Infinite Terms Research Articles

Applications of Taylor Series in Approximate Calculation and Limits

Taylor series plays a significant role in mathematics, especially in the fields of mathematical analysis, calculus and applied mathematics. It is not only a theoretical tool, but also an indispensable method in practical applications. For instance, it plays an important role in the fields of physics, engineering and economics. Taylor function uses function which is represented by a series of infinite additive terms, which are calculated by the derivative of the function at a certain point. This paper mainly uses literature research method and empirical research method to introduce the basic information and practical application of Taylor series. Some methods are given to solve related problems. This article demonstrates a brief introduction of Taylor series, then illustrates the steps of proof for Taylor series and provides the solutions of several practical examples. In the end, the paper focuses on the application of Taylor series on limits and analyzes some typical and interesting examples.

Space, body and presence

Abstract The trend towards remote interpreting has caused a radical change for interpreters worldwide, one that has shifted their workplace from well-known physical spaces to new digital spaces. Research to date has documented specific settings, language combinations and/or interpreting modes, and it has usually focused on certain forms of remote interpreting (e.g., video remote interpreting, video relay service). The combinations of different characteristics and factors in remote interpreting are almost infinite, however, and single terms fail to depict and cover all possible variations. This article proposes an analytical framework that uses conceptualisations of space, body and presence to analyse interpreting assignments in digital spaces, regardless of language combination, setting and mode. Two examples are used to illustrate the application of this framework: a hybrid conference and a meeting in virtual reality (VR), both of them involving sign language interpreting. The use of VR has undergone a substantial increase in the past few years, having developed rapidly. It is therefore possible that future interpreting assignments might take place in this new digital space. The application of these two scenarios shows that the analytical framework can be used both to reflect on past and to anticipate future assignments.

Enhancing Software Defect Prediction accuracy using Modified Entropy Calculation in Random Forest Algorithm

Imagine you are trying to classify software defect for a large dataset. How will you choose the best algorithm to do that? For the above problem we have various algorithms like Random Forest, Support Vector Machine, Neural Networks, Naive Bayes, K-Nearest Neighbours, Decision Tree, Logistic Regression etc. One of the most used methods is Random Forest algorithm, which uses multiple Decision Trees to make predictions. However, this algorithm relies on a complex calculation called Entropy, which measures the uncertainty in the data. Entropy is a function that uses natural logarithm which may be time consuming calculation. Is there a better way to calculate entropy? In this research, we have explored a different way to calculate the natural logarithm using the Taylor series expression. It is a series consisting of sum of infinite terms that approximates any function by using its derivatives. We further modified the Random Forest algorithm by replacing the natural logarithm with the Taylor series expression in the Entropy formula. We tested our modified algorithm on dataset and compared its performance with the original Entropy formula. We found that our modification in the algorithm has improved the accuracy of the algorithm on software defect prediction

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Solving Saint Venant torsion problems for rectangular beams using single finite Fourier sine transform method

This research presents the single Fourier sine transform method (SFSTM) for solving the Saint Venant torsion problem of rectangular prismatic bars. The problem is a common theme in the theory of elasticity of unrestrained torsion which was previously expressed by Prandtl using Prandtl stress functions ϕ(x,y) as a Poisson type nonhomogeneous partial differential equation (PDE) called the stress compatibility equation. In this work the SFSTM was applied to the stress compatibility equation, converting the PDE to an easier to solve ordinary differential equation (ODE) in the transformed domain. The boundary conditions were used to find the integration constant and inversion was used to find the solution in the physical domain. The non vanishing stresses and torsional moments were thus found as a single series of infinite terms with rapid convergence. The maximum stresses and moments were found in standard form in terms of torsional parameters which were tabulated for various ratios of the cross-sectional dimensions. A comparison of the torsional parameters with previous results show that the present results are identical with previous results illustrating the accuracy of the SFSTM used. The sine kernel of the SFSTM satisfies the boundary conditions of the problem and contributed to the exact solution obtained. The SFSTM simplified the PDE to an ODE which is simpler to solve.

On the uniform stability of a thermoelastic Timoshenko system with infinite memory

<abstract><p>The present research is aim at investigating a thermoelastic Timoshenko system with an infinite memory term on the shear force while the bending moment is under the influence of a thermoelastic dissipation governed by Fourier's law. We prove that the system's stability holds for a broader class of relaxation functions. Under this class of relaxation functions $ h $ at infinity, we establish a relation between the decay rate of the solution and the growth of $ h $ at infinity. Moreover, we drop the boundedness assumptions on the history data. We employ Neumann-Dirichlet-Neumann boundary conditions for our result. In comparison to the bulk of results in the literature, which frequently enforce the "equal-wave-speed" constraint, the present result shows that the infinite memory of the beam and the thermal damping are strong enough to guarantee stability without any conditions on the parameters.</p></abstract>

Query Model Framework Design for Conservation History and Endowments Database: A Case Study on the Digitization of the Sumedang Larang Kingdom’s History and Endowments in Indonesia

