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The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients are related to each other. A relation between the Euler–Frobenius polynomial and B‐spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and special finite sums are derived. Moreover, a few recurrence relations containing these particular finite sums are given. Using these recurrence relations, we give a solution of the problem which was given by Charalambides. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums. [ABSTRACT FROM AUTHOR]

In this study, the effects of fin tip thickness and fin root thickness of integral rolled spiral finned tube bundles on flow resistance, heat transfer performance and heat transfer and flow exergy destruction were investigated via mathematical simulation. The correlations between heat transfer and flow resistance performance were fitted with dimensionless numbers. The optimized parameters with performance evaluation criteria (PEC) as the objective were obtained using methods involving computational fluid dynamics and machine learning. The results show the effects of fin tip thickness and fin root thickness on the Nusselt number (Nu), Euler number (Eu), PEC, heat transfer exergy destruction (ExT) and flow exergy destruction (ExP) as obtained via mathematical simulation. A new mathematical correlation is proposed for predicting the Nu and Eu of integral rolled spiral finned tube bundles. Among the four optimization models tested, the random forest regression algorithm was the most accurate algorithm for PEC prediction models. In the studied range, the optimal parameters were a fin tip thickness of 2 mm and a fin root thickness of 3.5 mm. Compared with the initial parameters, when the Reynolds number was 20,380, the PEC increased by 2.53%, the ExP increased by 2.37% and the ExT decreased by 7.96%. [ABSTRACT FROM AUTHOR]

We consider a family of Euler-type polynomials, depending on a real parameter α 6= 0, 1. The case α = 2 corresponds to standard Euler polynomials. We show some properties of these polynomials, and show also two generalized recurrences. As consequences of these results, we obtain several explicit formulas for Euler numbers and polynomials.

In formulating a non-orientable analogue of the Milnor Conjecture on the 4-genus of torus knots, Batson [Math. Res. Lett. 21 (2014), pp. 423–436] developed an elegant construction that produces a smooth non-orientable spanning surface in B^4 for a given torus knot in S^3. While Lobb [Math. Res. Lett. 26 (2019), pp. 1789] showed that Batson's surfaces do not always minimize the non-orientable 4-genus, we prove that they do minimize among surfaces that share their normal Euler number. We also determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson's surfaces are non-orientable 4-genus minimizers. [ABSTRACT FROM AUTHOR]

This paper designs a modified weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws. Using the switching principle based on inflection points, the new scheme automatically adapts between linear upwind and WENO schemes. If there is at least one inflection point in the largest stencil available for reconstruction, a symmetrical WENO (SWENO) scheme is considered for the reconstruction of the numerical flux; otherwise the numerical flux is directly approximated by the reconstruction polynomial. By comparing the new scheme introduced in this paper with the classical WENO scheme and another improved scheme that has been proposed recently D. Chai, G. Xi, Z. Sun, Z. Wangand Z. Huang,Comput. Fluids, vol. 170, pp. 176–186, 2018), we can point out the robustness and better efficiency of this scheme. To examine and explain the features of the new scheme, a number of examples such as Euler equations have been considered. [ABSTRACT FROM AUTHOR]

We use Macaulay2 for several enriched counts in GW (k) . First, we compute the count of lines on a general cubic surface using Macaulay2 over F p in GW (F p) for p a prime number and over Q in GW (Q) . This gives a new proof for the fact that the A 1 -Euler number of Sym 3 S ∗ → Gr (2 , 4) is 15 ⟨ 1 ⟩ + 12 ⟨ - 1 ⟩ . Then, we compute the count of lines in P 3 meeting 4 general lines, the count of lines on a quadratic surface meeting one general line and the count of singular elements in a pencil of degree d-surfaces. Finally, we provide code to compute the EKL-form and compute several A 1 -Milnor numbers. [ABSTRACT FROM AUTHOR]

Several trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established. [ABSTRACT FROM AUTHOR]

