Differential Transformation Research Articles

Boundedness of Differential Transforms for Poisson Semigroups Generated by Bessel Operators

Abstract In this paper we analyze the convergence of the following type of series: $$ T_N \,\,f(x)=\sum_{j=N_1}^{N_2} v_j\Big({\mathcal P}_{a_{j+1}} \,\,f(x)-{\mathcal P}_{a_{j}} \,\,f(x)\Big),\quad x\in \mathbb R_+, $$ where $\{{\mathcal P}_{t} \}_{t\gt0}$ is the Poisson semigroup associated with the Bessel operator $\displaystyle \Delta_\lambda:=-{d^2\over dx^2}-{2\lambda\over x}{d\over dx}$, with λ being a positive constant, $N=(N_1, N_2)\in \mathbb Z^2$ with $N_1 \lt N_2,$ $\{v_j\}_{j\in \mathbb Z}$ is a bounded real sequence and $\{a_j\}_{j\in \mathbb Z}$ is an increasing real sequence. Our analysis will consist in the boundedness, in $L^p(\mathbb{R}_+)$ and in $BMO(\mathbb{R}_+)$, of the operators TN and its maximal operator $\displaystyle T^*\,\,f(x)= \sup_N \left\vert T_N \,\,f(x)\right\vert.$ It is also shown that the local size of the maximal differential transform operators is the same with the order of a singular integral for functions f having local support.

Irreversibility of Al2O3-Ag hybrid nanoparticles in mixture base fluid on microchannel with variable viscosity, buoyancy forces, and suction/injection effects: An analytical study

The development of inherent irreversibility in the system is caused by single phase Poiseuille flow considering hybrid nanoparticles (Al2O3-Ag) and mixture fluid (water and ethylene glycol H2O50%-C2H6O250%) in the upright microchannel with unequal viscosity. Taking into account the buoyancy force, suction/injection at the walls, and the form factor and geometry of the nanoparticles. The modeling is based on nonlinear PDEs such as continuity, momentum, and heat equations, which are then transformed to a system of nonlinear ODEs using similarity transformations and solved numerically and analytically. The analytical solution was built using the Differential Transform Method (DTM), and the current results in specific cases are compared to results obtained by the HAM-based Mathematica package, the Runge-Kutta Fehlberg 4th-5th order (RKF45), and those available in literature. The effects of active parameters are investigated on the velocity and temperature, entropy generation and Bejan number. The outcomes of the present analysis reveal that the nanoparticles volume fraction, thermal radiation and Biot number acts to enhance the cooling of the system through the release of thermal energy. in addition, the enhancement in variable viscosity parameter, causes a rise in the irreversibility rate, which in turn boosts the rates of entropy generation and Bejan number. On the other hand, Irreversibility due to heat transfer is dominant in the centerline of the microchannel, while fluid friction irreversibility dominates its walls.

Spatiotemporal feature learning for no-reference gaming content video quality assessment

Recently, over-the-top live gaming content video (GCV) services have significantly contributed to the overall internet traffic. Consequently, there is a growing demand of GCV quality assessment (GCVQA) to maintain service quality. Although recent literature has proposed a few GCVQA methods, these mainly focus on extracting spatial features and temporal fusion separately, limiting their performance due to the neglect of spatiotemporal feature learning, which is crucial for GCV as it typically shares spatial and temporal features across frames. To address this, we propose a novel GCVQA model, focusing on GCV spatiotemporal feature learning. First, we employ a multi-task self-supervised learning spatiotemporal pyramid convolutional neural network (STP-CNN) model to extract short-term spatiotemporal quality feature representations (STQFR) of GCVs. Our STP-CNN model specifically extracts multiscale spatiotemporal features from various temporal scales of multi-frames in pyramid mode, enabling dynamic learning of diverse spatiotemporal cues. Subsequently, we propose the differential Transformer model to process all short-term STQFR within a GCV, extracting global spatiotemporal features of GCV to assess the overall quality of GCV. To evaluate the effectiveness of our proposed method, we conducted experiments using four GCVQA datasets. The results demonstrate that our method outperforms existing approaches in predicting the perceived quality of GCV.

