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We propose new two-sided estimates with sharp constants for the number e. Minorants and majorants in the estimates are expressed as continued fractions. To justify the estimate, we essentially use the integral representation of a special analytic function obtained earlier by the authors. We discuss the validity of similar inequalities and relationship with known results. [ABSTRACT FROM AUTHOR]

In this study, we concentrate on the synchronizability of multilayer totally homogeneous networks. The eigenvalue spectrum of four typical multilayer networks, including multilayer nearest-neighbor regular networks, multilayer small-world networks, multilayer random networks and multilayer totally homogeneous networks, has been rigorously derived and the synchronizability of four typical multilayer networks has been compared due to the master stability function. Furthermore, numerical simulations are performed to validate the synchronizability of these multilayer networks. The results indicate that multilayer totally homogeneous networks exhibit optimal synchronizability. The Euler characteristic numbers and Betti numbers of these networks are derived, and their relationships with the synchronizability of multilayer networks are analyzed. We can observe that a negative correlation is exhibited between the Euler characteristic number and the synchronizability of multilayer networks, while a positive correlation is exhibited between the Betti number and the synchronizability of multilayer networks. [ABSTRACT FROM AUTHOR]

The Petrie conjecture asserts that if a homotopy \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbb {CP}^n. Petrie proved this conjecture in the case where the manifold admits a T^n-action. An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective T^n-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbb {CP}^n, the graph of M agrees with the graph of a linear T^n-action on \mathbb {CP}^n. Consequently, M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch \chi _y-genus, Todd genus, and signature as \mathbb {CP}^n, endowed with the standard linear action. Furthermore, if M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbb {CP}^n also agree. [ABSTRACT FROM AUTHOR]

Brugia malayi are thread-like parasitic worms and one of the etiological agents of Lymphatic filariasis (LF). Existing anthelmintic drugs to treat LF are effective in reducing the larval microfilaria (mf) counts in human bloodstream but are less effective on adult parasites. To test potential drug candidates, we report a multi-parameter phenotypic assay based on tracking the motility of adult B. malayi and mf in vitro. For adult B. malayi, motility is characterized by the centroid velocity, path curvature, angular velocity, eccentricity, extent, and Euler Number. These parameters are evaluated in experiments with three anthelmintic drugs. For B. malayi mf, motility is extracted from the evolving body skeleton to yield positional data and bending angles at 74 key point. We achieved high-fidelity tracking of complex worm postures (self-occlusions, omega turns, body bending, and reversals) while providing a visual representation of pose estimates and behavioral attributes in both space and time scales. [ABSTRACT FROM AUTHOR]

This paper presents a preliminary study of a novel type of turbulizers based on the jet pump working principle and designed to intensify heat transfer in double-tube heat exchangers. To determine the potential for their use, Computational Fluid Dynamics (CFD) and realizable k-ε turbulence model were applied to four turbulizers differing in the angles and diameters of the nozzle and diffuser. During studies, a number of parameters were determined, including: (i) Euler number Eu, (ii) overall heat transfer coefficient kT and (iii) reduced ratio of heat duty to mechanical power 𝜀r; and then compared with data available for strip turbulizers. The tests yielded the following maximum values of: Eu - 1.5 times higher, kT - 2 times higher, and 𝜀𝑟 - 27 times higher than those obtained by classical strip turbulizers, which indicates the high potential of presented jet pump turbulizers and justifies their further development. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the truncations of three classical theta series of Euler and Gauss, and analyze their combinatorial properties which play a key role in proving these truncated identities. Several interesting partition identities are established bijectively. [ABSTRACT FROM AUTHOR]

In this study, we introduced several types of higher-order difference equations involving q-SINE Euler (QSE) and q-COSINE Euler (QCE) polynomials. Depending on the parameters selected, these higher-order difference equations exhibited properties of trigonometric functions or related Euler numbers. Approximate root construction focused on the QSE polynomial, which was the solution of the q-difference equations obtained earlier. We also showed the structure of the approximate roots of higher-order polynomials among the QSE polynomials, understood them, and considered the associated conjectures. [ABSTRACT FROM AUTHOR]