Relating image contours and regions and their attributes according to connectivity based on incidence or adjacency is a crucial task in numerous applications in the fields of image processing, computer vision and pattern recognition. In this paper, the crucial incidence topological information of 2-dimensional images is extracted in an efficient manner through the computation of a new structure called the HomDuRAG of an image; that is, the dual graph of the HomRAG (a topologically consistent extended version of the classical RAG). These representations are derived from the two traditional self-dual square grids (in which physical pixels play the role of 2-dimensional cells) and encapsulate the whole set of topological features and relations between the three types of objects embedded in a digital image: 2-dimensional (regions), 1-dimensional (contours) and 0-dimensional objects (crosses). Here, a first version of a fully parallel algorithm to compute this new representation is presented, whose timing complexity order (in the worst case and supposing one processing element per 0-cell) is O (l o g (M × N)) , M and N being the height and width of the image. Efficient implementations of this parallel algorithm would allow images to be processed in real time, as well as permit us to uncover fast algorithms for contour detection and segmentation, opening new perspectives within the image processing field. [ABSTRACT FROM AUTHOR]

Vortex flow has been demonstrated to be an effective way of process intensification for interphase mass transfer. However, the underlying principles of this phenomenon are not yet fully known. To understand the nature of gas vortex flow in improving process intensification from a fluid flow perspective, this work conducts an experimental investigation and numerical simulations to compare the differences in energy loss, static pressure, main velocity, and turbulent kinetic energy between the vortex and the axial gas flow inside a countercurrent contactor operated at a flow rate of 50–250 L/min. The results indicate that the energy loss increases with increasing gas flow rate, while the Euler number remains stable at 2.49 ± 0.17, which is 21.05% higher than the conventional axial flow. The vortex flow displays higher values and gradients in the distribution of static pressure, tangential, axial velocity, and turbulent kinetic energy, which is directly linked to the enhanced interphase contact, mixing, and mass transfer processes. Additionally, the tangential velocity of vortex flow exhibits a decaying behavior, but it also has an extra tangential dimension, which was a critical factor for process enhancement compared to conventional axial flow. Moreover, semi-empirical models are developed to characterize the parameters of the maximum tangential velocity and its radial position for the vortex flow with R2 = 0.892 and 0.919, respectively. The results may provide a positive reference for the design, optimization, and operation of countercurrent vortex contactors. [ABSTRACT FROM AUTHOR]

An important phenomenon in the gas–liquid two-phase mixtures is a sudden change in the flow that may lead to transition of the flow regime from non-homogeneous slip flow to homogenous flow. This phenomenon is called mixing shock and has been investigated by several researchers. In the present paper, a more comprehensive model is proposed by including the entrainment ratio in the governing equations. Moreover, parametric studies are performed to assess the importance of this parameter in different conditions. The results of the present study indicate that at Euler numbers less than four the effect of gas mass flow is insignificant. However, at higher Euler numbers the differences of the present and previous models are noticeable. Since two possible solutions for the aftershock state exist, four criteria—entropy change across the shock, possibility of an expansion shock, positivity of the Euler number, and choking flow condition—are considered to identify the correct solution. The results indicate that the flow after the shock could only be of a subsonic type, and the mixing shock is compressive. A comparison of the possible realizable zones for the developed model and the previous models indicates that the developed model predicts a larger area for the realizable solution of the mixing shock. A comparison between analytical results and experimental data shows that the developed model predicts reasonable results. [ABSTRACT FROM AUTHOR]

We evaluate in closed form a large class of power series where the coefficients are products of special numbers, or products of special numbers and Laguerre polynomials. A number of examples is presented involving harmonic, hyperharmonic, Fibonacci, Lucas, Bernoulli, Euler, and derangement numbers. We also obtain a new convolution formula for the Laguerre polynomials. [ABSTRACT FROM AUTHOR]

