Rational Coefficients Research Articles

RATIONAL SHIFTS OF THE BASIC RACK’S PROFILE FOR WHEELS OF A CYLINDRICAL SPUR GEAR PAIR TO DECREASE TEETH WEAR

Purpose. It is to create a method for determination of such shift coefficients of the basic rack’s profile for driving and driven wheels of a cylindrical spur gear pair, that during gear pair operation time the wear rate of a surface layer on the most wearing sections of the teeth’ involute surfaces will be minimal. Research methods are based on integrated application of the involute gear pair theory and tribological laws. The golden section method is used to minimize the function. Results. We have obtained a dimensionless quantity, which is a function of shift coefficients of the basic rack’s profile for driving and driven wheels of a cylindrical spur gear pair. Values of this function are proportional to the greatest wear rate in the neighborhood of characteristic points on teeth’ involute profiles of these wheels. The method is created for determination of rational shift coefficients of the basic rack’s profile, which minimize the obtained dimensionless function, and which maximize the service life of the spur gear pair until the greatest worn-layer thickness on the teeth’ active surfaces reaches limiting permissible value. Scientific novelty. It is proved, that increasing of the worn-layer thickness during spur gear pair operation time will be occurring most rapidly in the neighborhood of certain characteristic points on teeth’ involute profiles of the wheels. Calculation model is elaborated for determination of the worn-layer thicknesses in the neighborhood of extreme active points on teeth’ profiles of the cylindrical spur gear pair and in the neighborhood of the lowest and the highest bounding points of single-pair contact. This model takes into account the influence of specific slides and numbers of wheels’ teeth as well as hardness values of teeth’ surface layer on the wear rates at characteristic points of teeth’ profiles. In addition, the model takes into account sharing of total force of the wheels interaction between two teeth pairs in double-pair contact and also a change of force transmitted by single pair of teeth when it comes into or go out of engagement. Practical value. Application of the created method for determination of shift coefficients of the basic rack’s profile is expedient in design of cylindrical spur gear pairs for machines and equipment, which operate in conditions when ingress of abrasive particles in the teeth engagement region is possible. An example of application of this method in design calculation of the cylindrical spur gear pair is given.

DSM Reconstruction from Uncalibrated Multi-View Satellite Stereo Images by RPC Estimation and Integration

In this paper, we propose a 3D Digital Surface Model (DSM) reconstruction method from uncalibrated Multi-view Satellite Stereo (MVSS) images, where Rational Polynomial Coefficient (RPC) sensor parameters are not available. While recent investigations have introduced several techniques to reconstruct high-precision and high-density DSMs from MVSS images, they inherently depend on the use of geo-corrected RPC sensor parameters. However, RPC parameters from satellite sensors are subject to being erroneous due to inaccurate sensor data. In addition, due to the increasing data availability from the internet, uncalibrated satellite images can be easily obtained without RPC parameters. This study proposes a novel method to reconstruct a 3D DSM from uncalibrated MVSS images by estimating and integrating RPC parameters. To do this, we first employ a structure from motion (SfM) and 3D homography-based geo-referencing method to reconstruct an initial DSM. Second, we sample 3D points from the initial DSM as references and reproject them to the 2D image space to determine 3D–2D correspondences. Using the correspondences, we directly calculate all RPC parameters. To overcome the memory shortage problem while running the large size of satellite images, we also propose an RPC integration method. Image space is partitioned to multiple tiles, and RPC estimation is performed independently in each tile. Then, all tiles’ RPCs are integrated into the final RPC to represent the geometry of the whole image space. Finally, the integrated RPC is used to run a true MVSS pipeline to obtain the 3D DSM. The experimental results show that the proposed method can achieve 1.455 m Mean Absolute Error (MAE) in the height map reconstruction from multi-view satellite benchmark datasets. We also show that the proposed method can be used to reconstruct a geo-referenced 3D DSM from uncalibrated and freely available Google Earth imagery.