In the paper, the authors intrinsically observe that Hoffman’s combinatorial identity discovered in 1992 and Genčev’s combinatorial identities extended in 2024 can be reformulated in terms of complete Bell polynomials. They provide alternative proofs and offer some elementary generalizations of these combinatorial identities. [ABSTRACT FROM AUTHOR]

The pore geometry and topology properties of rocks are significant for a better understanding of the complex hydrologic and elastic properties. If the complexity of porous media is considered, reliable results can achieve. In three-dimensional imaging of cavities, these complexities are taken into account. In this study, by conducting different methods deep understanding from some sandstones pore space is undertaken. Three series of 2D microcomputed tomography sandstones binary images have been considered, each of them imagined as a 3D binary image. Using novel skeletonization and pore-throat partitioning algorithms some network properties have been evaluated and compared for three cases. Those properties are pore and grain size distribution, throat length frequency, and coordination number frequency. Also, geometric measures in 2D and 3D have been considered using Minkowski functionals. The area, the perimeter and the 2D Euler number of 2D binary images and the volume, the surface area, the mean breadth also known as integral of the mean curvature, and the 3D Euler Number of 3D binary images are considered. [ABSTRACT FROM AUTHOR]

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C²-smoothable. [ABSTRACT FROM AUTHOR]

Three-dimensional topological relationships serve as a theoretical foundation for quality control, update processing, and spatial analysis of three-dimensional spatial data in real-world three-dimensional GIS. The existing 3D topological relationship models are all basic relationship models that cannot distinguish the refined topological relationship between the line and the body with multiple intersections. In this study, we develop a 3D refined topological relationship description framework that draws from the two-dimensional refined topological relationship model, defines the unit intersection between the line and the body based on manifold topology, and proposes a method for describing the unit intersections between the line and the body considering Euler numbers and adjacency types. In total, 23 basic types between the line and the body are deduced. An example is provided to illustrate the distinguished refined topological relationship between the line and the body with multiple intersections. Subsequently, an algorithm for determining the basic type of line/body is developed. Finally, a line/body refined topological relationship computation prototype system is developed using the Nef polyhedron model, C++ language, and an open-source geometric algorithm library, and the effectiveness of our method is verified using actual building and pedestrian data. [ABSTRACT FROM AUTHOR]

The engineering of tubes with surface corrugations is recognized as an effective method for enhancing heat transfer within the tube. Yet the impact of surface corrugation on the flow and heat transfer around the tube's exterior remains inadequately explored. This study investigates the crossflow and heat transfer characteristics in banks of periodically inward-corrugated tubes using computational fluid dynamics. Numerical simulations were performed for both in-line and staggered tube arrangements, covering Reynolds numbers from 1000 to 10,000. The aim was to examine how various corrugation parameters affect heat transfer and flow dynamics in tube banks configured in both in-line and staggered layouts. The results show that the heat transfer and the pressure drop in crossflow across tube banks are substantially influenced by changes in corrugation parameters. Specifically, in the in-line arrangement, both the Nusselt number and Euler number decrease significantly as the corrugation height increases. In contrast, in the staggered arrangement, the Nusselt number and Euler number exhibit less variation in response to surface corrugation. A comparative analysis of performance criteria suggests that a staggered arrangement is more advantageous for improving thermal–hydraulic efficiency in crossflow through corrugated tube banks. [ABSTRACT FROM AUTHOR]