Soil desiccation and cracking can be one of the most common physical phenomena during natural and man-made damage processes. There is the a significant effect on the mechanical and hydraulic properties of soil in modern agriculture. This study aimed to reveal the development and formation mechanism of desiccation cracking in farmland soil along the depth direction. The Hookean spring model proposed by Vogel was improved to construct the triangular prism network structure with the nodes under the gravity. A three-dimensional soil desiccation cracking model was established to simulate the crack depth. The crack along the depth direction was assumed as the vertical development during simulation. The farm soil desiccation cracking experiment was carried out in the Tongxin County (105°54 ′E, 36°58 ′N), Ningxia, China. The ponded water was exerted at the soil surface, until the soil was saturated at the upper 10 cm layer. Camera photography was used to capture the images of crack formation and development at 2-hour intervals for three days. The steps of digital image processing included: the non-experimental part of the image was cropped, the image size was processed to 2048×2048 pixels, while the image was applied geometrically corrected, and a color image of the crack patterns was converted into a gray image, which was then converted into a binary image using the threshold division, and finally the image was processed through the noise removal. The white areas and black pixels in the binary image were represented by the aggregates and crack skeleton, respectively. After that, the Minkowski densities were adopted to quantify the surface crack patterns. Three basic characteristics of cracks were divided into the area A, the length L, and the Euler number E of the crack (representing connectivity in the two-dimensional crack structure). The difference between the Minkowski density and crack depth of the experimental and simulated image was quantified by four evaluation indicators: the coefficient of determination, R ², the consistency index, IA, and the root mean square error, RMSE. Finally, a systematic analysis was implemented on the influence of the longitudinal elastic coefficients, Kz (representing the elasticity of soil along the depth direction on the model output. Parameters The parameters of the model were calibrated and verified by the field experimental measurement and photograph. The precision of the simulation was also evaluated. The results show that: The coefficients of determination, R 2 for the area, length, and Euler number density were between 0.849-0.959. RMSE was between 0.005-0.083, while the BIAS was between 0.103-0.190, and the consistency index IA was 0.965-0.988. As such, the improved 3D model calibration was fully met the requirement of calibration. The model was effective to simulate the formation and development of cracks in farmland soils along the depth direction. The simulated surface crack morphology was consistent with the development of natural cracks. The smaller the longitudinal elastic coefficient was, the more outstanding the trend of crack development along the depth direction was. The peak of crack depth frequency was at soil depth of 6 cm, and the proportion of deep cracks (soil depth of 5-10 cm) was larger. On the contrary, the crack generation and development along the depth direction were inhibited, with the peak of crack depth frequency at (soil depth of 1cm) and a greater proportion of shallow cracks (soil depth of1-4 cm). The finding can provide an ideal way to predict the development and formation of soil cracks along the depth direction. [ABSTRACT FROM AUTHOR]

Meier and Zupan proved that an orientable surface in S 4 admits a tri-plane diagram with zero crossings if and only if is unknotted, so that the crossing number of is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in S 4 , proving that c ( n , m) = max { 1 , | n − m | } , where n , m denotes the connected sum of n unknotted projective planes with normal Euler number + 2 and m unknotted projective planes with normal Euler number − 2. In addition, we convert Yoshikawa's table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers. [ABSTRACT FROM AUTHOR]

In this paper we establish a connection between the Gauss factorials and Iwasawa's cyclotomic λ -invariant for an imaginary quadratic field K. As a result, we will explain a correspondence between the 1-exceptional primes of Cosgrave and Dilcher [2] , [3] for m = 3 and m = 4 , and the primes for which the λ -invariants for K = Q (− 3) and K = Q (i) are greater than one, respectively. We refer to the latter primes as "non-trivial" for their respective fields. We will also see that similar correspondences are true for K = Q (− d) when d = 2 , 5 and 6. As a corollary we find that primes p of the form p 2 = 3 x 2 + 3 x + 1 are always non-trivial for K = Q (− 3). Last, we show that the non-trivial primes p for K = Q (i) and K = Q (− 3) are characterized by modulo p 2 congruences involving Euler and Glaisher numbers respectively. [ABSTRACT FROM AUTHOR]

Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli numbers, and Euler numbers. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove the following conjecture of Sun[Congruences involving binomial coefficients and Apery-like numbers, Publ. Math. Debrecen 96 (2020), pp. 315–346]: Let p>3 be a prime. Then ∑ k = 0 p − 1 ( 2 k k ) 3 k + 1 (− 16) k f k ≡ (− 1) (p − 1) / 2 p + p 3 E p − 3 (mod p 4) , where f n = ∑ k = 0 n ( n k ) 3 and E n stand for the nth Franel number and nth Euler number, respectively. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give formulas and Recurrence relations for the Apostol-Euler numbers and polynomials of order with complex numbers with the aid of the Euler operator and partial derivatives of the generating function. Relations among the these numbers and polynomials of negative integer order, the beta-type rational functions, finite combinatorial sums, the Stirling numbers, and the Lah numbers are given. Finally, new classes of polynomials and modification exponential Euler type splines are constructed. [ABSTRACT FROM AUTHOR]