Improve Solutions of Differential Transform Method for Jeffrey Hamel Magneto-hydrodynamic Non-Newtonian Casson Fluid Flow Problem

In this article, Magneto-hydrodynamic flow of a non-Newtonian Casson fluid in converging and diverging channels subject to stretching -shrinking cases is an important problem in the field of fluid flow that has been highlighted by the combination of the differential transformation method and the optimal method so-called optimal differential transformation method (ODTM). In this synthesis that is used achieves improved analytical results for this problem. This study examines the impact of several developing characteristics, including channel angle, stretching-shrinking parameter, Casson fluid parameter, Hartmann number, and Reynolds number with the velocity profiles. Results obtained from the DTM and the numerical method BVP4c are compared to those resulting from ODTM which consist of the collocation differential transform method (CDTM), the sub-domain differential transform method (SDTM), the least squares differential transform method (LSDTM), and the Galerkin differential transform method (GDTM). The comparison shows that the percentage errors of the optimal methods (ODTM) are more accurate than those of the DTM for a similar number of parts of the repeated series over a wide range of physical parameters. Finally, it can be said that the results of the proposed method for third-order nonlinear Jeffrey Hamel flow in the non-Newtonian Casson fluid problem are high accuracy.

Bernoulli wavelet application to the numerical solution of Jeffery–Hamel flow problem

In this study, we introduced the Bernoulli wavelet method (BWM) to address the nonlinear Jeffery–Hamel flow problem. Utilizing a newly devised operational matrix of integration with the Bernoulli wavelet, we apply the Bernoulli wavelet expansion in conjunction with the collocation method to convert the given differential equation into a series of nonlinear equations. These equations are subsequently solved using an appropriate iterative technique. The BWM demonstrates superior accuracy when compared to existing methods like differential transformation, homotopy perturbation, variational iteration, and Runge–Kutta (R–K). Our results indicate the effectiveness of BWM in providing more precise solutions for the Jeffery–Hamel flow, a significant phenomenon with wide applications in various engineering fields, including chemical, aerospace, civil, biomechanical, mechanical, and environmental engineering. The study also delves into the influence of Reynolds number and convergent/divergent channel configurations on Jeffery–Hamel flow applications.

Effect of variations in concrete quality on the crack width in rigid pavement

Cracks pose a significant issue in rigid pavement, leading to substantial damage. The manual mixing and casting of road pavement concrete underscore the importance of concrete quality as a critical parameter. This study investigates crack behavior by varying concrete quality. Loading is performed statically using line loads, shedding light on the impact of concrete quality on crack development. The concrete to be used has a quality fc' of 15 MPa, 25 MPa, and 35 MPa. The fine aggregate used in this study was Lumajang black sand, while the coarse aggregate used was machine crushed stone, and Portland Composite Cement (PCC) was used in all concrete mixes. The reinforcing steel used had a quality fy of 480 MPa, with a reinforcement ratio of ρ=0.010, which was converted to 5-D16 reinforcement. The subgrade density used to support the specimens had a CBR value of 10 %. Specimen dimensions were 2×0.6×0.2 m for length, width, and thickness. Pavement plates, 30 cm thick, were placed on leveled subgrade soil in a steel box set to achieve a 6 % CBR reading. Hydraulic jacks, monitored by a load cell, applied monotonic static loading with 2 kN intervals, reaching a maximum load of 200 kN. Steel tension and plate settlement were measured using a tension sensor and Linear Variable Differential Transformer (LVDT), respectively. A data logger recorded readings, and crack widths were captured by a digital microscope with 0.01 mm accuracy. Experimental results show that low concrete compressive strength values result in larger crack widths, and vice versa. Cracks also occur at earlier loading of concrete quality fc' 15 MPa. In addition, experiments show that the reinforcement stress value has a significant influence on crack width in specimens with low concrete quality