The phase behavior of gas condensate in reservoir formations differs from that in pressureevolume etemperature (PVT) cells because it is influenced by porous media in the reservoir formations. Sandstone was used as a sample to investigate the influence of porous media on the phase behavior of the gas condensate. The pore structure was first analyzed using computed tomography (CT) scanning, digital core technology, and a pore network model. The sandstone core sample was then saturated with gas condensate for the pressure depletion experiment. After each pressure-depletion state was stable, realtime CT scanning was performed on the sample. The scanning results of the sample were reconstructed into three-dimensional grayscale images, and the gas condensate and condensate liquid were segmented based on gray value discrepancy to dynamically characterize the phase behavior of the gas condensate in porous media. Pore network models of the condensate liquid ganglia under different pressures were built to calculate the characteristic parameters, including the average radius, coordination number, and tortuosity, and to analyze the changing mechanism caused by the phase behavior change of the gas condensate. Four types of condensate liquid (clustered, branched, membranous, and droplet ganglia) were then classified by shape factor and Euler number to investigate their morphological changes dynamically and elaborately. The results show that the dew point pressure of the gas condensate in porous media is 12.7 MPa, which is 0.7 MPa higher than 12.0 MPa in PVT cells. The average radius, volume, and coordination number of the condensate liquid ganglia increased when the system pressure was between the dew point pressure (12.7 MPa) and the pressure for the maximum liquid dropout, Pmax (10.0 MPa), and decreased when it was below Pmax. The volume proportion of clustered ganglia was the highest, followed by branched, membranous, and droplet ganglia. This study provides crucial experimental evidence for the phase behavior changing process of gas condensate in porous media during the depletion production of gas condensate reservoirs. [ABSTRACT FROM AUTHOR]

The diameter of a graph is the maximum distance between pairs of vertices in the graph. A pair of vertices whose distance is equal to its diameter is called diametrically opposite vertices. The collection of shortest paths between diametrically opposite vertices is referred as diametral paths. In this work, we enumerate the number of diametral paths for Fibonacci cubes, Lucas cubes and alternate Lucas cubes. We present bijective proofs that show that these numbers are related to alternating permutations and are enumerated by Euler numbers. [ABSTRACT FROM AUTHOR]

In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf–Zeilberger method. One of them is, for any prime p > 3, 4 F 3 [ 7 6 1 2 1 2 1 2 1 6 1 1 | - 1 8 ] p - 1 2 = p ( - 2 p ) + p 3 4 ( 2 p ) E p - 3 (mod p 4) , where (· p) stands for the Legendre symbol, and En is the n-th Euler number. [ABSTRACT FROM AUTHOR]

The fine dead fuel load on forest floors is the most critical classification feature in fuel description systems, affecting several parameters in the manifestation of wildland fires. An accurate determination of this fine dead fuel load contributes substantially to effective wildland fire prevention, monitoring, and suppression. This study investigated the viability of incorporating image Euler numbers into predictive models of fine dead fuel load via the taking photos method. Pinus massoniana needles and Quercus fabri broad leaves—typical fuel types in karst areas—served as the research subjects. Accurate field data were collected in the Tianhe Mountain forests, China, while artificial fine dead fuelbeds of differing loads were constructed in the laboratory. Images of the artificial fuelbeds were captured and uniformly digitized according to various conversion thresholds. Thereafter, the Euler numbers were extracted, their relationship with fuel load was analyzed, and this relationship was applied to generate three load-prediction models based on stepwise regression, nonlinear fitting, and random forest algorithms. The Euler number had a significant relationship with both P. massoniana and Q. fabri fuel loads. At low conversion thresholds, the Euler number was negatively correlated with fuel load, whereas a positive correlation was recorded when this threshold exceeded a certain value. The random forest model showed the best prediction performance, with mean relative errors of 9.35% and 14.54% for P. massoniana and Q. fabri, respectively. The nonlinear fitting model displayed the next best performance, while the stepwise regression model exhibited the largest error, which was significantly different from that of the random forest model. This study is the first to propose the use of image features to predict the fine fuel load on a surface. The results are more objective, accurate, and time-saving than current fuel load estimates, benefiting fuel load research and the scientific management of wildland fires. [ABSTRACT FROM AUTHOR]

In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf–Zeilberger method. One of them is, for any prime p > 3, 4 F 3 [ 7 6 1 2 1 2 1 2 1 6 1 1 | - 1 8 ] p - 1 2 = p ( - 2 p ) + p 3 4 ( 2 p ) E p - 3 (mod p 4) , where (· p) stands for the Legendre symbol, and En is the n-th Euler number. [ABSTRACT FROM AUTHOR]

