Dimensionless Points Research Articles

An efficient flux-reconstructed lattice boltzmann flux solver for flow interaction of multi-structure with curved boundary

Recently, the escalating computational capability provided by advanced GPU technology for numerical simulations is well-suited for tackling large-scale engineering challenges. The lattice Boltzmann flux solver (LBFS), as a relatively new fluid-solving method, combines the parallelization characteristics of the lattice Boltzmann method (LBM) with the ability to handle non-uniform grids. Building upon these advantages, this study aligns its flux computational steps with the demands of the GPU parallel computing environment, thus developing an efficient flux-reconstructed lattice Boltzmann flux solver (FRLBFS), the improved algorithm not only ensures precision but also achieves a significant enhancement in efficiency, reaching up to a remarkable 500-fold improvement. Additionally, this work combines the immersed boundary method, significantly improving the efficiency of addressing hydrodynamic problems with multiple structures. Through numerical validation, under identical GPU hardware conditions, the proposed method in this paper outperform LBM in terms of computational efficiency. Lastly, simulations of flow past an array of eight cylinders at different Reynolds numbers are conducted, instantaneous contour plots at various dimensionless time points and time-dependent curves of drag and lift coefficients for different cylinders are provided, to demonstrate the complex vortex shedding surrounding multiple cylinders and its extraordinary computational efficiency.

Is Holland a Lumpy Country? An application of the Lens-Condition to Dutch Cities

Traditional trade theory assumes that countries are dimensionless points. Recent research, however, shows that the internal geography of countries is important for the trade structure of a country. One aspect of internal geography is the uneven spatial distribution of factors of production, which especially concentrate in urban locations. The so-called lens-condition (based on the Heckscher-Ohlin model) tests whether the (urban) distribution of factors of production is uneven enough to affect the national structure of trade and welfare. Our analysis, using firm export data and applying the condition to 22 cities and 4 regions within The Netherlands for 2007–2017, shows that the condition is fulfilled. We explain why.

Is Holland a Lumpy Country? An Application of the Lens-Condition to Dutch Cities

Traditional trade theory assumes that countries are dimensionless points. Recent research shows, however, that the internal geography of countries is important for the effects on trade. One aspect of internal geography is the uneven spatial distribution of factors of production. Factors of production especially concentrate in urban locations. The so-called lens-condition tests whether the (urban) distribution of factors of production is uneven enough to affect the national structure of trade. Using detailed data and applying the condition to 22 cities and 4 regions within The Netherlands for 2007-2017, shows that the condition is fulfilled. We explain why.

The Comparative Advantage of Dutch Cities

The trade literature often treats countries as dimensionless points, which is a strong assumption. Agglomeration or lumpiness of production factors within countries can affect the national pattern of trade. In this paper we analyze comparative advantage patterns for 22 cities and 4 regions for (a selection of) 83 sectors within The Netherlands. Our findings are as follows. First, analysis of the lens condition indicates that the regional concentration of production factors (lumpiness) does not affect the Dutch national trade pattern. This suggests that the mobility of firms and factors of production is consistent with the so-called welfare maximizing integrated equilibrium. Second, despite the fact that the lens condition is verified, comparative advantage patterns across locations differ significantly from each other. We show this by comparing location specific distributions of the Balassa-Index (BI). Third, the differences across locations of comparative advantage patterns is determined by the interaction of local skill-abundance and sector skill-intensity, in line with the predictions of the factor abundance model. Moreover, at the sectoral level, location-specific variables such as market access or density, have limited effects. Fourth, most locations that house sectors that have a strong comparative (dis-) advantage relative to the Netherlands also have a strong comparative (dis-) advantage relative to the world. Only a few locations house sectors that are locally strong, but globally weak, and vice versa. The results indicate that international trade policies and disputes, such as Brexit or the US-China trade war, can have strong local consequences.

The Barycenter Theorem: Averaging Possible-Paths to Produce Optimal Discrete Straight-line Segments

In the world of Euclidean geometry, with its dimensionless points and infinitely divisible, continuous straight lines, everyone knows that the shortest path between two points is a straight line. B...

Analyzing linear spatial features in ecology.

The spatial analysis of dimensionless points (e.g., tree locations on a plot map) is common in ecology, for instance using point-process statistics to detect and compare patterns. However, the treatment of one-dimensional linear features (fiber processes) is rarely attempted. Here we appropriate the methods of vector sums and dot products, used regularly in fields like astrophysics, to analyze a data set of mapped linear features (logs) measured in 12×1-ha forest plots. For this demonstrative case study, we ask two deceptively simple questions: do trees tend to fall downhill, and if so, does slope gradient matter? Despite noisy data and many potential confounders, we show clearly that topography (slope direction and steepness) of forest plots does matter to treefall. More generally, these results underscore the value of mathematical methods of physics to problems in the spatial analysis of linear features, and the opportunities that interdisciplinary collaboration provides. This work provides scope for a variety of future ecological analyzes of fiber processes in space.