As an important topological property for a 3D binary image, the Euler number can be computed by finding specific a voxel block with 2 × 2 × 2 voxels, named the voxel pattern, in the image. In this paper, we introduce three strategies for enhancing the efficiency of a voxel-pattern-based Euler number computing algorithm used for 3D binary images. The first strategy is taking advantage of the voxel information acquired during computation to avoid accessing voxels repeatedly. This can reduce the average number of accessed voxels from 8 to 4 for processing a voxel pattern. Therefore, the efficiency of computation will be improved. The second strategy is scanning every two rows and processing two voxel patterns simultaneously in each scan. In this strategy, only three voxels need to be accessed when a voxel pattern is processed. The last strategy is determining the voxel accessing order in the processing voxel pattern and unifying the processing of the voxel patterns that have identical Euler number increments to one group in the computation. Although this strategy can theoretically reduce the average number of voxels accessed from 8 to 4.25 for processing a voxel pattern, it is more efficient than the above two strategies for moderate- and high-density 3D binary images. Experimental results demonstrated that the three algorithms with each of our proposed three strategies exhibit greater efficiency compared to the conventional Euler number computing algorithm based on finding specific voxel patterns in the image. [ABSTRACT FROM AUTHOR]

Motivated by the recent work on the symmetric domains, this article investigates certain features of symmetric domain which are caused by the secant hyperbolic functions. Geometric characteristics of analytic functions associated with secant hyperbolic functions are discussed, which include the inclusion results, structural formula, certain sharp radii results such as radius of starlikeness and convexity of order α. It also finds a radius for ratios of analytic functions associated with Euler numbers. [ABSTRACT FROM AUTHOR]

We obtain lacunary recurrent relations with gaps of length four for the Bernoulli and Euler polynomials and, as a consequence, known and new lacunary relations for the Bernoulli and Euler numbers. [ABSTRACT FROM AUTHOR]

The physical or morphological integrity of the coffee bean during post-harvest processing directly influences the economic value and sensory quality of the coffee beverage. Breakdowns in the outer layers of the beans are characteristics observed for the morphological and economic classification of coffee beans during the commercialization of this product. However, physical changes in the inner layers of the beans that are not seen with the naked eye can also influence the sensory quality of the coffee. Therefore, the objective of this study was to relate changes in the physical structure of coffee beans roasted by four different processes (light, medium, dark, and baked) with the sensory attributes of the beverage. The analyses of the physical characteristics of the coffee beans were carried out by X-ray microtomography and the sensory profile was determined using the Specialty Coffee Association of America protocol. The roasting profile with the highest sensory scores showed higher values for total pore space volume and a negative Euler number. However, the roasting profiles that fluctuated between the highest and lowest of scores of the sensory attributes did not present standardized behavior for the connectivity, Euler number, and total pore space volume. Hence, morphological or physical changes in the coffee beans caused by the different types of roasting correlate with changes in the sensorial profile. Furthermore, the sensory discrimination of these coffee beans among the different roast profiles may be observed by the joint analysis of the flavor and fragrance scores. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove some congruences involving Apéry-like numbers S n = ∑ k = 0 n ( n k ) ( 2 k k ) ( 2 n − 2 k n − k ) : Let p>3 be a prime. Then S p ≡ S 1 + (− 1) (p − 1) / 2 8 p 2 E p − 3 (mod p 3) , S 2 p ≡ S 2 + (− 1) (p − 1) / 2 128 p 2 E p − 3 (mod p 3) , S 2 p − 1 ≡ (− 1) (p − 1) / 2 32 2 (p − 1) S 1 + 20 p 2 E p − 3 (mod p 3) , S 3 p ≡ S 3 + (− 1) (p − 1) / 2 1440 p 2 E p − 3 (mod p 3). where E n stands for the nth Euler number. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove some congruences involving Apéry-like numbers S n = ∑ k = 0 n ( n k ) ( 2 k k ) ( 2 n − 2 k n − k ) : Let p>3 be a prime. Then S p ≡ S 1 + (− 1) (p − 1) / 2 8 p 2 E p − 3 (mod p 3) , S 2 p ≡ S 2 + (− 1) (p − 1) / 2 128 p 2 E p − 3 (mod p 3) , S 2 p − 1 ≡ (− 1) (p − 1) / 2 32 2 (p − 1) S 1 + 20 p 2 E p − 3 (mod p 3) , S 3 p ≡ S 3 + (− 1) (p − 1) / 2 1440 p 2 E p − 3 (mod p 3). where E n stands for the nth Euler number. [ABSTRACT FROM AUTHOR]