The evolution of the liquid–vapour interface plays a crucial role in multiphase flow, heat and mass transfer, and fluid phase change in porous media. A thorough investigation of the interface under varying degrees of saturation is necessary and crucial to fully understanding the key mechanism of soil water evaporation. The pore voids and fluids are characterized using X-ray microtomography and image processing. Salt solutions usually replace pure water for better contrast and image development. Machine learning algorithms were employed to identify and extract the different phase and their interface accurately. Then, variations in the geometrical and topological features of the interface at varying saturation during evaporation were analysed to quantitatively describe the connectivity of the liquid phase and the morphological change in the liquid–vapour interface. Topological analysis reveals that normalized Euler characteristic numbers quantify the complementary connectivity of liquid and vapour phase. The curvatures of the liquid–vapour interface of the samples under various saturations classify the liquid–air interface curvature of samples under various saturations for quantitatively describing the migration progress and quantity distribution of typical interface along with drying. [ABSTRACT FROM AUTHOR]

In this study, we use discrete Appell convolution to define the sequence of Laguerre type twice-iterated Appell polynomials. We obtain explicit representation, recurrence relation, determinantal representation, lowering operator, integro-partial raising operator and integro-partial differential equation. In addition, the special cases of this new family are investigated using Euler and Bernoulli numbers. We also state their corresponding characteristic properties. [ABSTRACT FROM AUTHOR]

In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2) a1,b1 (p1, k), where a1, a2, b1, b2 are given complex numbers, a1, a2 ?= 0, and p1, p2 are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1,p2 (x1, x2). By using results for S(1,x2,p2) 1,x1 (p1, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case. [ABSTRACT FROM AUTHOR]

This article, published in Mathematical Notes, discusses lacunary recurrence relations with gaps of length 8 for the Bernoulli and Euler polynomials. The authors use standard notations for these polynomials and numbers and provide formulas and expansions for them. They also mention the history of lacunary recurrence relations for these numbers and polynomials and highlight their own contributions in finding such relations. The authors describe the method of proof for their main result, Theorem 1, and provide corollaries that extend the relations to the Bernoulli and Euler numbers. The article concludes by acknowledging funding support and declaring no conflicts of interest. [Extracted from the article]

Motivated by their role in M‐theory, F‐theory, and S‐theory compactifications, all possible complete intersections Calabi‐Yau five‐folds in a product of four or less complex projective spaces are constructed, with up to four constraints. A total of 27 068 spaces are obtained, which are not related by permutations of rows and columns of the configuration matrix, and determine the Euler number for all of them. Excluding the 3909 product manifolds among those, the cohomological data for 12 433 cases are calculated, i.e., 53.7% of the non‐product spaces, obtaining 2375 different Hodge diamonds. The dataset containing all the above information is available here. The distributions of the invariants are presented, and a comparison with the lower‐dimensional analogues is discussed. Supervised machine learning is performed on the cohomological data, via classifier, and regressor (both fully connected and convolutional) neural networks. h1, 1 can be learnt very efficiently, with very high R2 score and an accuracy of 96% is found, i.e., 96% of the predictions exactly match the correct values. For h1,4,h2,3,η$h^{1,4},h^{2,3}, \eta$, very high R2 scores are also found, but the accuracy is lower, due to the large ranges of possible values. [ABSTRACT FROM AUTHOR]

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0). [ABSTRACT FROM AUTHOR]

