Euler Number Research Articles

Truncated Hecke-Rogers type series

The recent work of George Andrews and Mircea Merca on the truncated version of Euler's pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. The main purpose of this paper is to generalize the study to Hecke-Rogers type double series, which are associated with some interesting partition functions. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu on the q-partial differential equations and q-series.

Enhancement of Heat Transfer in Perforated Circular Finned-Tube Heat Exchangers: A Numerical Investigation

The effects of employing perforations of circular shape in circular finned-tube heat exchangers (CFTHEs) are numerically investigated in the present study using Computational Fluid Dynamics (CFD) software. Simulations performed on the validated CFD model shows considerable increase in the Average Nusselt Number for perforated circular finned-tube heat exchangers (PCFTHEs) as compared to their solid counterpart, with only a slight increase in Euler Number for 10,000 ≤ Re ≤ 50,000. Diameter of perforations (dp ) is kept equal to 6mm and optimum range of number of perforations per fin (np ) by analysis is found to be 16 to 20. For this range, the performance of PCFTHEs improves with the increase in number of perforations per fin. The maximum increase in Average Nusselt Number is 24% for dp = 6mm and np = 20 at Re = 10,000. The minimum increase in Average Nusselt Number is 3% for dp = 6mm and np = 14 at Re = 40,000. This increase in Average Nusselt Number on introducing perforations is due to (i) flow of localized air jets through the perforations, and (ii) reduction in recirculation region caused by the separation of flow. This improvement in the performance of CFTHEs on introducing perforations is achieved with the maximum and minimum reduction in weight of 41% and 29% respectively.

Experimental and numerical investigation on the performance of square cyclones with different vortex finder configurations

The aim of this study was to analyse the effect of varying the vortex finder diameter and its insertion length on the flow field and overall performance in a square cyclone. A total of 15 different vortex finder configurations with 5 different diameters, each having 3 different insertion lengths, were analyzed. An in-depth analysis of flow field inside the separator was conducted by using stereoscopic particle image velocimetry (Stereo-PIV) and large eddy simulation (LES). The former makes use of two cameras that focused simultaneously over the region of interest to record the details of the flow field, whereas the latter approach directly resolves a large part of the flow and provides the three-dimensional time-dependent details of the turbulent flows. The results extracted from PIV and LES – viz. mean tangential, mean axial, and mean radial velocities – were compared, and good agreement was observed between the two. In addition, the separation efficiency and Euler number (Eu) were evaluated experimentally and by using LES. Conclusive results indicate that the geometric shapes of the vortex finder have a significant effect on the performance of square cyclone separators. Decreasing the diameter of the vortex finder resulted in a considerable increase in Eu compared to the base variant (maximal increase in Eu amounted to about 660%). However, increasing the diameter of the vortex finder decreased Eu by about 55%. An optimally configured geometry of the vortex finder increased overall separation efficiency by 16%.

Influence of the dipleg and dustbin dimensions on performance of gas cyclones: An optimization study

Gas cyclones have numerous industrial applications. Typically, each cyclone has a dustbin to collect the trapped particles and the dimensions of the dustbin affect the cyclone performance. This paper aims to optimize the dustbin geometry via numerical simulations. The surrogate-based optimization approach has been applied in this study. The Latin-hyper cube sampling plan is used to generate thirty test cases. An artificial neural network with radial basis function has been used as a surrogate model trained by the CFD simulations. Here three design parameters (the dipleg length, the dustbin height, and the dustbin diameter) and two performance parameters, namely, the Euler number and the Stokes number are used. The fitted surrogate model shows that the variations of the dustbin geometry have a larger effect on the Stokes number than that on the Euler number. Both single-objective and bi-objective optimization studies are carried out using the artificial neural network. It is shown that the resulting optimum design of the dustbin and the dipleg leads to better performance than the conventional cyclones.

Hypergeometric Euler numbers

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors.

The Euler number of hyper-K¨ahler manifolds of OG10 type

Using the Laza-Sacca-Voisin construction, we give a simple proof for the fact that the Euler characteristic of a hyper-Kahler manifold of OG10 type is 176,904, a result previously established by Mozgovoy.