The nonorientable 4-genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and Ozsváth, Stipsicz, and Szabó. Given a nonorientable surface F ⊂ B4 with αF = K ⊂ S³ a knot, an analysis of the existing methods for bounding and computing the nonorientable 4-genus reveals relationships between the first Betti number β1 of F and the normal Euler class e of F. This relationship yields a geography problem: given a knot K, what is the set of realizable pairs (e(F), β1(F)) where F ⊂ B4 is a nonorientable surface bounded by Kl We explore this problem for families of torus knots. In addition, we use the Ozsváth-Szabó d-invariant of two-fold branched covers to give finer information on the geography problem. We present an infinite family of knots where this information provides an improvement upon the bound given by Ozsváth, Stipsicz, and Szabó using the Upsilon invariant. [ABSTRACT FROM AUTHOR]

The main objective in this paper is to study on special numbers and polynomials that contain finite sums of powers of binomial coefficients. By using generating function methods, some formulas and relations related to these numbers and the Apostol-Bernoulli and Apostol-Euler numbers of negative higher order, the Bernoulli and Euler numbers, the Stirling type numbers, the combinatorial numbers, the Bell polynomials, the Fubini type polynomials, and the Legendre polynomials are presented. Moreover, asymptotic expressions of the finite sums of powers of binomial coefficients for these numbers are given. Some numeric values of these asymptotic expressions are illustrated by the tables. Finally, some inequalities for these numbers are given. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, β(2k+1), where k is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses basic calculus and telescoping series. By a similar procedure, we also yield an integral representation of β(2k). [ABSTRACT FROM AUTHOR]

Simulations describing the spin physics underpinning nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) spectroscopy play an important role in the design of new experiments. When experiments are performed in the solid state, samples are commonly composed of powders or glasses, with molecules oriented at a large number of angles with respect to the laboratory frame. These powder angles must be represented in simulations to account for anisotropic interactions. Numerical techniques are typically used to accurately compute such powder averages. A large number of Euler angles are usually required, leading to lengthy simulation times. This is particularly true in broad spectra, such as those observed in EPR. The combination of the traditionally separate techniques of EPR and magic angle spinning (MAS) NMR could play an important role in future electron detected experiments, combined with dynamic nuclear polarization, which will allow for exceptional detection sensitivity of NMR spin coherences. Here, we present a method of reducing the required number of Euler angles in magnetic resonance simulations by analytically performing the powder average over one of the Euler angles in the static and MAS cases for the TEMPO nitroxide radical in a 7 T field. In the static case, this leads to a 97.5% reduction in simulation time over the fully numerical case and reproduces the expected spinning sideband manifold when simulated with a MAS frequency of 150 kHz. This technique is applicable to more traditional NMR experiments as well, such as those involving quadrupolar nuclei or multiple dimensions. [ABSTRACT FROM AUTHOR]

Optical fiber image transmission bundle is a passive device that using a certain number of regularly arranged optical fibers to transmit images. In order to ensure its imaging quality, the detection system for the number of optical fibers is optimized based on the machine vision technology. In view of the incomplete image acquisition of the end face of the optical fiber image transmission bundle or the defect of the edge, this paper proposes an edge optimization processing algorithm based on Euler number calculation, which can identify, mark and classify all the edge contours in the image. It not only realizes the accurate detection of the number of optical fibers, but also improves the performance of the algorithm through Graphic Processing Unit (GPU) processing. The algorithms in this paper are debugged and run in the environment of visual studio 2013, Opencv 3.1 and CUDA 10.1. The experimental results show that the detection error rate of the total number of optical fibers in the image is about 0.27%, and the error rate of optimized detection is about 0.03%, which basically meets the design requirements. Moreover, compared with the conventional detection methods, the optimization algorithm proposed in this paper can greatly reduce the selection requirements of threshold and have higher detection efficiency. With the support of Compute Unified Device Architecture (CUDA), the speed of bilateral filtering is increased by about 20 times, and the optimization algorithm and overall operation time are reduced by nearly 1/2.