In this paper, we express a generalization of the Ramanujan integral I ⁢ ( α ) {I(\alpha)} with the analytical solutions, using the Laplace transform technique and some algebraic relation or the Pochhammer symbol. Moreover, we evaluate some consequences of a generalized definite integral ϕ * ⁢ ( υ , β , a ) {\phi^{*}(\upsilon,\beta,a)} . The well-known special cases appeared, whose solutions are possible by Cauchy’s residue theorem, and many known applications of the integral I ⁢ ( a , β , υ ) {I(a,\beta,\upsilon)} are discussed by the Leibniz rule for differentiation under the sign of integration. Further, one closed-form evaluation of the infinite series of the F 0 1 ⁢ ( ⋅ ) {{}_{1}F_{0}(\,\cdot\,)} function is deduced. In addition, we establish some integral expressions in terms of the Euler numbers, which are not available in the tables of the book of Gradshteyn and Ryzhik. [ABSTRACT FROM AUTHOR]

We investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting. [ABSTRACT FROM AUTHOR]

We study polytopes defined by inequalities of the form ∑ i ∈ I z i ≤ 1 for I ⊆ [ d ] and nonnegative z i where the inequalities can be reordered into a matrix inequality involving a column-convex { 0 , 1 } -matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Vergès, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs G with a Hamiltonian path, which we call spinal graphs. We show that the volumes of these flow polytopes are given by the number of upper (or lower) G-cyclic orders defined by the graphs G. As a special case we recover results on volumes of consecutive coordinate polytopes. We study the combinatorics of k-Euler numbers, which are generalizations of the classical Euler numbers, and which arise as volumes of flow polytopes of a special family of spinal graphs. We show that their refinements, Ramassamy's k-Entringer numbers, can be realized as values of a Kostant partition function, satisfy a family of generalized boustrophedon recurrences, and are log concave along root directions. Finally, via our main integral equivalence and the known formula for the h ∗ -polynomial of consecutive coordinate polytopes, we give a combinatorial formula for the h ∗ -polynomial of flow polytopes of non-nested spinal graphs. For spinal graphs in general, we present a conjecture on upper and lower bounds for their h ∗ -polynomial. [ABSTRACT FROM AUTHOR]

With the development of gas-condensate reservoirs, the phase behavior of gas condensate undergoes continuous changes. The phase behavior of gas condensate in PVT cylinders is different from that in fractured-vuggy gas-condensate reservoirs because multiscale pore structures can affect the phase behavior. By vuggy, we mean the formation of a reservoir contains cavity structures. To elucidate the influence of fractured-vuggy porous media on phase behavior of gas condensate, a microfluidic chip and a fractured-vuggy carbonate sample were prepared for pressure depletion experiments. The phase behavior of gas condensate in the chip during depressurization was observed via brightfield optical microscopy. The real-time computed tomography scanning was used to analyze pore structure of the sample and to obtain the phase behavior of gas condensate under different pressures. The results showed that the dew point pressure of the gas condensate was increased from 12.0 MPa in PVT cylinders to 14.8 MPa in nanochannels of the chip. The condensate liquid appeared in micropores at 14.4 MPa. The change of the average radius, coordination number, and tortuosity of condensate liquid ganglia were calculated and analyzed using pore network models. The shape factor and the Euler number were used to classify the condensate liquid into four types including clustered, branched, membranous, and droplets ganglia. The condensate liquid was generally distributed as clustered ganglia. This study provided evidence of the effect of fractured-vuggy porous media on phase behavior of gas condensate and clarified that the phase behavior of gas condensate varies in multi-scale pores. [ABSTRACT FROM AUTHOR]

Our main focus here is a new type of degenerate poly-Euler polynomials and numbers. This focus stems from their nascent importance for applications in combinatorics, number theory and in other aspects of applied mathematics. We construct a new type of degenerate poly-Euler polynomials by using the degenerate polylogarithm functions. We also show several combinatorial identities related to this polynomials and numbers. [ABSTRACT FROM AUTHOR]

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable. [ABSTRACT FROM AUTHOR]

In this paper, we construct a crepant resolution for the quotient singularity A4/A4 in characteristic 2, where A4 is the alternating group of degree 4 with permutation action on A4. By computing the Euler number of the crepant resolution, we obtain a new counter-example to an analogous statement of McKay correspondence in positive characteristic. [ABSTRACT FROM AUTHOR]