Effect of Surface Roughness on Vortex Length and Efficiency of Gas-oil Cyclones through CFD Modelling

Separation of suspended droplets in a fluid flow has been a great concern for scientists and technologists. In the current study, the effect of the surface roughness on flow field and the performance of a gas-oil cyclone is studied numerically. The droplets and the turbulent airflow inside the cyclone are considered to be the discrete and continuous phases respectively. The Reynolds stress model (RSM) is employed to simulate the complex, yet strongly anisotropic, flow inside the cyclone while the Eulerian-Lagrangian approach is selected to track droplet motion. The results are compared to experimental studies; according to the results, the tangential and axial velocities, pressure drop, and Euler number decrease when the surface roughness increases. Moreover, the cyclone efficiency drops when the vortex length decreases as a result of a rise in surface roughness. The differences between the numerical and experimental results become significant at higher flow rates. By calculating the impact energy of droplets and imposing the film-wall condition on the walls, splash does not occur.

Three-dimensional quantification of macropore networks of different compacted soils from opencast coal mine area using X-ray computed tomography

Heavy machinery compaction in mining area is widespread, and seriously jeopardizes soil restoration. Characterization of soil macropores with three-dimensional methods is essential for understanding the reclaimed minesoil (RMS) development. The aim of this study was to demonstrate the application of X-ray computed tomographic (CT) scanning for analyzing soil macropore structure and to explore how soil compaction affects soil macropore properties. To achieve this, soil samples collected from Pingshuo opencast coalmine were tested through a laboratory scheme, which was conducted to simulate the real soil pore situation under increasing soil bulk density treatments of 1.3, 1.4, 1.5, 1.6, 1.7, and 1.8 g cm−3 (T1.3-T1.8). Three-dimensional (3D) macropore networks were observed via CT, and macroporosity, macropore number, area, volume and its connectivity were characterized. Soil compaction had a major impact on soil macropore characteristics. The macropores of the 1.3 g cm-3 treatment were well connected, whereas those of the 1.8 g cm-3 treatment were poorly connected and isolated from each other. Macropore number and macroporosity decreased significantly as the soil compaction increased. Moreover, a 3D visualization of soil pore connectivity results suggested that the macropore connectivity markedly decreased with increasing compaction level, with average Euler number increasing from 0.677 in the 1.3 g cm-3 treatment to 0.999 in the 1.8 g cm-3 treatment. Soil bulk density was negatively correlated with macropore throat number and thickness, indicating that the soil compaction causes a loss of pore connectivity. Despite using a coarse resolution, the combination of laboratory simulation tests and 3D visualization of soil macropores provides valuable data for identifying the differences in soil pore structure among compacted RMSs and monitoring the persistent effects of soil compaction caused by heavy machinery in the mining area. Furthermore, we recommend a new method combining 3D visualization techniques and hydraulic analysis in the study of compacted soils, and the establishment of soil restoration measurements to avoid further compaction in mining areas.

Using micro focus industrial computed tomography to characterize the effects of soil type and soil depth on soil pore characteristics, morphology, and soil compression in Xi’an, China

Land subsidence has caused serious geological damage in many countries, including China. Soil pore number, size, shape, and pore size distribution affect soil deformation and thus land subsidence. Quantification of complex pore structures in different soil types and at various soil depths, however, remains cumbersome. The objective of this study was to quantify pore characteristics of three different soil types using computed tomography (CT) technique, investigate the effects of soil depth, examine the relationships between various parameters, and preliminarily determine the pore deformation pattern. Undisturbed soil samples (70 mm diameter × 50 mm height) of clay, silt, and silty clay soils were collected from two 300-m-deep engineering geological boreholes on both sides of the Xi’an D7 ground fissure. One hundred CT images were acquired for each soil sample. Soil pore characteristics and morphological parameters, including porosity, macroporosity, coarse mesoporosity, microporosity, number of pores, number of macropores (diameter > 1000 μm), number of coarse mesopores (diameter 200–1000 μm), number of micropores (diameter < 200 μm), circularity, structure model index, degree of anisotropy, and the Euler number, were analyzed. Micropores were generally regularly shaped and the main contributors to the total number of pores, whereas macropores were mostly elongated and contributed to total porosity. With increasing depth, the most noticeable changes appeared in silt soil and the least noticeable in clay soil. The alternate pore deformation pattern was significant in silt and silty clay soils. Soil depth significantly influenced all soil pore parameters, whereas soil type significantly affected the remaining parameters, except circularity and degree of anisotropy. The elongated macropores in silt and silty clay soils favor land subsidence in the Xi’an area. The present results can help optimize the groundwater exploitation layer, reducing soil compaction and land subsidence in this area.