Optical fiber image transmission bundle is a passive device that using a certain number of regularly arranged optical fibers to transmit images. In order to ensure its imaging quality, the detection system for the number of optical fibers is optimized based on the machine vision technology. In view of the incomplete image acquisition of the end face of the optical fiber image transmission bundle or the defect of the edge, this paper proposes an edge optimization processing algorithm based on Euler number calculation, which can identify, mark and classify all the edge contours in the image. It not only realizes the accurate detection of the number of optical fibers, but also improves the performance of the algorithm through Graphic Processing Unit (GPU) processing. The algorithms in this paper are debugged and run in the environment of visual studio 2013, Opencv 3.1 and CUDA 10.1. The experimental results show that the detection error rate of the total number of optical fibers in the image is about 0.27%, and the error rate of optimized detection is about 0.03%, which basically meets the design requirements. Moreover, compared with the conventional detection methods, the optimization algorithm proposed in this paper can greatly reduce the selection requirements of threshold and have higher detection efficiency. With the support of Compute Unified Device Architecture (CUDA), the speed of bilateral filtering is increased by about 20 times, and the optimization algorithm and overall operation time are reduced by nearly 1/2.

The generalized Prandtl boundary value problem is considered, which simulates a quasistatic technological compression process in one direction and rapid spreading in another for a thin incompressible perfectly rigid plastic layer (a plane strain state) corresponding to the Mises–Hencky plasticity criterion with a yield stress variable in thickness. Stratification can be either continuous or piecewise constant. In the latter case, the problem simulates the pressing of layered plastic composites (laminates, "sandwiches") and precisely bringing them to the desired thickness. On the basis of thin-layer singular asymptotics for a small geometric parameter, an approximate solution for kinematic and force quantities is found using the asymptotic integration method developed in this study. The applicability of the quasistatic approach to various time ranges of the compression process is discussed. [ABSTRACT FROM AUTHOR]

Quantitative data about the internal lung structure are needed to better understand normal and pathological lung development. Aberrant lung development causes deficits in alveolar and microvascular development; however, the normal temporal relationship between these processes is still not fully understood. We hypothesized that alveolar and capillary development show a differential time pattern. Lungs of rats aged 3, 7, 14, 21 days (d) or 3 mo (n = 8-10 each) were fixed by vascular perfusion and processed for light microscopy. Using design-based stereology number, the surface area and volume of alveoli, septal capillaries, and alveolar septa were quantified. The total number and the total volume of alveoli increased progressively during postnatal development. Interestingly, the numerical density of capillary loops was significantly higher in 14- and 21-d-old rats than before or after this age, causing a duplication of the total number of capillary loops between 1 and 2 wk of age. The mean thickness of alveolar septa started to decline slightly at the age of 14d and more pronounced at later stages. Although the septal epithelial surface area increased in proportion to alveolar number during the first 3 wk, the capillary endothelial surface area grew only slightly compared with the number of capillaries. In conclusion, the number of elements composing the alveolar capillary network expands massively during the first two postnatal weeks and exceeds the formation of alveoli. The thinning of the alveolar septa during further development suggests a reduction of the capillary network during alveolarization. [ABSTRACT FROM AUTHOR]

In this article, new families of the generalized Legendre-Laguerre-Appell polynomials are introduced using a combination of operational definitions and integral representations. The integral transformations and the appropriate operational rules are used to obtain the explicit summation equations, determinant definitions, and recurrence relations for the generalised Legendre-Laguerre-Appell polynomials. For the generalized Legendre-Laguerre-Bernoulli, Legendre-Laguerre-Euler, and Legendre-Laguerre-Genocchi polynomials, an equivalent investigation of these findings is offered. Additionally, a number of identities for these polynomials are derived by using suitable operational definitions. [ABSTRACT FROM AUTHOR]

Cyclone separators have been widely used for gas–particle separation in chemical engineering. However, their enhancement in separation performances usually increases the pressure drop, which inevitably leads to an increase in operating energy consumption. One of the challenging issues is how to reduce the cyclone pressure drop while improving separation performances. To gain insight into the pathways and impacts of cyclone pressure drop reduction, this work reviews the state-of-the-art technical principles, performances and effects, focusing on the processes, mechanisms, and characteristics of pressure drop reduction by inlet/outlet variations, additional auxiliary devices, local cyclone dimension improvement, and global optimization based on intelligent algorithms. The cyclone performances are assessed using a proposed index that combines the Euler number and the square-root particle cut-off Stokes number. It is suggested that the pressure drop and separation capability usually have a dynamic compromise. Considering the comprehensive performances, the technology using helical roof inlet, cross cone, increasing cylindrical height with h/D = 4.3 (H/D = 6.35), and globally optimized design by Sun et al. (2017) are respectively considered to the others. Particularly, the last one is recommended to be more representative in practice. Finally, the key issues to be considered in further research were also prospected. [ABSTRACT FROM AUTHOR]

