Differential Operators Research Articles

Extensions of Some Statistical Concepts to the Complex Domain

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under Hermitian-form constraints, and similar maxima/minima problems in the complex domain are discussed. Some vector/matrix differential operators are developed to handle the above types of problems. These operators in the complex domain and the optimization problems in the complex domain are believed to be new and novel. These operators will also be useful in maximum likelihood estimation problems, which will be illustrated in the concluding remarks. Detailed steps are given in the derivations so that the methods are easily accessible to everyone.

Symmetric Identities Involving the Extended Degenerate Central Fubini Polynomials Arising from the Fermionic p-Adic Integral on ℤp

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind.

Local invariants of noncommutative tori

The notion of a generic curved noncommutative torus is considered, which extends the notion of a conformally deformed noncommutative torus introduced by Connes and Tretkoff. For this manifold, an asymptotic expansion is established for the heat semigroup generated by the Laplace–Beltrami operator (in fact, for an arbitrary selfadjoint positive elliptic differential operator of order 2 2 ) and an algorithm is provided to compute the local invariants that arize as the coefficients in the expansion. This allows one to extend a series of previous results by several authors beyond the conformal case and/or for multidimensional tori.

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable.

This study suggests a strategy for calculating the fuzzy analytical solutions to a two-dimensional fuzzy fractional-order heat problem including a diffusion variable connected externally. We propose Sawi residual power series scheme (SRPSS) which is the amalgamation of Sawi transform and residual power series scheme under the Caputo fractional differential operator. We demonstrate three different examples to derive the fuzzy fractional series solution which is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity. The most significant aspect of this scheme is that it derives the results without time effort compared with the traditional residual power series approach. Our findings confirm that the SRPSS is a robust and valuable method for approximating the solution of fuzzy fractional problems. Furthermore, we provide 2D and 3D symbolic representations to present the physical behavior of fuzzy fractional problems under the lower and upper bounded solutions.

Two-dimensional optical multiple-order differentiations based on spatial spectrum modulation

Optical differential operations have recently attracted considerable attention owing to the capabilities of ultrafast speed and low power consumption. The transfer function, which embodies the frequency-domain characteristics of differential systems, plays an important role in differentiator design. Here, we report a super-Gaussian aperture differential filter, and we reveal unique characteristics of odd- and even-order transfer functions and corresponding differential effects via spatial spectrum modulation. We show that the feature of the transfer function is well maintained, and more precise differentiation can be achieved using the designed filter. Two-dimensional first- to fifth-order full and partial differentiations are implemented both theoretically and experimentally. Our work provides an approach for engineering customized multiple-order differentiators and promotes the advancements of related areas such as optical analog computing and image processing.

Hyperelastic material modeling using first- and second- order differential operations with complex step derivative approximation

Hyperelastic material modeling using first- and second- order differential operations with complex step derivative approximation

Subclass of Analytic Functions Associated with Differential Operator

Subclass of Analytic Functions Associated with Differential Operator

Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra

We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to the modular group and $q$-deformed rational numbers defined by Morier-Genoud and Ovsienko and lead to $q$-deformed Möbius transformations acting on the hyperbolic plane.

A Certain <i>q</i>-Sălăgean Differential Operator and Its Applications to Subclasses of Analytic and Bi-Univalent Functions Involving (<i>p</i>, <i>q</i>)-Chebyshev Polynomia

In the present investigation, we make use of the q- analogue of the Sălăgean differential operator and introduce a new subclass of analytic and bi-univalent functions involving the (P, Q)-Chebyshev polynomials. Furthermore, we derive coefficient inequalities and obtain the Fekete-Szegö problem for this new function class f ∈ of functions.

A comprehensive mathematical analysis of fractal–fractional order nonlinear re-infection model

In recent times, various real word problems including infectious diseases, engineering problems, and chemical processes, etc are modelled by using differential equations with fractals and fractional orders. The mentioned area provides a powerful tool to investigate the aforesaid problems from different perspectives like analysis, numerical investigation, and qualitative theory. Similarly, one of the global issue is devoted to re-infection of viral diseases like COVID-19. The said behaviour has produced very saviour impact on human life worldwide. Also, this is very critical for health system of a country as well as for its economics situation. To study the aforesaid phenomenon of re-infection, a hybrid type fractal–fractional three compartments model is formulated. The basic results related to boundedness, positivity, equilibrium points, basic reproductive number, and sensitivity analysis of the basic threshold are included in the theoretical aspect. Additionally, the Volterra–Lyapunov approach is used to establish both local and global stability analysis under fractals and fractional derivative. Furthermore, utilizing techniques from numerical functional analysis and fixed point theory, the existence and Hyers–Ulam stability of the solutions are also examined. A numerical algorithm based on Lagrange’s interpolation polynomials is extended to simulate the results of various compartments. Graphical interpretations corresponding to different fractals–fractional order values are presented. For verification of our numerical results, a comparison is provided between the real and simulated data. Here it should be kept in mind that the differential operator of fractals–fractional order is considered with Mittag-Leffler kernel. The mentioned operator is more general among the available fractals fractional differential operators in literature.