Algebraic Reduced Genus One Gromov–Witten Invariants for Complete Intersections in Projective Spaces

Abstract In [ 17, 18], Zinger defined reduced Gromov–Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). In [ 3], Chang and Li provided a proof of the comparison theorem for quintic Calabi–Yau three-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. In [ 5], Coates and Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger in [ 14] and proved the comparison theorem for every Calabi–Yau threefold. In this paper, we prove the comparison theorem for every (not necessarily Calabi–Yau) complete intersection of Dimension 2 or 3 in projective spaces by taking a definition of reduced GW invariants in [ 5].

Multi-Stage Filter Designs for a Gasifier System

Most of the previous investigations on flow control devices have been reported on single-stage perforated plate with variable porosity and circular holes. This is the reason that functional relationships for pressure loss coefficient or Euler number (Eu) variation reported earlier are a strong function of porosity. In this paper, multi-stage filter design with constant porosity has been investigated using an experimentally validated numerical model. Several researchers have worked on the design of the producer gas cleaning system by using different filter materials such as electrostatic precipitators, wet scrubber, ceramics, fabric, and sand bed separately. However, these methods are inefficient in the final stages of the gas purification. Hence, multi-stage filters designs are conceived and investigated. Effect on pressure loss coefficient variation has been investigated for different hole geometry having same porosity with multiple filters. In a first, four new correlations has been developed for Eu variation as function of number of filters and different hole shapes. The Eu variation has the form: Eu = a(N)b (t/dh) c where N is the number of filters, a, b and c are constants whose value depend on the type of hole geometry. The prediction from correlation agrees within 4% accuracy with the numerical data

Segre Indices and Welschinger Weights as Options for Invariant Count of Real Lines

Abstract In our previous paper [5] we have elaborated a certain signed count of real lines on real hypersurfaces of degree $2n-1$ in $P^{n+1}$. Contrary to the honest “cardinal” count, it is independent of the choice of a hypersurface and, by this reason, provides a strong lower bound on the honest count. In this count the contribution of a line is its local input to the Euler number of a certain auxiliary vector bundle. The aim of this paper is to present other, in a sense more geometric, interpretations of this local input. One of them results from a generalization of Segre species of real lines on cubic surfaces and another from a generalization of Welschinger weights of real lines on quintic three-folds.

A characterization of Fuchsian actions by topological rigidity

We prove that any rigid representation of $\pi_1\Sigma_g$ in $\mathrm{Homeo}_+(S^1)$ with Euler number at least $g$ is necessarily semi-conjugate to a discrete, faithful representation into $\mathrm{PSL}(2,\mathbb{R})$. Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rigidity property. Though independent, this work can be read as an introduction to the companion paper {\em rigidity and geometricity for surface group actions on the circle}, by the same authors.

Some congruences related to a congruence of Van Hamme

ABSTRACTWe establish some supercongruences related to a supercongruence of Van Hamme, such as where p is an odd prime and is the th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf–Zeilberger method, Whipple's transformation, and the Mathematica package Sigma developed by Schneider. We also put forward two related conjectures.

A new explicit formula for Bernoulli numbers involving the Euler number

We derive a new explicit formula for Bernoulli numbers in terms of the Stirling numbers of the second kind and the Euler numbers. As a corollary of our result, we obtain an explicit formula for the even Euler numbers in terms of the Stirling numbers of the second kind.

Linear Instability of Liquid Sheets Subjected to a Transverse Electric Field

AbstractA temporal linear instability analysis was performed for a liquid sheet moving around the inviscid gas in transverse electrical field. The fluid was described by the leaky-dielectric model, which is more complex and more comparable to the liquid electrical properties than existing models. As a result, the sinuous and the varicose modes exist, in which the dimensionless dispersion relation between wave number and temporal growth rate can be derived as a 3 × 3 matrix. According to this relationship, the effects of liquid properties on sheet instability were performed. It was concluded that, as the electrical Euler number (Eu), the ratio of gas-to-liquid density (ρ), Weber number (We), Reynolds number (Re), and the relative relaxation time (τ) increased, the instability of the sheet was enhanced. This work also compared the leaky-dielectric model with the perfect conductor model and found that the unstable growth rate in the leaky-dielectric model was higher than the one in the perfect conductor model. Moreover, as the ratio of gas-to-liquid improved, this difference decreased. Finally, an energy approach was adopted to investigate the instability mechanism for the two models.