Conservation agriculture is increasingly preferred to conventional methods due to its benefits in promoting more sustainable soil management. Our study aims to compare physical and morphological properties, at the microscale, of soils under long-term no tillage (NT) and minimum-tillage (MT) to adjacent 'natural' soils under long-term secondary forest (SF). Soil aggregates of c. 2 cm length were imaged by X-ray Computed Tomography (XCT). The three-dimensional (3D) images were segmented and analyzed in order to assess properties such as porosity, number of pores, degree of anisotropy, pore shape, volume classifications, Euler number for pore connectivity, and pore tortuosity. The pore architecture of soils under NT and MT, for c. 40 years, was similar to that from the SF in terms of imaged porosity, pore size, and shape distributions, as hypothesized in our study. However, we observed some important differences; for instance, SF had larger, more connected, and more complex pores, likely due to the greater biological activity. In addition, SF had more isotropic pores than NT and MT, i.e., without preferential flow paths for water redistribution. Therefore, we concluded that long-term conservation agriculture was efficient at reversing structural damage typically associated with conventional, intensive agriculture, but some large differences remain, particularly concerning the pore network complexity and connectivity. [ABSTRACT FROM AUTHOR]

Hydroxyl radical (·OH) is a key component that leads to the cleavage of the glycosidic bond in the process of chitosan (CS) degradation by hydrodynamic cavitation (HC). In this paper, methylene blue (MB) was selected as the trapping agent of ·OH and the yield of ·OH in an impact-jet hydraulic cavitator was investigated. The results showed that the cavitation intensity and the number of passes (N) were the two main factors affecting the yield of ·OH. A smaller cavitation number (Cv) or a larger N indicated that more ·OH can be produced. Based on the dimensionless number correlation method, the yield of ·OH was correlated with Cv, N, Euler number (Eu), Reynolds number (Re), and a dimensionless parameter (γ), and a prediction model of ·OH yield was established. The relative deviations between the experimental and calculated values of the ·OH yield were basically within 10% by the prediction model. On the basis of the prediction model, the yield of ·OH produced in the process of CS degradation by HC was obtained. The results showed that the predicted yield of ·OH was significantly correlated with the intrinsic viscosity reduction rate of CS. It was suggested that the prediction model of ·OH yield based on the MB solution can be used to calculate the ·OH yield during the degradation of low concentration CS by HC. [ABSTRACT FROM AUTHOR]

We establish congruences for higher order Euler polynomials modulo 720. We apply this result for constructing analogues of Stern congruences for Euler secant numbers E4n ≡ 5(mod 60); E4n+2 ≡-1(mod 60) to Euler tangent numbers and Genocchi numbers. We prove that Euler tangent numbers satisfy the following congruences E4n+1 ≡ 16(mod 720), and E4n+3 ≡-272(mod 720). We establish 12-periodic property of Genocchi numbers modulo 45. [ABSTRACT FROM AUTHOR]

The temporal linear instability of a viscoelastic liquid sheet moving around an inviscid gas in a transverse electrical field is analyzed. The fluid is described by the leaky dielectric model, which is more complex than existing models and enables a characterization of the liquid electrical properties. In addition, the liquid is assumed to be viscoelastic, and the dimensionless dispersion relation of the sinuous and varicose modes between the wavenumber and the temporal growth rate can be derived as a 3 × 3 matrix. According to this relationship, the effects of the liquid properties on the sheet instability are determined. The results suggest that, as the electrical Euler number and the elasticity number increase and the time constant ratio decreases, the sheet becomes more unstable. Finally, an energy budget approach is adopted to investigate the instability mechanism for the sinuous mode. [ABSTRACT FROM AUTHOR]

