Algebraic Properties for Molecular Structure of Magnesium Iodide

As an inorganic chemical,magnesium iodide has a significant crystalline structure.It is a complex and multifunctional substance that has the potential to be used in a wide range of medical advancements.Molecular graph theory,on the other hand,provides a sufficient and cost-effective method of investigating chemical structures and networks.M-polynomial is a relatively new method for studying chemical networks and structures in molecular graph theory.It displays numerical descriptors in algebraic form and highlights molecular features in the form of a polynomial function.We present a polynomials display of magnesium iodide structure and calculate several M-polynomials in this paper,particularly the M-polynomials of the augmented Zagreb index,inverse sum index,hyper Zagreb index and for the symmetric division index.

Two kinds of B-basis of the algebraic hyperbolic space

In this paper, two new kinds of B-basis functions called algebraic hyperbolic （AH） Bézier basis and AH B-Spline basis are presented in the space Гk=span{ l,t ……f^k-3,sinht,cosht}, in which K is an arbitrary integer larger than or equal to 3. They share most optimal properties as those of the Bézier basis and B-Spline basis respectively and can represent exactly some remarkable curves and surfaces such as the hyperbola, catenary, hyperbolic spiral and the hyperbolic paraboloid. The generation of tensor product surfaces of the AH B-Spline basis have two forms： AH B-Spline surface and AH T-Spline surface.

Norm-Based Adaptive Coefficient ZNN for Solving the Time-Dependent Algebraic Riccati Equation

Dear Editor, The time-dependent algebraic Riccati equation(TDARE) problem is applied to many optimal control industrial applications. It is susceptible to interference from measurement noises in the virtual environment, which current methods cannot effectively address. A normbased adaptive coefficient zeroing neural network(NACZNN) model to solve the TDARE problem is proposed.

Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms.We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system.Through the maximal orthogonal projection concept,to minimize a merit function within a selected interval of splitting parameters,the optimal parameters can be quickly determined.In each step,a linear system is solved by the Gaussian elimination method,and the whole iteration procedure is convergent very fast.Several numerical tests show the high performance of the optimal split-linearization iterative method(OSLIM).

Solving Algebraic Problems with Geometry Diagrams Using Syntax-Semantics Diagram Understanding

Solving Algebraic Problems with Geometry Diagrams(APGDs)poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects.Problems typically involve both textual descriptions and geometry diagrams,requiring a joint understanding of these modalities.Although considerable progress has been made in solving math word problems,research on solving APGDs still cannot discover implicit geometry knowledge for solving APGDs,which limits their ability to effectively solve problems.In this study,a systematic and modular three-phase scheme is proposed to design an algorithm for solving APGDs that involve textual and diagrammatic information.The three-phase scheme begins with the application of the statetransformer paradigm,modeling the problem-solving process and effectively representing the intermediate states and transformations during the process.Next,a generalized APGD-solving approach is introduced to effectively extract geometric knowledge from the problem’s textual descriptions and diagrams.Finally,a specific algorithm is designed focusing on diagram understanding,which utilizes the vectorized syntax-semantics model to extract basic geometric relations from the diagram.A method for generating derived relations,which are essential for solving APGDs,is also introduced.Experiments on real-world datasets,including geometry calculation problems and shaded area problems,demonstrate that the proposed diagram understanding method significantly improves problem-solving accuracy compared to methods relying solely on simple diagram parsing.

Temperature-Triggered Hardware Trojan Based Algebraic Fault Analysis of SKINNY-64-64 Lightweight Block Cipher