Thermal and solutal stratified Heimanz flow of AA7072-deionized water over a wedge in the presence of bioconvection

The bioconvective Heimanz flow of nanofluid across a wedge with thermal stratification is analyzed in this article. The wedges are often seen in glider aircraft, rocket climbing frames, etc. The nanofluid considered in this study is composed of aluminum alloys of AA7072 and deionized water. The AA7072 alloys are specially manufactured materials composed of Aluminum and Zinc in the ratio of 98 : 1 along with metals like silicon, ferrous, and copper so that they possess enhanced heat transfer features. The mathematical model is formed using the modified Buongiorno’s model that includes the discussions related to slip mechanisms and volumetric analysis in terms of the weight of the nanoparticle. The model is in the form of partial differential equations and is later converted to ordinary differential equations with the assistance of similarity transformation. This set of equations is solved by the Differential Transformation Method (DTM) and the outcomes are discussed through graphs. ,

Abstract A063: CancerTools.org: Research Tools Supporting Breast Cancer Studies

Abstract CancerTools.org is the latest, non-profit cancer-focused biorepository. Driven by a mission to accelerate cancer research discoveries, the platform allows scientists to deposit and access tangible laboratory research materials. Till date, researchers from over 175 academic institutes globally have already deposited their tools, generating a cutting-edge portfolio of over 5,000 reagents, including a range of cell-based models and antibodies. Breast cancer is one of the most frequent malignancies in women worldwide. A comprehensive collection of tools has been curated at CancerTools.org to support this important area of research. Cell-based models include the RMAM007-3 breast cancer organoid which recapitulates in vivo tumour characteristics and is suitable for disease modelling, 3D imaging, and anticancer drug discovery. Cell lines such as human anti-oestrogen resistant lines developed from MCF-7 and T47D cells, demonstrate resistance to hormone-dependent treatments and are useful for studying therapy resistance mechanisms and biomarker development for breast cancer detection. NCI-HPN-F murine cell lines with recurrent chromosomal aneuploidies, have enabled investigation and identification of tumorigenesis related cancer-specific genes and signalling pathways. Further models include gene-edited cell lines to explore BRCA1 hypermethylation pathways; reporter breast cancer cell lines to assess apoptosis, necrosis, and autophagy; and immortalised human breast epithelial cell lines to study differentiation and phenotypic transformation. To provide a physiologically relevant cell culture environment, PlasmaxTM, a defined media that mirrors the concentration of over 50 components to human plasma, is also part of the repertoire. It has been used to culture triple-negative breast cancer cell lines to recapitulate the tumour metabolic environment. Antibodies to study breast cancer biology, including tumour suppressors, angiogenesis and metastasis, form part of the portfolio. Antibodies targeting tumour suppressors include RB1, which is often lost by mutation, deletion, transcriptional silencing and hyper-phosphorylation in malignancies- including breast cancer. Deletion and rearrangement of RB1 is estimated in ~20-25% of breast cancer cell lines. Tools identifying specific RB1 phosphorylation sites, include Anti-Phospho-Rb for Ser608 and Thr356. Additional insights into breast cancer prognosis are offered by Anti-Maspin [16F7] which recognises wildtype and denaturated forms of the protein. Maspin is an inhibitor of serine proteases with tumour suppressor activity in breast cancer. Its nuclear staining associates with favourable outcomes in contrast to cytoplasmic staining. MUC1 is also aberrantly overexpressed in 90% of human breast cancer, thus, representing a promising marker for breast cancer diagnosis and prognosis. Further research support is offered by an extensive range of antibodies for MUC1’s different isoforms. Visit our stand to explore more about depositing and accessing tools and technologies that support breast cancer research. Citation Format: Vera Moiseeva, Sara Omenetti, Abby Henney, Katie Valencia. CancerTools.org: Research Tools Supporting Breast Cancer Studies [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: Advances in Breast Cancer Research; 2023 Oct 19-22; San Diego, California. Philadelphia (PA): AACR; Cancer Res 2024;84(3 Suppl_1):Abstract nr A063.

