Transformation Formula Research Articles

Non-Inertial Dynamic Analysis of 3-SPS/U Parallel Platform by Screw Theory and Kane’s Method

This paper presents an improved method for the non-inertial dynamic analysis of the 3-SPS/U parallel platform (3-SPS/U PM), employing the screw theory and Kane’s method, where S, P, and U denote spherical, prismatic, and universal joints, respectively. The proposed method extends the traditional inertial dynamic analysis to non-inertial systems. First, the generalized screw method is introduced, followed by the derivation of a transformation formula that adapts the screw method to various co-ordinate systems. Subsequently, the velocities and accelerations of each rigid body within the platform under non-inertial conditions are examined by combining the extended screw method with the system’s inverse kinematics model. The extended screw method is not only conceptually simple, but also adaptable to other non-inertial systems. Finally, the standard non-inertial dynamic model of the 3-SPS/U PM is derived through the Kane’s method and validated by the co-simulations with RecurDyn (V9R5) and MATLAB/Simulation (2019b).

Lattice sums of I-Bessel functions, theta functions, linear codes and heat equations

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{Z}}^n$$\\end{document} whose initial condition is given by a linear code.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

X-y duality in topological recursion for exponential variables via quantum dilogarithm

For a given spectral curve, the theory of topological recursion generates two different families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ of multi-differentials, which are for algebraic spectral curves related via the universal x-yx−y duality formula. We propose a formalism to extend the validity of the x-yx−y duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form e^x=F(e^y)ex=F(ey) or e^x=F(y)e^{a y}ex=F(y)eay with FF rational and aa some complex number, which was in principle already observed in [Commun. Number Theory Phys. 13, 763 (2019); J. Lond. Math. Soc. 109, e12946 (2024)]. From topological recursion perspective the family \omega_{g,n}^\veeωg,n∨ would be trivial for these curves. However, we propose changing the n=1n=1 sector of \omega_{g,n}^\veeωg,n∨ via a version of the Faddeev’s quantum dilogarithm which will lead to the correct two families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ related by the same x-yx−y duality formula as for algebraic curves. As a consequence, the x-yx−y symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, Gromov-Witten invariants of \mathbb{C}^3ℂ3 (or, equivalently, triple Hodge integrals), orbifold Hurwitz numbers, and stationary Gromov-Witten invariants of \mathbb{P}^1ℙ1. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Algebraic analysis of pairwise comparisons with non-reciprocal property

The law of comparative judgement usually implies the reciprocal property of pairwise comparisons. But time error and position preference could yield some judgements that contradict this identity assumption. This paper proposes an algebraic approach to dealing with the challenge of non-reciprocal property in pairwise comparisons. The concept of non-reciprocal pairwise comparison matrix (NrPCM) is used to characterize the situation of non-reciprocal property. Algebraic analysis of NrPCMs is carried out, where a linear space is constructed by proposing a set of basis. Then the linear space is decomposed into the direct sum of some subspaces. The properties of the space are analyzed such as inner product, decomposition and coordinate. Finally, we apply the algebraic method to investigate the Condorcet paradox under the impact of indifferent voters. A novel transformation formula of non-reciprocal pairwise comparisons is proposed, where the non-reciprocal property holds. It is revealed that the occurrence of voting paradox can be understood clearly according to the transitivity property of NrPCMs.

Kinematics analysis and simulation of bionic five-finger manipulator

Abstract To address the complexity of traditional manipulator structures, a bionic five-finger manipulator has been designed and developed. After introducing the manipulator’s structure and functionality, the kinematic model was established using the D-H parameter method, followed by the analysis of forward and inverse kinematics. The fingertip pose was derived using the homogeneous transformation formula, and the accuracy of the theoretical analysis was validated by comparing the mathematical and MATLAB simulation results. The Monte Carlo method was employed to determine the accessible motion space of the mechanical fingertip. Subsequently, trajectory planning for the manipulator was performed based on known initial and end finger angles. A comparison with ADAMS simulation results demonstrated the rationality of the mechanical finger’s movement space, stable speed, and compliance with actual grasping requirements. This study lays a theoretical foundation for the subsequent parameter optimization of the manipulator.

A correction method for calculating sky view factor in urban canyons using fisheye images

The Sky View Factor (SVF) is a critical parameter for studying the thermal environment of buildings theoretically. However, there are two main definitions of SVF, one based on the visible area ratio and the other based on the view factor. When utilizing fisheye images for SVF calculation, various issues often arise, such as conceptual confusion and algorithmic errors, significantly impeding the accurate scientific analysis of thermal environmental problems. Therefore, in this study, we employed methods such as theoretical model deduction, field analysis, and software simulation and validation to investigate the calculation methods for fisheye images with equidistant projection, equisolid angle projection, and stereographic projection. We proposed the Image Distance Ratio Transformation Method (IDRTM), a correction method for calculating the SVF of fisheye images with the three projection types and corresponding calculation formulas. Finally, two cases demonstrate the computational advantages of simplicity and high accuracy associated with the orthogonal transformation formulas applied to the three types of fisheye images. In this study, the error rates of SVF calculation using three non-orthographic fisheye images directly are all higher than 29 %. After applying IDRTM, the average error rates of the transformed IDRs of three types of fisheye images are all less than 0.58 %, and the error rates of the corrected SVF of three types of fisheye images are all less than 1 %. This study elucidates the scientific definition and algorithm of SVF and provides a scientific basis for the accurate selection of SVF calculation methods corresponding to fisheye image types.