Use of Stereology in Forest Inventories—A Brief History and Prospects for the Future

Several forest inventory techniques utilize approaches that are similar to stereological approaches often applied in microscopy and other fields. Stereology is characterized by the description and estimation of properties of objects based on samples of lower dimension than the object, e.g., 2-dimensional slices from 3-dimensional objects, 1-dimensional probes from 3-dimensional or 2-dimensional objects and dimensionless points from higher dimensional objects. The stereological character of many forest inventory methods was historically developed independently of recognition of a relationship with stereology. Strip sampling of forests, common in the late 19th and early 20th century, can be considered as a sterelogical approach if the strip centerline is viewed as a 1-dimensional probe of tree inclusion zones on a land area. The stereological character of plot sampling and Bitterlich sampling becomes evident if one views these methods as samples of 1-dimensional probes for volume within tree inclusion zones, or dimensionless points sampling for basal area in inclusion zones. Traditional methods of estimation of tree stem volume include samples of 2-dimensional cross-sectional area at fixed points along the tree stem to estimate 3-dimensional volume. Though these traditional methods usually use a shape assumption (e.g., parabolic frustum) for short stem segments, we show how a random-systematic start estimator of stem cross-sections can provide a design-unbiased estimate of stem volume without using any stem shape assumptions. Monte Carlo integration estimators of tree volume such as importance sampling that are designed to depend on only a few (usually one or two) tree upper-stem height or cross-sectional samples can also be viewed as stereological methods. Several forest inventory methods such as Matern’s individual tree basal area estimator and sector sampling can be viewed as local stereology, in which sample lines or slices pass through a central point. Finally, we suggest potential applications of stereological principles in the emerging “big data” era characterized by lidar and other remote sensing data and the assemblage of large tree and stand datasets. We suggest a new stem volume estimator which may have potential for future use with terrestrial lidar.

A Monte-Carlo simulation of ionic conductivity and viscosity of highly concentrated electrolytes based on a pseudo-lattice model.

Simulating three transport phenomena-ionic conductivity, viscosity, and self-diffusion coefficient-in a common Monte-Carlo framework, we discuss their relationship to the intermolecular interactions of electrolyte solutions at high concentrations (C/mol l-1 ∼ 1). The simulation is predicated on a pseudolattice model of the solution. The ions and solvents (collectively termed "molecules") are considered dimensionless points occupying the lattice sites. The molecular transport is realized by a repetition of swapping two adjacent molecules by the stochastic Gibbs sampling process based on simple intermolecular interactions. The framework has been validated by the fact that the simulated ionic conductivity and dynamic viscosity of 1:1- and 2:1-salts qualitatively well represent the experimental data. The magnitude of the Coulombic interaction itself is not reflected in the ionic conductivity, but the extent to which the Coulombic interaction is shielded by the dielectric constant has a significant influence. On the other hand, the dielectric constant barely influences the viscosity, while the magnitude of the Coulombic interaction is directly reflected in the viscosity.

Irrational images – the visualization of abstract mathematical terms

In this article we would like to draw attention to the cognitive potential hidden in an image and in the art which employs it. We will focus on the visualization of basic mathematical objects i.e. irrational numbers. Our starting point will be the easy and intuitive case of the square root of two, as it is observed in the diagonal of a square. Next we will move over to the golden ratio hidden in a regular pentagon. With the visualization of the irrational number ϕ Normal 0 21 false false false PL ZH-CN X-NONE we will use a looped, endless animation. Finally, we will have a closer look at the famous number π Normal 0 21 false false false PL ZH-CN X-NONE and we will suggest an attempt at its clearly visual representation. In the last section of the article we will consider the possibility of indicating rational and irrational real numbers represented by dimensionless points on a straight line. We will also try to present a straight line on a flat surface which - as we know has length - but has no width. The above issues will enable us to see the extent to which mathematics may be inspirational for art, as well as how art may familiarize us with mathematical issues and explain them.

Spatial analyses for nonoverlapping objects with size variations and their application to coral communities.