Procedure for Determining the Upper Bound on Absolute Error for LVDT Sensors

This paper presents the procedure for determining the upper bound on absolute error (UBAE) for linear variable differential transformer (LVDT) sensors without inter-winding capacitance. This procedure is based on the results of a parametric identification of the sensor, from which the coefficient values of the corresponding transfer function are obtained. The UBAE is then determined by a simulation experiment using dedicated computational software. The error value and the shape of the input signal, which is constrained in magnitude, are determined. The calculations and additional validation of the procedure presented in this article are carried out using the MathCad 5.0 program. As the basis for the proposed procedure, we review issues related to the modelling of LVDT sensors without inter-winding capacitance. The solutions presented here can be used to determine the UBAE, which can then be used to assess the accuracy of these types of sensor and their mutual comparability in terms of dynamic accuracy.

In Silico and In Vitro Mapping of Receptor-Type Protein Tyrosine Phosphatase Receptor Type D in Health and Disease: Implications for Asprosin Signalling in Endometrial Cancer and Neuroblastoma

Background: Protein Tyrosine Phosphatase Receptor Type D (PTPRD) is involved in the regulation of cell growth, differentiation, and oncogenic transformation, as well as in brain development. PTPRD also mediates the effects of asprosin, which is a glucogenic hormone/adipokine derived following the cleavage of the C-terminal of fibrillin 1. Since the asprosin circulating levels are elevated in certain cancers, research is now focused on the potential role of this adipokine and its receptors in cancer. As such, in this study, we investigated the expression of PTPRD in endometrial cancer (EC) and the placenta, as well as in glioblastoma (GBM). Methods: An array of in silico tools, in vitro models, tissue microarrays (TMAs), and liquid biopsies were employed to determine the gene and protein expression of PTPRD in healthy tissues/organs and in patients with EC and GBM. Results: PTPRD exhibits high expression in the occipital lobe, parietal lobe, globus pallidus, ventral thalamus, and white matter, whereas in the human placenta, it is primarily localised around the tertiary villi. PTPRD is significantly upregulated at the mRNA and protein levels in patients with EC and GBM compared to healthy controls. In patients with EC, PTPRD is significantly downregulated with obesity, whilst it is also expressed in the peripheral leukocytes. The EC TMAs revealed abundant PTPRD expression in both low- and high-grade tumours. Asprosin treatment upregulated the expression of PTPRD only in syncytialised placental cells. Conclusions: Our data indicate that PTPRD may have potential as a biomarker for malignancies such as EC and GBM, further implicating asprosin as a potential metabolic regulator in these cancers. Future studies are needed to explore the potential molecular mechanisms/signalling pathways that link PTPRD and asprosin in cancer.

Design and Application of a Miniature Pneumatic Bellows Loading Device for In-Situ Tensile Testing inside the Scanning Electron Microscope

A miniature pneumatic bellows device was designed for in-situ testing inside the scanning electron microscope. The device uses a pneumatic actuator to load the specimen and displacement is directly monitored with a linear variable differential transformer sensor. Its application allowed the direct monitoring of microstructural and defect evolution in materials at the micro scale. The data produced by this testing device, in combination with measurements from micro digital image correlation, were successfully used to model the crystal plasticity of a martensitic/bainitic steel at the microstructural length scale.

Estimation of Soil Moisture Content Based on Fractional Differential and Optimal Spectral Index