Eulerian series as Hecke-type double sums through the transformation formulas

Recently, establishing the Hecke-type double sums forms for Eulerian series has received a great deal of attention to seek special cases of [Formula: see text]-hypergeometric series. In this paper, with the aid of some basic hypergeometric series identities given by Chen and Wang, and Liu, we discover many new Eulerian series and express them in terms of Hecke-type double sums and Appell–Lerch sums. Based on the modern sense for the mock theta functions, we find that some of these series are mock theta functions and the others are mixed mock theta functions. Furthermore, from the viewpoint of Hecke-type double sums, we find three string function identities with levels [Formula: see text], [Formula: see text], and [Formula: see text].

Multiple Sound Scattering by Combinations of Cylindrically Symmetric and Linearly Invariant Anomalies

A number of previous papers have used the coupled-mode method to assess three-dimensional scattering by cylindrically symmetric anomalies (seamounts, hills) or anomalies which are invariant in a horizontal direction (wedges, ridges). This paper makes an extension to combinations of these two anomaly types. The upper and lower depth boundaries of the anomalies may be flat or irregular, and the sound source may be anywhere in the medium. After a discretization of the anomalies of the two types with laterally homogeneous rings and strips, respectively, an adaptation of the coupled-mode method yields the solution of the pertinent Helmholtz equation. The adaptation involves a combination of Fourier-series summations to handle the ring anomalies and adaptive wavenumber integrations to handle the strip structure. For each anomaly, recursively computed reflection matrices relate the expansion coefficients for incoming and outgoing normal modes. Iterative solution of a linear equation system for the amplitudes of the scattered cylindrical waves from the ring anomalies, involving formulas for transformation between plane and cylindrical waves, yields an expansion of the field. Related expansions allow isolation of partial waves, multiply scattered among the anomalies. The paper includes examples from underwater acoustics and atmospheric acoustics.

Hydrate-based gas separation model considering hydrate structure transformation

The study introduces a pioneering model for hydrate-based gas separation technology, which enhances the prediction method for the coexistence of multiple structure hydrates. The calculated results demonstrate strong agreement with 96 sets of experimental data, exhibiting a lower average relative deviation of 8.00 % compared to the traditional model’s average relative deviation of 60.00 %. Furthermore, it offers a comprehensive discussion on the composition of hydrate structures in (H2 + CH4 + C2H6 + C3H8) during gas–water-hydrate equilibrium, providing a more precise depiction of hydrate structure evolution. Additional gas-hydrate equilibrium experiments were conducted for the (H2 + CH4 + C2H6 + C3H8) system to optimize the model parameters. The fitting goodness of the hydrate structure transformation formula (niI-Si formula) is enhanced from 0.9162 to 0.9816.

Some 𝑞-identities derived by the ordinary derivative operator

In this paper, we investigate applications of the ordinary derivative operator, instead of the q q -derivative operator, to the theory of q q -series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the q q -binomial theorem, Ramanujan’s 1 ψ 1 {}_1\psi _1 formula, the quintuple product identity, Gasper’s q q -Clausen product formula, and Rogers’ 6 ϕ 5 {}_6\phi _5 formula, etc. Among these results is a finite form of the Rogers-Ramanujan identity and a short way to Eisenstein’s theorem on Lambert series.

Darboux transformation-based LPNN generating novel localized wave solutions

Darboux transformation method is one of the most essential and important methods for solving localized wave solutions of integrable systems. In this work, we introduce the core idea of Darboux transformation of integrable systems into the Lax pairs informed neural networks (LPNNs), which we proposed earlier. By fully utilizing the data-driven solutions, spectral parameter and spectral function obtained from LPNNs, we present the novel Darboux transformation-based LPNN (DT-LPNN). The notable feature of DT-LPNN lies in its ability to solve data-driven localized wave solutions and spectral problems with high precision, and it also can employ Darboux transformation formulas of integrable systems and non-trivial seed solutions to discover novel localized wave solutions that were previously unobserved and unreported. The numerical results indicate that, by utilizing the single-soliton solutions as the non-trivial seed solutions, we obtain novel localized wave solutions for the Kraenkel–Manna–Merle (KMM) system by employing DT-LPNN, in which solution u changes from original bright single-soliton on zero background wave to new dark single-soliton dynamic behavior on a variable non-zero background wave. Moreover, by treating the two-soliton solutions as the non-trivial seed solutions, DT-LPNN generates novel localized wave solutions for the KMM system that exhibit completely different dynamic behaviors from prior two-soliton solutions. DT-LPNN combines the Darboux transformation theory of integrable systems with deep neural networks, offering a new approach for generating novel localized wave solutions using non-trivial seed solutions.