Spatial distributions of individuals are conventionally analysed by representing objects as dimensionless points, in which spatial statistics are based on centre-to-centre distances. However, if organisms expand without overlapping and show size variations, such as is the case for encrusting corals, interobject spacing is crucial for spatial associations where interactions occur. We introduced new pairwise statistics using minimum distances between objects and demonstrated their utility when examining encrusting coral community data. We also calculated the conventional point process statistics and the grid-based statistics to clarify the advantages and limitations of each spatial statistical method. For simplicity, coral colonies were approximated by disks in these demonstrations. Focusing on short-distance effects, the use of minimum distances revealed that almost all coral genera were aggregated at a scale of 1-25 cm. However, when fragmented colonies (ramets) were treated as a genet, a genet-level analysis indicated weak or no aggregation, suggesting that most corals were randomly distributed and that fragmentation was the primary cause of colony aggregations. In contrast, point process statistics showed larger aggregation scales, presumably because centre-to-centre distances included both intercolony spacing and colony sizes (radius). The grid-based statistics were able to quantify the patch (aggregation) scale of colonies, but the scale was strongly affected by the colony size. Our approach quantitatively showed repulsive effects between an aggressive genus and a competitively weak genus, while the grid-based statistics (covariance function) also showed repulsion although the spatial scale indicated from the statistics was not directly interpretable in terms of ecological meaning. The use of minimum distances together with previously proposed spatial statistics helped us to extend our understanding of the spatial patterns of nonoverlapping objects that vary in size and the associated specific scales.

Weighting Ripley’s K-Function to Account for the Firm Dimension in the Analysis of Spatial Concentration

The spatial concentration of firms has long been a central issue in economics under both the theoretical and the applied point of view mainly due to the important policy implications. A popular approach to its measurement, which does not suffer from the problem of the arbitrariness of the regional boundaries, makes use of micro data and looks at the firms as if they were dimensionless points distributed in the economic space. However, in practical circumstances the points (firms) observed in the economic space are far from being dimensionless and are conversely characterized by different dimension in terms of the number of employees, the product, the capital, and so on. In the literature, the works that originally introduce such an approach disregard the aspect of the different firm dimension and ignore the fact that a high degree of spatial concentration may result from the case of both many small points clustering in definite portions of space and only few large points clustering together (e.g., few large firms). We refer to this phenomenon as clustering of firms and clustering of economic activities. The present article aims at tackling this problem by adapting the popular K-function to account for the point dimension using the framework of marked point process theory.

FOURIER ANALYSIS AND THE KINETIC THEORY OF GASES

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior force is acting on them, (4) there is no collision between the particles, (5) the collisions against the walls of the container are according to the law of elastic collision, we deduce from Newtonian mechanics two basic probabilistic limit laws about particle-counting. The first one is a global central limit theorem, and the second one is a local Poisson law. The key fact is that we can prove the “realistic limit”: first the time interval is fixed and the number of particles tends to infinity (where the density particle/volume remains a fixed constant), and then, in the second step, the time tends to infinity.

SU‐E‐I‐166: Investigation and Development of Point‐Based Image Reconstruction Algorithm Capable of Handling Non‐Rigid Motion

Purpose: To evaluate PET image reconstruction algorithm capable of handling moving, deformable objects, particularly unrestrained conscious rodents. Methods: The proposed point‐based image reconstruction algorithm is based on 2D filtered back‐projection. Reconstruction took place in 2D virtual space filled with dimensionless sampling points. During the backprojection, points acted as spatial count bins, accumulating values of the backprojected LORs. To correct for motion, sampling points in the image space were moved in accord with the scanned object. Thus, motion correction and deformation correction were performed via image bin repositioning. After the backprojection step, images of activity distributions were obtained by fitting a smooth surface over the sampling points. Results: To evaluate the performance of the image reconstruction algorithm, stationary and moving NEMA image quality phantoms were scanned on Siemens Focus 120 microPET scanner, and data was reconstructed using conventional 2D filtered back‐projection and point‐based algorithms. Images obtained using two reconstruction algorithms were in remarkable quantitative agreement, with deviations mostly in the background and on the order of few percent. The location of points during reconstruction had little influence as long as Nyquist sampling criteria was satisfied. For the point‐ based reconstruction, motion‐corrected images preserved quantification and demonstrated good performance in removing the motion artifact. The SNR of motion‐corrected images also improved compared with shorter but motion‐free frames. Conclusions: It had been demonstrated that point reconstruction can provide accurate, quantitative PET images. As opposed to voxels, points can be moved freely relative to one another. By moving points during the backprojection process, almost any kind of deformation can be taken into account. Plans for the future include further method development of the method to enable full 3D list‐mode reconstruction, which could ultimately lead to unrestrained conscious rodent imaging.

