ1、SQ2. By calibrating the synthetic seismogram, the seismic sequence, well drilling and logging sequences are united. Based on the works above, this paper chooses seven primary cross-sections and builds connecting-well stratigraphic correlation of seven main connecting-well sections. Ultimately, the high-resolution sequence stratigraphic frameworks in the lower member of the lower Ganchaigou Formation, which are uniform to logging and seismic data, are figured out. In terms of study on each sequence features, the main style of the base-level cycle overlay which forms the third-order sequence is confirmed. It contains asymmetric “becoming deep upward” style and symmetry style. Researching on the spreading characters of sequence stratigraphy indicates that SQ1 and SQ2 are rather thicker near northwest well Shashen 20 and Shaxin1 while they are quite thiner near Hongcan 1, Yuejin, Qie 4 and Dong8-Wu3, and the thickness of SQ1 is thicker than SQ2.Based on the deep analysis of the marks for depositional facies, it is proposed that the lake facies and braid river deltas facies mainly occurred in study areas. Besides, the sorts of sub-facies and micro-facies model are divided and described. Under the control of high-resolution sequence stratigraphic framework, three source directions from Arlarer Mountain、Qimantage Mountain and Dongchai Mountain are identified by using the features of heavy mineral assemblage and paleogeomorphy. In addition, regularities of distribution sedimentary facies in sequence stratigraphic framework are studied in accordance with research thinking of the "point" (single well) "line" (section) "face" (plane). In the stage of lower member in the lower Ganchaigou Formation in the southwestern of Qaidam Basin, it is at the early phrase of evolution of the lake basin with the gradual outspread and the rise of the lake level. Combined with physical analysis of reservoir sands formed in different sedimentary environment, the paper studies the style of favorable sandstone bodies that are underwater distributary channel of braided rive delta front, coarse sand in mouth bar and the sand body in sand flat of shore-shallow lacustrine facies. Finally, this article comprehensively analyzes the distribution relationship between sedimentary facies and favorable sandstone body and proposes the ideas that sequence SQ1 Yuejin area, well east 8-wu3 area, well qie4-qie1 area and well hongcan2 area are distributed areas of favorable sandstone.

东营凹陷成藏组合体研究与勘探实践

With the development of petroleum exploration, subtle reservoir has become the main exploration object in Dongying Depression, which requires some new technologies and methods to further reveal the geological characteristics in step with the mature exploration stage. In this paper, on the references to the studies of petroleum system and multiple oil-gas accumulation belt with flexible maneuverability, and the application of systematic theory, the concept of reservoir assemblage is initially defined as "the association of active source rock(s) and hydrocarbon reservoir(s) that are genetically related, with the bridge of pathway system in an oil and gas bearing basin". Compared with the theories of petroleum system and multiple oil-gas accumulation belts, it emphasizes on the processes of petroleum migration and accumulation and the correlation among active source rock, trapped hydrocarbon and migration pathway, and has been confirmed to be more suitably applied to high maturely explored basin. In the first study of this paper, sequence stratigraphy and subtle analytical technology of source rock have been employed to find that two categories of source rock with their characteristic types of organic matter and substantial states occurred in Dongying Depression. The first category, consisting of the oil shales within the third-order sequences of lacustrine expanding system tracts in the upper interval of the fourth Member of Shahejie Formation and both in the middle and lower intervals of the third Member of Shahejie Formation, is featured with the highest abundance of total organic matter (TOC) and the strongest abilities of hydrocarbon generation and expulsion, which is classified into the standard of good hydrocarbon source rock. Exploration assessment confirmed that about 70-80% of hydrocarbon in Dongying Depression came from this set of source rock for which the low sedimentary rate and strong oxygen-free environment would play the key role during its generation. The second category, composed of organic matter of dark mudstone in high stand system tracts in the upper and middle intervals of the third Member of Shahejie Formation, has been characterized by low content of total organic matter which mostly dispersedly distributes, and formed in the pre-delta to delta front environments. In classification, it belongs to the ordinary standard of source rocks. In the second research part, through the studies of high frequency sequence stratigraphy, fault geometry and active history combining with geochemistry of fluid inclusion and nitrogen compound and simulation test of hydrocarbon migration and accumulation, the faults have been thought to be the principal conduits, and the sandy bodies and unconformities might played the complementary pathways for hydrocarbon migration and accumulation in Dongying Depression of the continental faulted basin. Therefore, the fault activities may mainly constrain on the development of hydrocarbon pathways in space and time. Even more, using homogenization temperatures of fluid inclusion in digenetic minerals, three critical moments for hydrocarbon accumulation have been determined as well in Dongying Depression, which happened during the late stage of Dongying Formation (Ed), the early stage of Guantao Formation (Nig) and the early stage of Minghuazhen Formation (Nim), respectively. Comparatively, the last stage is looked as the main forming-reservoir period, which has also been supported by the results of geochemical analysis and simulation experiments of hydrocarbon generation and expulsion. Clearly, the times of hydrocarbon migration and accumulation are consistent with those of the fault activities in Dongying Depression, which indicate that tectonic activities would control the forming-reservoir. A conceptual model of faulting-episodic expulsion coupled with episodic forming-reservoir has then been established in this study. In the third part of this paper, some focusing areas were selected for the fine descriptions of pathway distribution and forming-reservoir, which has given four types of reservoir assemblage in terms of the main pathway and its correlation with the reservoir and trap: (1) mainly consisted of sandy bodies; (2) mainly consisted of faults; (3) mainly consisted of unconformities; and (4) their complex with two or three types of pathways. This classified criteria has also been applied to access the risk of some prospected traps in Dongying Depression. Finally, through the application of reservoir assemblage integrated with pathway distribution to all the prospective targets in Dongying Depression, the new favorably hydrocarbon accumulated belts have been figured out, and more subtle reservoirs have also been found. For examples, during 2000 and 2002, in the mature exploration areas, such as Liangjialou and Shengtuo structural closures etc., newly proved reserves were 2274 * 104t, and forecasted oil reserves 5660-5860xl04t; and in the predicted favorable areas, newly additional controlled oil reserves was 3355xl04t. Besides those, many other favorable exploration areas need to be further appraised.

Higher-order effects and ultrashort solitons in left-handed metamaterials

Starting from Maxwell's equations, we use the reductive perturbation method to derive a second-order and a third-order nonlinear Schrodinger equation, describing ultrashort solitons in nonlinear left-handed metamaterials. We find necessary conditions and derive exact bright and dark soliton solutions of these equations for the electric and magnetic field envelopes.

Polynomial Model for High-Order and Multi-Carrier Passive Intermodulation Products

Passive intermodulation (PIM) often limits the performance of communication systems, particularly in the presence of multiple carriers. Since the origins of the apparently multiple physical sources of nonlinearity causing PIM in distributed circuits are not fully understood, the behavioural models are frequently employed to describe the process of PIM generation. In this paper, a memoryless nonlinear polynomial model, capable of predicting high-order multi-carrier intermodulation products, is deduced from the third-order two-tone PIM measurements on a microstrip transmission line with distributed nonlinearity. The analytical model of passive distributed nonlinearity is implemented in Keysight Technology’s ADS simulator to evaluate the adjacent band power ratio for three-tone signals. The obtained results suggest that the costly multi-carrier test setups can possibly be replaced by a simulation tool based on the properly retrieved nonlinear polynomial model.