STEF: a Swin Transformer-Based Enhanced Feature Pyramid Fusion Model for Dongba character detection

The Dongba manuscripts are a unique primitive pictographic writing system that originated among the Naxi people of Lijiang, China, boasting over a thousand years of history. The uniqueness of the Dongba manuscripts stems from their pronounced pictorial and ideographic characteristics. However, the digital preservation and inheritance of Dongba manu manuscripts face multiple challenges, including extracting its rich semantic information, recognizing individual characters, retrieving Dongba manuscripts, and automatically interpreting the meanings of Dongba manuscripts. Developing efficient Dongba character detection technology has become a key research focus, wherein establishing a standardized Dongba detection dataset is crucial for training and evaluating techniques. In this study, we have created a comprehensive Dongba manuscripts detection dataset covering various commonly used Dongba characters and vocabularies. Additionally, we propose a model named STEF. Firstly, the Swin Transformer extracts the complex structures and diverse shapes of Dongba manuscripts’ features. Then, by introducing a Feature Pyramid Enhancement Module, features of different sizes are cascaded to preserve multi-scale information. Subsequently, all features are fused in a FUSION module, resulting in features of various Dongba manuscript styles. Each pixel’s binarisation threshold is dynamically adjusted through a differentiable binarisation operation, accurately distinguishing between foreground Dongba manuscripts and background. Lastly, deformable convolution is introduced, allowing the model to dynamically adjust the convolution kernel’s size and shape based on the Dongba manuscripts’ size, thereby better capturing the detailed information of Dongba characters of different sizes. Experimental results show that STEF achieves a recall rate of 88.88%, a precision rate of 88.65%, and an F-measure of 88.76%, outperforming other text detection algorithms. Visualization experiments demonstrate that STEF performs well in detecting Dongba manuscripts of various sizes, shapes, and styles, especially in blurred handwriting and complex backgrounds.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

Spectral reordering for faster elasticity simulations

We present a novel method for faster physics simulations of elastic solids. Our key idea is to reorder the unknown variables according to the Fiedler vector (i.e., the second-smallest eigenvector) of the combinatorial Laplacian. It is well known in the geometry processing community that the Fiedler vector brings together vertices that are geometrically nearby, causing fewer cache misses when computing differential operators. However, to the best of our knowledge, this idea has not been exploited to accelerate simulations of elastic solids which require an expensive linear (or non-linear) system solve at every time step. The cost of computing the Fiedler vector is negligible, thanks to an algebraic Multigrid-preconditioned Conjugate Gradients (AMGPCG) solver. We observe that our AMGPCG solver requires approximately 1 s for computing the Fiedler vector for a mesh with approximately 50K vertices or 100K tetrahedra. Our method provides a speed-up between 10%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10\\%$$\\end{document} – 30%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$30\\%$$\\end{document} at every time step, which can lead to considerable savings, noting that even modest simulations of elastic solids require at least 240 time steps. Our method is easy to implement and can be used as a plugin for speeding up existing physics simulators for elastic solids, as we demonstrate through our experiments using the Vega library and the ADMM solver, which use different algorithms for elasticity.

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

On Some Properties of a Class of Analytic Functions Defined by Opoola Differential Operator

The Fekete-Szego functional upper bounds and the Second Hankel Determinant upper bounds for a class of analytical functions defined by the Opoola Differential Operator are found in this study. The estimations made are accurate with the theorems proved.

Solving second order homogeneous differential equations in terms of Heun's general function

In this paper, we present an algorithm that checks if a second-order differential operator L∈C(x)[∂] can be reduced to the general Heun's differential operator. The algorithm detects the parameters of the transformations in C(x)[∂] that transfer the general Heun's differential operator to the operator L whose solutions are of the form(1)exp(∫rdx)⋅HeunG(a,q;α,β,γ,δ;f(x)), where {α,β,δ,γ}∈Q∖Z, the functions r,f(x)∈C(x), and C(f(x)) is a subfield of C(x) of index 2 or 3 or f(x)=axn+bcxn+d for some n in N∖{1}.