Let $ \{S_n\} $ be the Apéry-like sequence given by $ S_n = \sum_{k = 0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k} $. We show that for any odd prime $ p $, $ \sum_{n = 1}^{p-1}\frac {nS_n}{8^n}{\equiv} (1-(-1)^{\frac{p-1}2})p^2\ (\text{ mod}\ {p^3}) $. Let $ \{Q_n\} $ be the Apéry-like sequence given by $ Q_n = \sum_{k = 0}^n\binom nk(-8)^{n-k}\sum_{r = 0}^k\binom kr^3 $. We establish many congruences concerning $ Q_n $. For an odd prime $ p $, we also deduce congruences for $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k}\ (\text{ mod}\ {p^3}) $, $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(k+1)^2}\ (\text{ mod}\ {p^2}) $ and $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(2k-1)}\ (\text{ mod}\ p) $, and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.

As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin–Ma–Ma–Zhou (2021). Various intriguing connections and bijections for weakly increasing trees have already been found and the purpose of this paper is to present yet more bijective combinatorics on this unified object. Two of our main contributions are • extension of an equidistribution result on plane trees due to Eu–Seo–Shin (2017), regarding levels and degrees of nodes, to weakly increasing trees; • a new interpretation of the multiset Schett polynomials in terms of odd left/right chains on weakly increasing binary trees. Interesting consequences are discussed, including new tree interpretations for the Jacobi elliptic functions and Euler numbers. Relevant enumerative results are also presented, involving recurrence relations, exponential generating functions and context-free grammars. [ABSTRACT FROM AUTHOR]

To realize the transition of our society to a low-carbon future with innovative subsurface energy solutions, understanding the dynamic behavior of gas invading multi-fluid systems in underground pore space is critical. In this work, a joint approach of flow imaging and digital image processing is employed to investigate the fingering dynamics of gas invading multi-fluids in porous media. We examined various gas (G) invasion scenarios of a high-viscosity defending liquid (HL), low-viscosity defending liquid (LL), and their co-existing multi-fluid system, focusing on the viscosity effect. Quantification of phase saturation shows that the displacement efficiency follows the order of G → (L → L) > L → L > G → L, regardless of the varieties in injection flow rate in the viscous-dominated flow regime. In other words, the enhancement in displacement efficiency and potential energy savings are achieved by solely introducing a third phase without the cost of the higher pumping power. When gas invades the HL and LL multi-liquid system, the fingering pattern in G → (HL → LL) and G → (LL → HL) significantly differs and highly depends on the sequential occupation of HL and LL in the pore spaces. The previously unobserved yarn-liked gas pattern in G → (LL → HL) is suspected as the main reason for the fast gas displacement. Through Local dynamics analysis, we identified that the preferential invasion into interconnected LL channels and the inhibitory effect of scattered HL on bypass invasion are the primary mechanisms behind the formation of yarn-liked fingers. We classified two distinct categories of ganglia mobilization and connection in G → (LL → HL), i.e. "catch up to connect" and "expand to connect". Finally, the topological connectivity of the gas finger in G → (LL → HL) is evaluated using Euler number. Euler number shows an ascending trajectory before breakthrough, followed by a rapid descent and stabilization at steady state. This signifies that disconnected ganglia emerge before breakthrough and subsequently expand and reconnect. Our new findings are of great importance for subsurface extraction/storage strategy innovation through enriching multi-fluids injection scenarios. • Investigate various viscous-dependent gas invasion scenarios of a high-viscosity and low-viscosity liquid as well as their co-existing multi-fluid system. • Demonstrate the highest displacement efficiency in three-phase displacement followed by the liquid–liquid and gas–liquid two-phase displacements, regardless of the varieties in injection flow rate. • Discover a novel yarn-liked gas pattern in three phase displacement and reveal the formation mechanism by local invading analysis. • Identify two types of ganglia mobilization and analyze time evolution of the topological ganglia connectivity. [ABSTRACT FROM AUTHOR]

We consider the L-fractional derivative, which has been proposed in the literature to study fractional differentials in geometry and processes in mechanics. Our context is population growth and epidemiology, for which the use of L-derivatives is motivated by transitions. Using power series, we solve the logistic differential equation model under this fractional derivative. Several conclusions on the method, the derivative, and the singularity of the associated kernel are reached. Fractional Euler numbers, related to the logistic map and the famous Riemann zeta function, are also introduced. [ABSTRACT FROM AUTHOR]