A note on transcendental analytic functions with rational coefficients mapping $\mathbf{Q}$ into itself

A note on transcendental analytic functions with rational coefficients mapping $\mathbf{Q}$ into itself

Ground Control Point Chip-based Kompsat-3A Rational Polynomial Coefficient Bias Compensation Using Both Intensity- and Edge-based Matching Methods

Ground Control Point Chip-based Kompsat-3A Rational Polynomial Coefficient Bias Compensation Using Both Intensity- and Edge-based Matching Methods

On a case of -linear conjugation problem on the unit circle

The R -linear conjugation problem on the unit circle with rational coefficients is solved. The problem is equivalently reduced to the vector–matrix C -linear conjugation problem for which the canonical matrix is effectively constructed. The applied method is based on the two-fold application of G.N.Chebotarev's approach to the factorization of rational matrix functions.

The Chow ring of the moduli space of degree 2 quasi-polarized K3 surfaces

Abstract We study the Chow ring with rational coefficients of the moduli space F 2 \mathcal{F}_{2} of quasi-polarized K3 surfaces of degree 2. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group is A 17 ⁢ ( F 2 ) ≅ Q \mathsf{A}^{17}(\mathcal{F}_{2})\cong{\mathbb{Q}} . We prove that the Chow ring consists of tautological classes and is isomorphic to the even cohomology. The Chow ring is not generated by divisors and does not satisfy duality with respect to the pairing into A 17 ⁢ ( F 2 ) \mathsf{A}^{17}(\mathcal{F}_{2}) . The kernel of the pairing is a 1-dimensional subspace of A 9 ⁢ ( F 2 ) \mathsf{A}^{9}(\mathcal{F}_{2}) which we calculate explicitly. In the appendix, we revisit Kirwan–Lee’s calculation of the Poincaré polynomial of F 2 \mathcal{F}_{2} .

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

AbstractWe compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the rational homology of the unit tangent bundles of the surfaces. These coefficients define a covariant functor over the classical category associated to mapping class groups, rather than a contravariant one, and are thus out of the scope of the traditional framework to study twisted cohomological stability. A remarkable property is that the computed stable twisted cohomology is not free as a module over the stable cohomology algebra with constant rational coefficients. For comparison, we also compute the stable cohomology group with coefficients in the first rational cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

Rational Polynomial Coefficient Estimation via Adaptive Sparse PCA-Based Method

The Rational Function Model (RFM) is composed of numerous highly correlated Rational Polynomial Coefficients (RPCs), establishing a mathematical relationship between two-dimensional images and three-dimensional spatial coordinates. Due to the existence of ill-posedness and overparameterization, the estimated RPCs are sensitive to any slight perturbations in the observation data, particularly when handling a limited number of Ground Control Points (GCPs). Recently, Principal Component Analysis (PCA) has demonstrated significant performance improvements in the RFM optimization problem. In the PCA-based RFM, each Principal Component (PC) is a linear combination of all variables in the design matrix. However, some original variables are noise related and have very small or almost zero contributions to the construction of PCs, which leads to the overparameterization problem and makes the RPC estimation process ill posed. To address this problem, in this paper, we propose an Adaptive Sparse Principal Component Analysis-based RFM method (ASPCA-RFM) for RPC estimation. In this method, the Elastic Net sparsity constraint is introduced to ensure that each PC contains only a small number of original variables, which automatically eliminates unnecessary variables during PC computation. Since the optimal regularization parameters of the Elastic Net vary significantly in different scenarios, an adaptive regularization parameter approach is proposed to dynamically adjust the regularization parameters according to the explained variance of PCs and degrees of freedom. By adopting the proposed method, the noise and error in the design matrix can be reduced, and the ill-posedness and overparameterization of the RPC estimation can be significantly mitigated. Additionally, we conduct extensive experiments to validate the effectiveness of our method. Compared to existing state-of-the-art methods, the proposed method yields markedly improved or competitive performance.

The volume of an isocanted cube is a determinant*

In any dimension d ≥ 2 , we give exact volume formulas of two mutually polar dual convex d-polytopes. The primal body is called isocanted cube of dimension d, depending on two real parameters 0 < a < ℓ . The limit case a = 0 yields a d-cube of edge-length ℓ. We prove that the volume of such a body is the determinant of the matrix of order d having diagonal entries equal to ℓ and a elsewhere. We also compute the volume of the polar dual body, getting a rational expression in ℓ , a , homogeneous of degree − d with rational coefficients. Isocanted cubes are origin-symmetric zonotopes. Zonoids (defined as the limits of families of zonotopes) satisfy the Mahler conjecture; in particular, zonotopes do. Nonetheless, we confirm (by elementary methods) that the Mahler conjecture holds for isocanted cubes.