Global Gevrey solvability of a complex of differential operators associated with an involutive system

We consider a complex of differential operators associated with an involutive system of vector fields on the torus. We present a complete characterization of the global Gevrey solvability in terms of Diophantine conditions involving exponential Liouville forms.

A peridynamic differential operator modeling approach for ceramic matrix composites microstructure with uniform or non-uniform discretization

Based on the theory of peridynamic differential operator (PDDO), a novel three-dimensional (3D) peridynamic (PD) model for isotropic and orthotropic material is proposed to investigate the elastic mechanical behaviors and stress distribution of Ceramic Matrix Composites (CMC) microstructure. The non-local expressions of strain and stress tensors are derived by employing PDDO and the classical constitutive equations. The bond forces in interface region crossing two different constituent materials are obtained by converting the partial differential terms into the non-local forms. The weak form of PD equations of motion is derived by using the principle of virtual work and PDDO. The validity of this approach is verified by predicting the elastic response and stress distributions of isotropic and orthotropic materials under uniaxial tension and pure shear deformation. Finally, simulations are conducted on a CMC microstructure with fiber and matrix under periodic boundary conditions. It is demonstrated that the current approach can effectively build 3D CMC microstructure with uniform or non-uniform discretization, and accurately predict the stress fields and effective elastic properties.

Physics-informed boundary integral networks (PIBI-Nets): A data-driven approach for solving partial differential equations

Partial differential equations (PDEs) are widely used to describe relevant phenomena in dynamical systems. In real-world applications, we commonly need to combine formal PDE models with (potentially noisy) observations. This is especially relevant in settings where we lack information about boundary or initial conditions, or where we need to identify unknown model parameters. In recent years, Physics-Informed Neural Networks (PINNs) have become a popular tool for this kind of problems. In high-dimensional settings, however, PINNs often suffer from computational problems because they usually require dense collocation points over the entire computational domain. To address this problem, we present Physics-Informed Boundary Integral Networks (PIBI-Nets) as a data-driven approach for solving PDEs in one dimension less than the original problem space. PIBI-Nets only require points at the computational domain boundary, while still achieving highly accurate results. Moreover, PIBI-Nets clearly outperform PINNs in several practical settings. Exploiting elementary properties of fundamental solutions of linear differential operators, we present a principled and simple way to handle point sources in inverse problems. We demonstrate the excellent performance of PIBI-Nets for the Laplace and Poisson equations, both on artificial datasets and within a real-world application concerning the reconstruction of groundwater flows.

An unmanned aerial vehicle autonomous flight trajectory planning method and algorithm for the early detection of the ignition source during fire monitoring

ABSTRACT Timely identification of a fire’s origin in its initial stages is essential for minimizing human as well as material losses. Hence, formulating a methodology and algorithm for autonomously planning the flight path of an unmanned aerial vehicle (UAV) during fire monitoring to identify ignition sources early is a pressing objective. The proposed ignition detection method comprises three flight plans (FPs); Flight Plan A (FPA) involves surveying the monitored area with tacks, and assessing harmful substance concentrations in each pixel. Upon identifying a pixel surpassing the threshold concentration, Flight Plan B (FPB) directs the UAV control for targeted navigation, employing local planning based on calculating local differential operators within a nine-element mask. FPB enables the UAV to directly reach and pinpoint the fire source coordinates. Flight Plan C (FPC) outlines the UAV’s return journey to the departure point from any pixel within the monitoring zone. A multi-criteria algorithm for UAV flight path control has been devised, facilitating the determination of local target pixels and subsequent Local Flight Plan (LFP) construction. The guiding principle for constructing an LFP is the ‘at least three pixels on the tack’ rule, yielding a nine-element matrix with known harmful substance concentrations in the target pixel. It enables the identification of pixels within the LFP where obtaining this matrix is unattainable. Mathematical modelling of the UAV flight control algorithm, as per the proposed methodology, was executed using MATLAB® R2019b, demonstrating control stability and accelerated attainment of ignition source pixel coordinates. The achieved speed surpassed the set goal by 1.5 to 2 times, contingent on the ignition source’s location concerning the monitored territory’s flyby direction.