Previous Smoothed Particle Hydrodynamics (SPH) study on water entry issues has primarily been conducted for the load analysis of impact phase rather than the cavity oscillation effect because the calculation and simulation of this complex physical process are more complicated and time consuming. In order to increase computational efficiency and accuracy, the multiphase δ + -SPH model is combined with Adaptive Particle Refinement technology to investigate the whole process of the wedge's water entry. The hydrodynamic phenomena in the stages before cavity closure for the four cases with different Froude numbers (Fn) are compared and analyzed. After the cavity is pinched off, the wedge exhibits kinematic oscillation. Our test shows that the adoption of sound speed has a significant influence on the oscillation period and peak value of closed cavities in weakly compressible SPH calculations. Then, a suitable sound speed adoption is selected to simulate the oscillatory phenomenon accurately. Comparing the pressure profile with the surface pressure and acceleration of the wedge at the same time, it can be concluded that the oscillation of the hydrodynamic load on the wedge is caused by the pressure oscillation in the closed cavity. Especially for the case of low Fn, the pressure peak on the wedge's surface in the oscillation stage is even greater than the pressure load in the impact stage. The peak pressure of closed cavity is positively correlated with Fn and negatively correlated with Euler number (Eu). Finally, by analyzing the influence of wedge width and impact velocity, it is found that the oscillation period of the closed cavity is related to the morphology of the cavity. The larger the aspect ratio of the closed cavity, the longer the oscillation period. [ABSTRACT FROM AUTHOR]

A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G. A graph G is acyclically k-choosable if for any list assignment L = {L(v): v ∈ V(G)} with ∣L(v)∣ ≥ k for each vertex v ∈ V(G), there exists an acyclic proper vertex coloring ϕ of G such that ϕ(v) ∈ L(v) for each vertex v ∈ V(G). In this paper, we prove that every graph G embedded on the surface with Euler characteristic number ε = −1 is acyclically 11-choosable. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic p-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind. [ABSTRACT FROM AUTHOR]

In a binary image, the Euler number is a crucial topological feature that holds immense significance in image understanding and image analysis owing to its invariance under scaling, rotation, or any arbitrary rubber-sheet transformation of images. This paper focuses on the Euler number computing problem in a binary image. The state-of-the-art Euler number computing algorithms are reviewed, which obtain the Euler number through different techniques, such as definition, features of binary images, and special data structures representing forms of binary images, and we explain the main principles and strategies of the algorithms in detail. Afterwards, we present the experimental results to bring order of the prevailing Euler number computing algorithms in 8-connectivity cases. Then, we discuss both the parallel implementation and the hardware implementation of algorithms for calculating the Euler number and present the algorithm extension for 3D image Euler number computation. Lastly, we aim to outline forthcoming efforts concerning the computation of the Euler number. [ABSTRACT FROM AUTHOR]

Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler number and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This provides a weighted count of the number of plane conics meeting 8 general lines, where the weight of each conic is determined the geometry of its intersections with the 8 given lines. As a corollary, real conics meeting 8 general lines come in two families of equal size. [ABSTRACT FROM AUTHOR]

Bernoulli and Euler numbers and polynomials are well known and find applications in various areas of mathematics, such as number theory, combinatorial mathematics, series expansions, and the theory of special functions. Using fractional exponential functions, we extend the classical Bernoulli and Euler numbers and polynomials to introduce their fractional-index-based types. This reveals a symmetry in relation to the classical numbers and polynomials. We demonstrate some examples of these generalized mathematical entities, which we derive using the computer algebra system Mathematica©. [ABSTRACT FROM AUTHOR]