Generalisations of multiple zeta values to rooted forests

We show that any convergent (shuffle) arborified zeta value admits a series representation. This justifies the introduction of a new generalisation to rooted forests of multiple zeta values, and we study its algebraic properties. As a consequence of the series representation, we derive elementary proofs of some results of Bradley and Zhou for Mordell-Tornheim zeta values and give explicit formulas. The series representation for shuffle arborified zeta values also implies that they are conical zeta values. We characterise which conical zeta values are arborified zeta values and evaluate them as sums of multiple zeta values with rational coefficients.

Rational enriched motivic spaces

Enriched motivic A-spaces are introduced and studied in this paper, where A is an additive category of correspondences. They are linear counterparts of motivic Γ-spaces. It is shown that rational special enriched motivic Cor˜-spaces recover connective motivic bispectra with rational coefficients, where Cor˜ is the category of Milnor–Witt correspondences.

CoCoSo framework for multi-attribute decision-making with triangular fuzzy neutrosophic sets: “Innovation and entrepreneurship” evaluation case

The “innovation and Entrepreneurship” education focuses on cultivating students’ “innovation and Entrepreneurship” abilities, reflecting the fundamental requirements of economic progress and social development for the knowledge, ability, and quality structure of talents. Local universities should actively promote the reform of talent cultivation models, establish a scientific “innovation and Entrepreneurship” education target system based on the actual situation of the school, strengthen the construction of “innovation and Entrepreneurship” education courses, build various “innovation and Entrepreneurship” practice platforms, pay attention to the interaction between the first and second classrooms, as well as practical and training platforms, serve the growth and talent development of students, and achieve a comprehensive improvement in the quality of school education, teaching, and talent cultivation. The “innovation and Entrepreneurship” education quality evaluation of local colleges could be considered as multiple attribute decision-making (MADM). Recently, the Combined Compromise Solution (CoCoSo) method and information entropy method was employed to deal with MADM. The triangular fuzzy neutrosophic sets (TFNSs) are employed as a better tool for expressing uncertain information during the “innovation and Entrepreneurship” education quality evaluation of local colleges. In this paper, the triangular fuzzy neutrosophic number CoCoSo (TFNN-CoCoSo) based on the TFNN grey rational coefficients (TFNNGRC) from TFNN positive ideal solution (TFNNPIS) is constructed to cope with the MADM under TFNSs. The information entropy method is employed to compute the weight values based on the TFNNGRC from TFNNPIS under TFNSs. Finally, a numerical example of “innovation and Entrepreneurship” education quality evaluation of local colleges is constructed and some decision comparisons are constructed to verify the TFNN-CoCoSo method.

Double-virtual NNLO QCD corrections for five-parton scattering. I. The gluon channel

We compute the two-loop helicity amplitudes for the scattering of five gluons, including all contributions beyond the leading-color approximation. The analytic expressions are represented as linear combinations of transcendental functions with rational coefficients, which we reconstruct from finite-field samples obtained with the numerical unitarity method. Guided by the requirement of removing unphysical singularities, we find a remarkably compact generating set of rational coefficients, which we display entirely in the manuscript. We implement our results in a public code, which provides efficient and reliable numerical evaluations for phenomenological applications. Published by the American Physical Society 2024

Double-virtual NNLO QCD corrections for five-parton scattering. II. The quark channels

We complete the computation of two-loop helicity amplitudes required to obtain next-to-next-to-leading order QCD corrections for three-jet production at hadron colliders, including all contributions beyond the leading-color approximation. The analytic expressions are reconstructed from finite-field samples obtained with the numerical unitarity method. We find that the reconstruction is significantly facilitated by exploiting the overlaps between rational coefficient functions of quark and gluon processes, and we display their compact generating sets. We implement our results in a public code, and demonstrate its suitability for phenomenological applications. Published by the American Physical Society 2024

Multivortex traveling waves for the Schrödinger map equation

We construct traveling wave solutions for the Schrödinger map equation in R2. These solutions have n(n + 1)/2 pairs of degree ±1 vortices. The locations of those vortices are symmetric in the plane and determined by the roots of a special class of Adler–Moser polynomials. With a few modifications, a similar construction allows for the creation of traveling wave solutions of the Schrödinger map equation in R3. These solutions have the shape of 2n + 1 vortex rings, whose locations are given by a sequence of polynomials with rational coefficients and are far away from each other.