SKINNY-64-64 is a lightweight block cipher with a 64-bit block length and key length,and it is mainly used on the Internet of Things(IoT).Currently,faults can be injected into cryptographic devices by attackers in a variety of ways,but it is still difficult to achieve a precisely located fault attacks at a low cost,whereas a Hardware Trojan(HT)can realize this.Temperature,as a physical quantity incidental to the operation of a cryptographic device,is easily overlooked.In this paper,a temperature-triggered HT(THT)is designed,which,when activated,causes a specific bit of the intermediate state of the SKINNY-64-64 to be flipped.Further,in this paper,a THT-based algebraic fault analysis(THT-AFA)method is proposed.To demonstrate the effectiveness of the method,experiments on algebraic fault analysis(AFA)and THT-AFA have been carried out on SKINNY-64-64.In the THT-AFA for SKINNY-64-64,it is only required to activate the THT 3 times to obtain the master key with a 100%success rate,and the average time for the attack is 64.57 s.However,when performing AFA on this cipher,we provide a relation-ship between the number of different faults and the residual entropy of the key.In comparison,our proposed THT-AFA method has better performance in terms of attack efficiency.To the best of our knowledge,this is the first HT attack on SKINNY-64-64.

Uncovering Patterns Beneath the Surface:A Glimpse into the Applications of Algebraic Topology

This paper explores the significant impact of algebraic topology on diverse real-world applications.Starting with an introduction to the historical development and essence of algebraic topology,it delves into its applications in neuroscience,physics,biology,engineering,data analysis,and Geographic Information Systems(GIS).Remarkable applications incorporate the analysis of neural networks,quantum mechanics,materials science,and disaster management,showcasing its boundless significance.Despite computational challenges,this study outlines prospects,emphasizing the requirement for proficient algorithms,noise robustness,multi-scale analysis,machine learning integration,user-friendly tools,and interdisciplinary collaborations.In essence,algebraic topology provides a transformative lens for uncovering stowed-away topological structures in complex data,offering solutions to perplexing problems in science,engineering,and society,with vast potential for future exploration and innovation.

WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY

Algebraic reflexivity introduced by Hadwin is related to linear interpolation.In this paper,the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced.Some properties of them are obtained and some relations between them revealed

On the growth of transcendental entire solutions of algebraic differential equations

In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.

Algebraic processing technique for extracting frequency-dependent shear-wave splitting parameters in an anisotropic medium

Based on the dual source cumulative rotation technique in the time-domain proposed by Zeng and MacBeth（1993）,a new algebraic processing technique for extracting shear-wave splitting parameters from multi-component VSP data in frequency-dependent medium has been developed.By using this dual source cumulative rotation technique in the frequency-domain（DCTF）,anisotropic parameters,including polarization direction of the shear-waves and timedelay between the fast and slow shear-waves,can be estimated for each frequency component in the frequency domain.It avoids the possible error which comes from using a narrow-band filter in the current commonly used method.By using synthetic seismograms,the feasibility and validity of the technique was tested and a comparison with the currently used method was also given.The results demonstrate that the shear-wave splitting parameters frequency dependence can be extracted directly from four-component seismic data using the DCTF.In the presence of larger scale fractures,substantial frequency dependence would be found in the seismic frequency range,which implies that dispersion would occur at seismic frequencies.Our study shows that shear-wave anisotropy decreases as frequency increases.

ON ALGEBRAIC SOLUTIONS OF THE SECOND-ORDER COMPLEX DIFFERENTIAL EQUATIONS WITH ENTIRE ALGEBRAIC ELEMENT COEFFICIENTS

The main purpose of this paper is to study the problems on the existence of algebraic solutions for some second-order complex differential equations with entire algebraic function element coeifficients. Several theorems on the existence of solutions are obtained, which perfect the solution theory of linear complex differential equations.

Vibrational Spectra of Nonrigid Molecule HCP in an Algebraic Model

An algebraic Harniltonian for the two coupled nonlinear vibrations of highly excited nonrigid molecule HCP was presented. The Hamiltonian reduces to the conventional one in a limit which was expressed in terms of harmonic oscillator operators. It showed that the algebraic model can better reproduce the data than the conventional model by fitting the observed data of HCP.

On the First-degree Algebraic Equation of the Generalized Quaternion

In this paper, by using the matrix representation of the generalized quaternion algebra, we discussed solution problem for two classes of the first_degree algebraic equation of the generalized quaternion and obtained critical conditions on existence of a unique solution, infinitely many solutions or nonexistence any solution for the two classes algebraic equation.

ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES

This article investigates the algebraic differential independence concerning the Euler Γ-function and the function F in a certain class F which contains Dirichlet L-functions,L-functions in the extended Selberg class, or some periodic functions. We prove that the EulerΓ-function and the function F cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions Ø with ρ(Ø) < 1.

Numerical Simulation of the Whole Three-Dimensional Flow in a Stirred Tank with Anisotropic Algebraic Stress Model

In accordance to the anisotropic feature of turbulent flow, ananisotropic algebraic stress model is adopted to predict theturbulent flow field and turbulent characteristics generated by aRushton disc turbine with the improved inner-outer iterativeprocedure. The predicted turbulent flow is compared with experimentaldata and the simulation by the standard k-ε turbulence model. Theanisotropic algebraic stress model is found to give better predictionthan the standard k-ε turbulence model. The predicted turbulent flowfield is in accordance to experimental data and the trend of theturbulence intensity can be effectively reflected in the simulation.

THE GROWTH OF SOLUTIONS OF SYSTEMS OF COMPLEX NONLINEAR ALGEBRAIC DIFFERENTIAL EQUATIONS

We investigate the problem of growth order of solutions of a type of systems of non-linear algebraic differential equations, and extend some results of the growth order of solutions of algebraic differential equations to systems of algebraic differential equations.

Algebraic Method‑Based Point‑to‑Point Trajectory Planning of an Under‑Constrained Cable‑Suspended Parallel Robot with Variable Angle and Height Cable Mast

To avoid impacts and vibrations during the processes of acceleration and deceleration while possessing flexible working ways for cable-suspended parallel robots(CSPRs),point-to-point trajectory planning demands an under-constrained cable-suspended parallel robot(UCPR)with variable angle and height cable mast as described in this paper.The end-effector of the UCPR with three cables can achieve three translational degrees of freedom(DOFs).The inverse kinematic and dynamic modeling of the UCPR considering the angle and height of cable mast are completed.The motion trajectory of the end-effector comprising six segments is given.The connection points of the trajectory segments(except for point P3 in the X direction)are devised to have zero instantaneous velocities,which ensure that the acceleration has continuity and the planned acceleration curve achieves smooth transition.The trajectory is respectively planned using three algebraic methods,including fifth degree polynomial,cycloid trajectory,and double-S velocity curve.The results indicate that the trajectory planned by fifth degree polynomial method is much closer to the given trajectory of the end-effector.Numerical simulation and experiments are accomplished for the given trajectory based on fifth degree polynomial planning.At the points where the velocity suddenly changes,the length and tension variation curves of the planned and unplanned three cables are compared and analyzed.The OptiTrack motion capture system is adopted to track the end-effector of the UCPR during the experiment.The effectiveness and feasibility of fifth degree polynomial planning are validated.

Algebraic Structure of the Dynamical Equations of Holonomic Mechanical System in Relative Motion

Aim To study an algebraic of the dynamical equations of holonomic mechanical systems in relative motion. Methods The equations of motion were presented in a contravariant algebraic form and an algebraic product was determined. Results and Conclusion The equations a Lie algebraic structure if any nonpotential generalized force doesn't exist while while the equations possess a Lie-admissible algebraic structure if nonpotential generalized forces exist .

Extended Fan's Algebraic Method and Its Application to KdV and Variant Boussinesq Equations

An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinearpartial differential equations.The key idea of this method is to introduce an auxiliary ordinary differential equationwhich is regarded as an extended elliptic equation and whose degree Υ is expanded to the case of r>4.The efficiency ofthe method is demonstrated by the KdV equation and the variant Boussinesq equations.The results indicate that themethod not only offers all solutions obtained by using Fu's and Fan's methods,but also some new solutions.

The Algebraic Criteria for the All-delay Stabilityof Two-dim ensional Degenerate Differential System swith Delay

This paper give the algebraic criteria for all delay stability of two dimensional degenerate differential systems with delays and give two examples to illustrate the use of them.