A method is proposed for solving the problem of a pulsating quasistationary flow in a channel, based on the use of calculation results for stationary flow. This approach is applicable at low relative oscillation frequencies (for Womersley numbers less than one). The solution to the system of stationary equations of motion and continuity in the initial section of a flat channel was carried out by the finite difference method using an iterative implicit unconditionally stable scheme. The hydrodynamic characteristics of a developing pulsating laminar flow in a flat channel have been studied. The results of calculating the longitudinal velocity component and the Poiseuille and Euler numbers are presented as a function of the relative amplitude of the oscillation of the cross-sectional average velocity A and dimensionless length of the channel. It was found that for A values exceeding unity, the period-averaged coefficients of hydraulic resistance and friction resistance near the inlet to the channel are significantly higher than these values for a stationary flow. It has been shown that in order to achieve a pulsating flow with large amplitudes of oscillations, it is necessary to create a time-average pressure drop approximately three times higher (at A = 5) than for a stationary flow. [ABSTRACT FROM AUTHOR]

Three-dimensional topological relationships serve as a theoretical foundation for quality control, update processing, and spatial analysis of three-dimensional spatial data in real-world three-dimensional GIS. The existing 3D topological relationship models are all basic relationship models that cannot distinguish the refined topological relationship between the line and the body with multiple intersections. In this study, we develop a 3D refined topological relationship description framework that draws from the two-dimensional refined topological relationship model, defines the unit intersection between the line and the body based on manifold topology, and proposes a method for describing the unit intersections between the line and the body considering Euler numbers and adjacency types. In total, 23 basic types between the line and the body are deduced. An example is provided to illustrate the distinguished refined topological relationship between the line and the body with multiple intersections. Subsequently, an algorithm for determining the basic type of line/body is developed. Finally, a line/body refined topological relationship computation prototype system is developed using the Nef polyhedron model, C++ language, and an open-source geometric algorithm library, and the effectiveness of our method is verified using actual building and pedestrian data.

3D‐printed vortex‐based devices have shown excellent performance in realizing hydrodynamic cavitation (HC) and have opened up new ways of effectively producing oil‐in‐water emulsions via cavitation. In this work, the performance of machined and 3D‐printed vortex‐based HC devices for producing liquid‐liquid emulsions was compared in terms of droplet breakage efficiency and energy consumption per kilogram of emulsion. The Euler numbers of 3D‐printed and VMC‐machined HC devices were 42 and 48, respectively. The differences in droplet size distribution and other characteristic diameters were insignificant after 50 passes. The present work confirms the potential of harnessing 3D printing technology for fabricating HC devices and can facilitate development of new device designs to improve overall emulsification performance. [ABSTRACT FROM AUTHOR]

The heat recovery steam generator (HRSG) was utilized to recover the waste heat resources of the catalyst's regenerated gas with the objective to reduce the energy consumption of the hydrocracking process. In this study, the flow resistance and heat transfer performance of the superheater tube bundles in the hydrocracking HRSG were investigated via numerical simulation. The performance evaluation criterion (PEC1) was applied to characterize the comprehensive heat transfer performance of superheater tube bundles. The results showed that as the transverse tube pitch increased, the Nusselt number (Nu) showed a monotonically increasing trend, the Euler number (Eu) showed a monotonically decreasing trend, and PEC1 showed a monotonically increasing trend. As the longitudinal tube pitch increased, Nu exhibited a monotonically increasing trend, Eu showed a monotonically decreasing trend, and PEC1 showed a monotonically increasing trend. In the scope of the simulated results, as the transverse and longitudinal tube pitches were 110 mm and 95 mm, respectively, PEC1 reached the maximum value. Compared with the primary structural parameters, PEC1 increased by 2.32% and 8.50%, respectively. Finally, a new correlation was proposed to predict Nu and Eu of the superheater tube bundles in the hydrocracking HRSG. [ABSTRACT FROM AUTHOR]

We use recent duality results of Eisenbud and Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We give example computations of quadratically enriched excess and residual intersections. [ABSTRACT FROM AUTHOR]

We observe that three context-free grammars of Dumont can be brought to a common ground, via the idea of transformations of grammars, proposed by Ma–Ma–Yeh. Then we develop a unified perspective to investigate several combinatorial objects in connection with the bivariate Eulerian polynomials. We call this approach the Dumont ansatz. As applications, we provide grammatical treatments, in the spirit of the symbolic method, of relations on the Springer numbers, the Euler numbers, the three kinds of peak polynomials, an identity of Petersen, and the two kinds of derivative polynomials, introduced by Knuth–Buckholtz and Carlitz–Scoville, and later by Hoffman in a broader context. We obtain a convolution formula on the left peak polynomials, leading to the Gessel formula. In this framework, we come to the combinatorial interpretations of the derivative polynomials due to Josuat-Vergès. [ABSTRACT FROM AUTHOR]