On the Number of Rational Roots of a Rational Polynomial

We propose an extension of Gauss’s lemma and Schönemann-Eisenstein’s irreducibility criterion to determine the impossibility of decomposing an integer polynomial into several polynomials with rational coefficients. This extension allows us to obtain some limitations on the number of rational roots and on the degrees of the factors of a rational polynomial.

The Impact of GCP Chip Distribution on Kompsat-3A RPC Bias Compensation

The vast potential of high-resolution satellite images, including Kompsat-3A, has been demonstrated across diverse applications, such as mapping and disaster monitoring. However, these images can only be utilized as reliable GIS (geographic information system) data when they possess precise geographical information. To achieve this, sensor model information, represented by RPCs (rational polynomial coefficients), requires bias compensation through GCPs (ground control points). Though having a substantial number of well-distributed GCPs across satellite images is ideal, the acquisition process is often restricted due to cost and inaccessibility. The uniform distribution of GCP chips is not guaranteed, necessitating an investigation into the impact of GCP distribution on the bias compensation process, which is the focus of this study. Experiments were meticulously conducted using Kompsat-3A data using dense GCP information. The dense GCP information was automatically generated from aerial orthoimages through a three-step process. Firstly, the GCP chips were extracted from the aerial images, focusing on feature points. Secondly, these chips were projected onto the target Kompsat-3A data to align them accurately. Lastly, precise satellite image coordinates of the chips were obtained through image matching between the chips and the target Kompsat-3A image. The dense GCPs enabled detailed bias analysis that exhibited skewness in most Kompsat-3A data. This necessitates the implementation of an affine model for proper bias compensation over the entire image space. Next, the study delved into the influence of GCP distribution on RPC bias compensation. To this end, each target satellite image space was divided into nine zones, with the dense GCPs assigned accordingly. The accuracy of bias compensation was analyzed across nine experimental cases, ranging from GCPs occupying only one zone to GCPs covering all nine zones. It was observed that GCPs covering at least four or five zones should be utilized for reliable RPC bias compensation of Kompsat-3A, especially when aiming for a high level of accuracy with an RMSE of one pixel. Finally, it was concluded that GCPs covering three zones yielded satisfactory results as a minimum GCP requirement, but this was contingent upon their distribution not following a straight zone pattern.

GPU Acceleration for SAR Satellite Image Ortho-Rectification

Synthetic Aperture Radar (SAR) satellite image ortho-rectification requires pixel-level calculations, which are time-consuming. Moreover, for SAR images with large overlapping areas, the processing time for ortho-rectification increases linearly, significantly reducing the efficiency of SAR satellite image mosaic. This paper thoroughly analyzes two geometric positioning models for SAR images. In order to address the high computation time of pixel-by-pixel ortho-rectification in SAR satellite images, a GPU-accelerated pixel-by-pixel correction method based on a rational polynomial coefficients (RPCs) model is proposed, which improves the efficiency of SAR satellite image ortho-rectification. Furthermore, in order to solve the problem of linearly increasing processing time for the ortho-rectification of multiple SAR images in large overlapping areas, a multi-GPU collaborative acceleration strategy for the ortho-rectification of multiple SAR images in large overlapping areas is proposed, achieving efficient ortho-rectification processing of multiple SAR image data in large overlapping areas. By conducting ortho-rectification experiments on 20 high-resolution SAR images from the Gaofen-3 satellite, the feasibility and efficiency of the multi-GPU collaborative acceleration processing algorithm are verified.

How to solve a binary cubic equation in integers

AbstractGiven any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.

A dynamical analogue of a question of Fermat

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter quadratic polynomials with orbits containing three consecutive squares. In addition, we show that there exists at least one polynomial of the form $x^2+c$ with a rational point whose orbit under this map contains four consecutive squares. This can be viewed as a dynamical analogue of a question of Fermat on rational squares in arithmetic progression. Finally, assuming a standard conjecture on exact periods of periodic points of quadratic polynomials over the rational field, we give necessary and sufficient conditions under which the orbit of a periodic point contains only rational squares.