Image Feature Extraction and Retrieval of the Euler Number to Chinese Herbal Medicine based on PCNN

In order to extract microscopic image feature information of traditional Chinese medicine effectively and simply and improve the precision of image retrieval, an image Euler number feature extraction and retrieval algorithm based on pulse coupled neural network (PCNN) was proposed. First of all, the PCNN model is introduced from the perspective of biological tissue image processing. Secondly, the CHM image is segmented into a series of binary correlation images using simplified PCNN. The Euler number of the edge information of each binary image in the generated binary image sequence is calculated, and a one-dimensional feature vector is constructed. Finally, the Euclidean distance is used to measure the similarity of the feature vector in the retrieval of CHM microscopic images. Euler number feature extraction and retrieval method is characterized by simple calculation and small amount of data. Compared with other methods, this algorithm has stronger robustness and retrieval accuracy, and certain feasibility.

Numerical analysis of wind velocity effects on fire-wind enhancement

Variation in flow characteristics triggered through the fire-wind interface can potentially damage the buildings during bushfires. Fire-wind enhancement which is referred to as the increase of wind velocity, caused by the fire-wind interaction, is one of the destructive phenomena in this regard. In spite of the significance, the underlying mechanism contributing to this phenomenon is still not well understood. This study employs computational fluid dynamic (CFD) simulation to fundamentally investigate the effects of free-stream wind velocity on fire-wind enhancement through analyzing the momentum and buoyancy of fluid. Fire-wind interaction is shown to cause the generation of fire-induced longitudinal negative pressure gradient which results in fire-induced pressure and viscous forces in longitudinal direction. These forces are further found as the prime reason for the distortion of the wind velocity profile. A module is implemented to the FireFOAM solver to calculate and extract these forces quantitatively. The results reveal that under a constant fire intensity, the level of distortion and/or enhancement in the wind velocity profile comparatively reduces with the increase of free-stream wind velocity. A new non-dimensional group (modified Euler number) is introduced to take into account dominant fire-induced forces causing fire-wind enhancement. Richardson number and the modified Euler number are employed to determine the influence of free-stream wind velocity and longitudinal distance from the fire source on wind velocity enhancement. Large-eddy simulation (LES) results indicate that while the level of enhancement generally depends on both Richardson and the modified Euler number, the location of the maximum level of enhancement along the plume centreline coincides with the maximum value of modified Euler number under a constant free-stream wind velocity scenario.

Novel morphometric analysis of higher order structure of human radial peri-papillary capillaries: relevance to retinal perfusion efficiency and age

We apply novel analyses to images of superficial capillaries that are located near and around the optic disc of the human retina: the radial peri-papillary capillaries (RPCs). Due to their unique perfusion of the nerve fibre layer the RPCs are particularly significant for optic-neuropathies. The inputs to the analysis were z-stacks from 3D confocal fluorescence microscopy from 62 human retinas aged 9 to 84 years. Our aim was to find morphometric correlates of age. The retinas had no ophthalmic history. The analysis was undertaken in two stages: (1) converting the z-stacks to 3D tubular networks of vessels, and (2) characterizing the tubular networks using features derived from the Minkowski functionals (MFs). The MFs measure: the capillary volume, surface area, mean breadth, and Euler number. The mean breadth is related to tortuosity, wall shear stress and resistance to flow, and the Euler number is related to the density of loops (collaterals). Features derived from the surface area, mean breadth and Euler number were most related to age (all p ≤ 0.006). The results indicate the importance of pressure-equalizing loops and tortuosity as quantitative measures related to perfusion efficiency. The novel morphometric analysis could quantify disease-related accelerated aging and vessel malformation.

Strain heterogeneity and micro-damage nucleation under tensile stresses in an Mg–5Al–3Ca alloy with an intermetallic skeleton

Strain heterogeneity at the microstructural level plays a vital role in the deformation and fracture behaviour of dual or multi-phase materials. In the present work, the strain heterogeneity, localisation and partitioning arising at the sub-micron scale during elevated temperature (170 °C) tensile deformation of an Mg–5Al–3Ca alloy was investigated using quasi in-situ μ-DIC experiments. The results reveal that the strain is mainly carried by the α-Mg phase, while the intermetallic Laves phase plays a critical role in that strain concentrations build up at the α-Mg matrix and Laves phase interfaces, hence, reducing the overall deformability of the alloy. In quasi in-situ and bulk material analysis at elevated temperature, cracks were observed to nucleate in the Laves phase, at i) the intersection points of slip lines in the α-Mg matrix with the Laves phase and ii) the twin intersections with α-Mg/Laves phase interfaces and iii) twin transmissions across α-Mg/Laves phase interfaces. Euler number analysis has shown that the (inter-)connectivity of the Laves phase decreases with deformation. Finally, cracks grow preferentially along the Laves phases until the material fractures.