Perforated plates are commonly used for flow control in pressurized systems. In different industrial applications, these devices are also used in series (multistage perforated plates) to manage high-pressure drop or reduce cavitation occurrence in industrial pipelines and enhance efficiency of gas turbines in power plants. In some specific conditions, the installation of perforated plates in series is a simple and cost-effective solution that increases plant efficiency. However, the large number of design parameters complicates the search for the optimal solution. With the aim of improving the knowledge of the most relevant design parameters, in the present investigation, the dissipation characteristics of multistage perforated plates are studied. Specifically, the analysis of the dependence of the pressure loss coefficient Eu on relative spacing and hole alignment for two subsequent identical plates is performed with CFD approach. Eight plates with different characteristics such as porosity, thickness, diameter, position, and number of holes are studied. The considered porosities vary in a range of 0.3-0.5, the ratio between plate thickness and hole diameter ranges between 0.7 and 1.3, and the number of holes between 4 and 26. A critical spacing between the plates, beyond which the pressure losses are independent of the alignment of the holes, has been calculated. The results obtained are relevant for a deeper understanding of the complex phenomenon of multistage perforated plates and can be used for the design and the installation of these devices commonly used in different engineering applications. [ABSTRACT FROM AUTHOR]

The restricted partition function p N (n) counts the number of partitions of n into at most N parts. In the nineteenth century, Sylvester showed that these partition functions can be expressed as a sum of k-periodic quasi-polynomials ( 1 ≤ k ≤ N ) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called q-partial fractions) of the generating function of p N (n) . In this paper, we show that the coefficients of these q-partial fractions can be expressed as a linear combination of the Ramanujan sums. In particular, we show, for the first time, an appearance of the Degenerate Bernoulli numbers, the Degenerate Euler numbers and a special generalization of the Ramanujan sums, which we term as a Gaussian–Ramanujan sum, in the formulae for certain waves. These coefficients not only provide a good approximation of p N (n) but they can also be used for obtaining good bounds. Further, we provide a combinatorial meaning to these sums. Our approach for partial fractions is based on a projection operator on the I-adic completion of the ring of polynomials, where I is an ideal generated by the cyclotomic polynomial. [ABSTRACT FROM AUTHOR]

In order to prevent the gas purified by wet desulfurization from being discharged into the atmosphere with droplets, the efficient demisters are required to separate droplets. This study proposes to regulate the baffle angle to improve the separation efficiency of the demister, and further improve the separation efficiency of small droplets. The influence of baffle angle on the total separation efficiency, grade separation efficiency, and pressure drop of the demister is analyzed. The instantaneous distribution characteristics of droplets and the action mechanism of the baffle are studied. The results indicate that the separation efficiency increases with the decrease of the baffle angle. Within the scope of this study, an optimal separation efficiency is achieved with a 40° baffle angle. Complete separation occurs for droplets larger than 30 µm, while the separation efficiency for droplets smaller than 20 µm is enhanced by 52%−80% in comparison to the wave-plate demister. The separation efficiency of the baffle demister for 10 µm droplets is 2.7–5.8 times that of the wave-plate demisters. The relationship between the baffle angle and the Euler number (Eu) is established. These relationships provide a reference for optimizing demister performance. [Display omitted] • The separation efficiency of 10 µm droplets is increased to 5.8 times. • The separation efficiency increases obviously with the decrease of baffle angle. • The separation efficiency of droplets (>30 µm) can reach 100%. • The mathematical relationship between pressure drop and baffle angle is established. • Vortices can increase the separation efficiency of small droplet sizes. [ABSTRACT FROM AUTHOR]

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]