Introduction to the Thukral-Determinantal Formula for Accelerating Convergence of Sequence

There are two objectives for this paper. Firstly, we shall introduce the Thukral-determinantal formula, and secondly, we shall demonstrate the similarities between the well-established algorithms, namely the Aitkin Δ2 algorithm and the Durbin sequence transformation. In fact, we have found that the solution of the Thukral-determinantal formula is equivalent to the Thukral-sequence transformation formula.

Darboux transformation of symmetric Jacobi matrices and Toda lattices

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = 𝔘𝔏) with L (or 𝔏) and U (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J(p) = UL (or J(d) = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.

Research on High Performance Apple Recognition Model Construction Strategy Based on Deep Learning

In order to improve the working efficiency and environmental adaptability of apple picking robots, which can realize accurate recognition and localization, ripening assessment, and quality assessment under complex situations such as "leaf occlusion", "branch occlusion", "fruit occlusion", "mixed occlusion", etc., this paper proposes a Faster R-CNN-based apple recognition method. "mixed occlusion" and other complex situations to achieve accurate recognition and localization, maturity assessment, quality assessment, the paper proposes an apple recognition method based on Faster R-CNN. The method combines the cv2.inRange and cv2.fitEllipse functions in OpenCV to optimize the model, and performs color segmentation and shape fitting by setting up color and shape thresholds to recognize apples. The trained model was tested to have a recall of 88.44% under the validation set, an accuracy of 90.35%, an F1 value of 89.38%, and a recognition time of about 0.3 s per image.The experimental results showed that the number of apples in the dataset showed a normal distribution, with the number distributed more in the range of 0 to 10. Candidate frames are screened by RPN, and apple coordinates are derived based on the coordinate transformation formula, and the experimental results show that the apples are more densely distributed near the range of x=100, y=100. The image color distribution was extracted through color histogram, and the entropy value of GLCM, contrast as the texture feature of apples, and the classification prediction of apple ripeness was carried out by using Random Forest Model, and the accuracy of the model prediction was 83.2%, the precision rate was 84.1%, and the recall rate was 83.5%, and the experimental results showed that 88.7% of the apples were ripe. Image edges were extracted by the Canny edge detection algorithm and the area values of pixel points within the edges were calculated. The ratio of pixel area converted to apple mass was estimated to be 0.007 based on the image resolution and apple density, and the test results indicated that the mass of apples was mostly concentrated in the range of 0~100g.

The Generalized Eta Transformation Formulas as the Hecke Modular Relation

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions.

A real-world pharmacovigilance study of amivantamab-related cardiovascular adverse events based on the FDA adverse event reporting system (FAERS) database

Amivantamab is the first dual-specificity antibody targeting EGFR and MET, which is approved for the treatment of locally advanced or metastatic non-small cell lung cancer (NSCLC) with EGFR exon 20 insertion mutations. Cardiovascular toxicities related to amivantamab have not been reported in the CHRYSALIS study. However, the occurrence of cardiovascular events in the real world is unknown. To comprehensively investigate the clinical characteristics, onset times, and outcomes of cardiovascular toxicities associated with amivantamab. The Food and Drug Administration Adverse Event Reporting System (FAERS) database from 1st quarter of 2019 to the 2nd quarter of 2023 was retrospectively queried to extract reports of cardiovascular adverse events (AEs) associated with amivantamab. To perform disproportionality analysis, the reporting odds ratios (RORs) and information components (ICs) were calculated with statistical shrinkage trans-formation formulas and a lower limit of the 95% confidence interval (CI) for ROR (ROR025) > 1 or IC (IC025) > 0 with at least 3 reports was considered statistically significant. A total of 20,270,918 eligible records were identified, among which 98 records were related to cardiovascular events associated with amivantamab. 4 categories of cardiovascular events exhibited positive signals: venous thrombotic diseases, abnormal blood pressure, arrhythmia, and pericardial effusion. Venous thrombotic diseases and abnormal blood pressure were the two most common signals. The median time to onset (TTO) for cardiovascular AEs was 33 days. The cumulative incidence within 90 days was 100% for cardiac failure, 75% for stroke, 63.16% for arrhythmia, 50% for sudden death, and 44.18% for venous thrombotic diseases. Death accounted for 16.3% of all cardiovascular AEs associated with amivantamab. The mortality rates for Major Adverse Cardiovascular Events (MACE) were up to 60%. This pharmacovigilance study systematically explored the cardiovascular adverse events of amivantamab and provided new safety signals based on past safety information. Early and intensified monitoring is crucial, and attention should be directed towards high-risk signals.

‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of 0 , 1 , … , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

ON MATRICES ARISING IN FINITE FIELD HYPERGEOMETRIC FUNCTIONS

Abstract Lehmer [‘On certain character matrices’, Pacific J. Math.6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math.10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.