Deterministic Approach to the Kinetic Theory of Gases

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior forces are acting on them, (4) there is no collision between the particles, (5) the collision against the walls of the container are according to the law of elastic reflection, we deduce from Newtonian mechanics two local probabilistic laws: a Poisson limit law and a central limit theorem. We also prove some global law of large numbers, justifying that “density” and “pressure” are constant. Finally, as a byproduct of our research, we prove the surprising super-uniformity of the typical billiard path in a square.

TRANSMISSION PROPERTIES OF QUASI-PERIODIC FIBONACCI ONE-DIMENSIONAL PHOTONIC CRYSTAL CONTAINING NEGATIVE PERMITTIVITY/PERMEABILITY MATERIAL

The transmission properties of one-dimensional quasi-periodic Fibonacci photonic crystal containing negative permittivity/permeability material are investigated by use of the transfer matrix method. We find that perfect transmission peaks exist at a series of dimensionless wavelength points in the transmission spectra, and the discrete localized modes appear inside the photonic band gap in the case of normal incidence. In the case of oblique incidence, the detailed numerical calculations show that the transmission peaks and angles strongly depend on the polarization of incident waves and the periodicity of the photonic crystals. In addition, we notice that the one-dimensional quasi-periodic photonic crystal can exhibit omni-directional reflection in a certain frequency region for all incident angles and polarizations. We also investigate the possibility of realizing the omni-directional reflection by means of the simple one-dimensional quasi-periodic Fibonacci photonic crystal.

Flow characterization and thermo-mechanical response of anisotropic conductive films

The paper reports investigations on the chip on glass (COG) bonding process using anisotropic conductive films (ACF). Experimental methods as well as theoretical analyses, by both analytical and numerical means, are applied. The assumptions concerning the thermo-mechanical and rheological properties of the polymer materials involved in the bonding process are characterized for dependence on temperature. The transient development of the temperature field during the bonding process is studied by finite element (FE) analysis for dependence on the upper and lower chuck temperatures. Analytical techniques of fluid mechanics are used to predict the flow of the conductive particles during bonding, treated as dimensionless points embedded in a viscous matrix. This analytical description allows one to estimate the number of conducting particles on a bump of a chip after bonding. Furthermore, numerical calculations are applied to characterize the influence of viscosity gradients on the particle flow. Finally, nonlinear finite element simulations are used to investigate the stress development and stress relaxation process within the ACF joints.

Representing the Shape and Orientation of Destinations in Spatial Choice Models

This paper illustrates the importance of spatial unit shape and orientation in spatial choice analysis. It also examines the implications of assuming spatial units to be dimensionless points in this context. Through theory and simulation experiments, it is shown how these aspects cannot be ignored and how the ordinary multinomial logit model applied at the spatially aggregate level is particularly vulnerable to such oversights. The aggregated spatial logit model in the end is recommended as a general formulation which addresses many of these fundamental issues of interest to geographers and regional scientists.

Partial orders for parallel debugging

Parallel programs differ from sequential programs primarily in that the temporal relationships between events are only partially defined. However, for a given distributed computation, debugging utilities typically linearize the observed set of events into a total ordering, thus losing information and allowing potentially capturable temporal errors to escape detection. We explore use of the partially ordered relation “happened before” to augment both centralized and distributed parallel debuggers to ensure that such errors are always detected and that the results produced by the debugger are unaffected by the non-determinism inherent in the partial ordering. This greatly reduces the number of tests required during debugging. Assertions are based on time intervals, rather than treating events as dimensionless points.

Nearest-Neighbour Relationships of Overlapping Circles and the Dispersion Pattern of Desert Shrubs

(1) Most techniques for analysing the dispersion of plant individuals in populations assume that the objects in question are points. When applied to populations of desert shrubs, these techniques often indicate significantly uniform dispersion. (2) The growth patterns of some plants may result in the overlap and fusion of separate individuals to form a clump that cannot be distinguished from that produced by a single individual. This inability to distinguish intimately fused individuals creates a bias toward perceiving uniformity as the techniques of dispersion analysis are based on theory relating to the dispersion of dimensionless points. (3) Simberloff's technique of analysis of nearest-neighbour relationships of nonoverlapping circles can be adapted to overlapping circles that become indistinguishable at some critical minimum separation distance and thus provides a technique for pattern analysis under these circumstances. (4) Analysis of the dispersion of Larrea tridentata in the Colorado Desert of California demonstrates that its clumped growth pattern creates a strong bias toward the perception of uniformity with techniques based on the point-dispersion theory.