Applying hyperspectral remote sensing technology to the prediction of soil moisture content (SMC) during the growth stage of soybean emerges as an effective approach, imperative for advancing the development of modern precision agriculture. This investigation focuses on SMC during the flowering stage under varying nitrogen application levels and film mulching treatments. The soybean canopy’s original hyperspectral data, acquired at the flowering stage, underwent 0–2-order differential transformation (with a step size of 0.5). Five spectral indices exhibiting the highest correlation with SMC were identified as optimal inputs. Three machine learning methods, namely support vector machine (SVM), random forest (RF), and back propagation neural network (BPNN), were employed to formulate the SMC prediction model. The results indicate the following: (1) The correlation between the optimal spectral index of each order, obtained after fractional differential transformation, and SMC significantly improved compared to the original hyperspectral reflectance data. The average correlation coefficient between each spectral index and SMC under the 1.5-order treatment was 0.380% higher than that of the original spectral index, with mNDI showing the highest correlation coefficient at 0.766. (2) In instances of utilizing the same modeling method with different input variables, the SMC prediction model’s accuracy follows the order: 1.5 order > 2.0 order > 1.0 order > 0.5 order > original order. Conversely, with consistent input variables and a change in the modeling method, the accuracy order becomes RF > SVM > BPNN. When comprehensively assessing model evaluation indicators, the 1.5-order differential method and RF method emerge as the preferred order differential method and model construction method, respectively. The R2 for the optimal SMC estimation model in the modeling set and validation set were 0.912 and 0.792, RMSEs were 0.005 and 0.004, and MREs were 2.390% and 2.380%, respectively. This study lays the groundwork for future applications of hyperspectral remote sensing technology in developing soil moisture content estimation models for various crop growth stages and sparks discussions on enhancing the accuracy of these different soil moisture content estimation models.

Analysis of Repetitive Positioning Accuracy of Parallel Mechanisms Based on Monte Carlo Method

Based on the inverse kinematics model of the Stewart parallel mechanism, a motion platform pose error model was constructed through differential transformation. The error model shows that the motion platform pose error comes from the length error of 6 struts and the position error of 12 hinge centers, with a total of 42 error terms. Monte Carlo simulation was used to analyze the effects of struts length error on the pose error of the moving platform. The results show that when the length error of the struts follows a normal distribution, the posture error of the moving platform approximately follows a normal distribution. Monte Carlo simulation is a highly effective error analysis method that can analyze the error distribution pattern of the mechanism before processing and assembly, which provides a method for error analysis of Stewart parallel mechanisms.

Modification of the Differential Transform Method With Search Direction Along the Spatial Axis

This paper presents a modification of the Differential Transform Method (ModDTM) formulated so that the search direction of the solution is along a spatial axis (x). It is shown that the method is applicable to a wide spectrum of (1 + 1) partial differential equations, integro partial differential equations, and integral equations. The solutions obtained are in the form of a Taylor series, the coefficients of which are determined by recursively operating a differential transform of the equation. These solutions are either exact and in closed form, or are good approximations. To illustrate the application of the method, as well as to show its versatility, examples are chosen from all the categories mentioned earlier. These include equations such as the nonlinear Fisher, combined KdV‐mKdV, Hunter–Saxton (H‐S), Fornberg–Whitham (FW), coupled systems such as the Whitham–Broer–Kaup (WBK) and the sine‐Gordon (s‐G) equations, integral equations, Volterra partial integro‐differential equations (PIDE), and linear and nonlinear versions of the (complex) Schröedinger equation. The Tables and 3D‐plots show the good rate of convergence of the obtained solutions with the exact ones. It is thus found that the new scheme is successful in producing satisfactory results as any other method in which the conventional approach of searching for solutions along the temporal axis is followed. Further, the procedures involved are simple, and hence may be operated with great ease.

Mathematical modeling of bending of a thin orthotropic plate clamped along the contour

Within the framework of Kirchhoff’s theory, a new approach to constructing a solution to the problem of modeling the bending of a thin rectangular orthotropic plate clamped along the contour, which is under the influence of a load normally distributed over its surface, is proposed. the solution to the inhomogeneous biharmonic equation for an orthotropic plate is obtained in the form of a partial sum of a double series in Chebyshev polynomials of the first kind. To find the coefficients in this expansion, the boundary value problem is reduced by the collocation method to a system of linear algebraic equations in matrix form using the properties of these polynomials. Based on matrix and differential transformations, expressions for bending moments and shearing forces are obtained. the results of calculations of the bending of the middle surface of the plate under different loads on the plate are presented, which demonstrate the effectiveness of the proposed approach.

SOFTWARE IMPLEMENTATION OF DECOMPOSITION METHODS FOR DETERMINING COMPLEX ONE-PARAMETER GENERALIZED INVERSE MOORE-PENROSE MATRICES (I)

Software implementations of analytical and numerical-analytical decomposition methods for determining complex one-parameter generalized inverse Moore-Penrose mat-rices are presented. Analytical methods are based on the first two Moore-Penrose con-ditions: in both cases the same solution to the problem is obtained, as it should be. Numerical-analytical methods are based on the obtained analytical relations and differential Pukhov transformations. Computational methods are implemented using modern infor-mation technologies, in particular Python programming language, NumPy and SymPy libraries. Software implementations consist of programs for entering the di-mensions and elements of a given matrix, implementation of an analytical method, de-termination of matrix elements derivatives and their values, calculation of matrix discretes and restoration of original matrices, as well as determining pseudo-inverse matrix. In addi-tion, the software implementation of the numerical-analytical method also contains a cus-tom implemented functions for determining the given matrix elements derivatives and for calculating the matrix discretes at the approximation center with a given scaling coeffi-cient. Algorithm flowcharts and Python source codes for all listed programs are presented. The software implementation is tested on a model example with a square matrix, for which an analytical solution is obtained using two approaches - the determination of the usual inverse one-parameter matrix and the decomposition method for determining the same matrix. This matrix is defined in exactly the same way as numerical-analytical decomposition methods.

Analysis of Random Difference Equations Using the Differential Transformation Method

The differential transformation method (DTM) is one of the best methods easily applied to linear and nonlinear difference equations with random coefficients. In this study, we apply the theorems related to the DTM to the given examples and investigate the behaviour of the approximate analytical solutions. The expected value, variance, coefficient of variation, and confidence intervals of the solutions of random difference equations obtained from discrete probability distributions such as uniform, geometric, Poisson, and binomial distributions will be calculated. Maple and MATLAB software packages are used to plot the solution graphs and also to interpret the solution behaviour.

LOCAL TIME FRACTIONAL REDUCED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING LOCAL TIME FRACTIONAL TELEGRAPH EQUATIONS

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency of the proposed method. The yielded results established the effectiveness of LTFRDTM as a reliable and solid approach for obtaining solutions of LTFTEs. The solutions coincided with the exact solution in the ordinary case when [Formula: see text]. It also required minimal amount of computational work and saved a lot of time.

Semi-analytical iteration techniques for solving differential equations

The main objective of this work is to use and compare some techniques, namely the variational Iteration method (VIM), Decomposition Method (ADM), Differential Transform Method (DTM), and Temimi and Ansari Method (TAM) for differential equations of the first and second orders that appeared in physics and engineering. Notably, the results indicate that the VIM gives a solution in series form, which converges to an exact solution;. At the same time, DTM is easy to apply but requires transformation, ADM does not need any transformation except the calculation of Adomain polynomials. TAM solved nonlinear problems in much easier iteration steps, which converge to the true solution without applying any assumption. The results obtained converge rapidly to the exact solution.

Development of a Creep Mechanical Frame Setting Based on Arcan Test

Abstract Structural adhesive bonding has several advantages compared to other assembly techniques such as welding and riveting. Yet, creep is an important long-term phenomenon that needs to be considered for the design of such joints. To investigate this aspect, experimental investigations are needed at the scale of the adhesive layer. At such a scale, Arcan setting presents several advantages compared to other existing tests (analysis of different loads, limited edge effects) and was thus chosen for the development of an experimental approach aiming at providing creep characterization of adhesive layers. A mechanically operated testing machine was thus developed and is presented herein. To measure local strains of the adhesive layer, two linear variable differential transformer sensors are fixed on a support placed on the beaks of the Arcan sample according to two directions: normal and tangential to the bonded surface. The developed system (mechanical system and strain measurement) proved to be successful in investigating the creep behavior of adhesive layers while varying several parameters (load levels, type of load, temperature, adhesive thickness).