The use of structured query formulation languages is a method for gaining access to data in information management systems. Writing structured queries is a powerful method of data retrieval, and end users can generate complex database queries by becoming proficient in a particular query language. Digitization of database information systems, history and endowment, and technology conservation are some examples of database conservation techniques in the process of displaying data that require optimal query techniques. Historical and endowment properties comply with a certain set of laws governing waqf recipients, which sets them apart from heritage and cultural assets. Trusted property typically comprises real estate, valuables, or both and has an infinite preservation term so long as it may still be put to use. A solid information technology infrastructure is essential for ensuring comprehensive data security. Given the availability of knowledge and the quick development of information technology, ensuring its sustainability is a challenge. This research is aimed at preserving historical and waqf databases based on the case study of the Kingdom of Sumedang Larang (KSL), Indonesia, through examination of the database, synchronization of digital data with physical data, as well as looking at its relationship with organizations and management. The proposed method, namely the Design and Analysis of Database Query Model for Preservation Information System (DAQMP), creates a recurring query model from the original tabular data placed in the next table as a special table check for data specifications containing important data. The results must be accurately aligned between digital and physical records regularly to serve as a record of any changes and as evidence to determine whether data has been corrupted, altered, or lost.

Evaluation of Electromagnetic Fields of Extremely Low-Frequency Horizontal Electric Dipoles at Sea–Air Boundaries

The technologies of undersea detection and communication, seabed sensor networks, and geophysical detection using electromagnetic waves have emerged as research focal points within the field of marine science and engineering. However, most studies have focused on the propagation of electromagnetic fields over long distances within the shallow “sea-seabed” environment. This paper introduces a quasi-static approximation method to address the Sommerfeld numerical integration challenge within the near-field region, employing the horizontal electric dipole (HED) as a model. It derives the Sommerfeld numerical integral expressions under conditions where the wave-number ratio at the “seawater-air” boundary does not adhere to the requirement of |k0/k1| << 1 (where subscripts 0 and 1 denote seawater and air media, respectively). Building upon this, the paper simplifies the Bessel-Fourier infinite integral term within the integral expression to obtain Sommerfeld numerical integral approximations for the propagation of electromagnetic fields in the near region of extremely low frequency (ELF) within seawater. The study further conducts simulations and calculations to determine amplitude variations in electromagnetic field intensity generated by an ELF HED at different frequencies, dipole heights, and observation point depths. It concludes with an analysis of electromagnetic field propagation characteristics at the seawater-air boundary. Experimental findings highlight the lateral wave as the primary mode of electromagnetic wave propagation at this interface.

Well-Posedness and Exponential Stability of the Von Kármán Beam With Infinite Memory

In the present work, we consider a one-dimensional Von kármán beam with infinite memory, we establish the well-posedness of the system using semigroup theory and prove the exponential stability under some conditions on the kernel of the infinite memory term.

On nonlinear stability analysis of saturated embedded porous nanobeams

In this work, the nonlinear stability behaviors of the saturated porous nanobeams embedded in an elastic foundation are investigated. The restrained nanobeam is modeled using geometrical nonlinear equations in conjunction with the constitutive law of saturation. Three patterns for saturation along the thickness of the nanobeam are considered as porous/monotonous, porous/nonlinear symmetric, and porous/nonlinear non-symmetric distributions. The infinite terms of the power series are discretized thanks to Stokes' transformation and trigonometric series. Then, the effects of saturation and nonlinearity on buckling loads are studied by considering the restrained boundary conditions. Moreover, the nonlinear results are validated with the saturated nanobeam with the rigid supporting conditions.

Stability analysis of arbitrary restrained nanobeam embedded in an elastic medium via nonlocal strain gradient theory

A novel stability model is analytically reformulated for the nano-sized beam resting on a one-parameter elastic foundation. The stability solution is based on the nonlocal strain gradient elasticity theory. To corporate the small size effects, two small scale parameters are introduced. The six-order ordinary differential form of the buckling equation, together with two force boundary conditions, are utilized to examine the stability equation in terms of lateral deflection. The infinite terms of linear equations are discretized with the help of the Stokes’ transformation and Fourier sine series. The present work can investigate the effects of elastic spring parameters at the ends, nonlocal properties, elastic medium properties, strain gradient parameter, and buckling behavior of the nanobeam. The predictions of the proposed analytical model with deformable boundary conditions are in agreement with those available in the scientific literature for the nanobeam on elastic foundation based on a closed form of solution. The presence of the deformable conditions, elastic foundation, nonlocal, and strain gradient properties change the buckling loads and buckling mode shapes.

Infinite memory effects on the stabilization of a biharmonic Schrödinger equation

This paper deals with the stabilization of the linear biharmonic Schrödinger equation in an $n$-dimensional open bounded domain under Dirichlet–Neumann boundary conditions considering three infinite memory terms as damping mechanisms. We show that depending on the smoothness of initial data and the arbitrary growth at infinity of the kernel function, this class of solution goes to zero with a polynomial decay rate like t − n depending on assumptions about the kernel function associated with the infinite memory terms.

Theoretical and computational decay results for a Bresse system with one infinite memory in the longitudinal displacement

<p style='text-indent:20px;'>In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory term acting in the third equation (longitudinal displacements). Under a general condition on the memory kernel (relaxation function), we establish a decay estimate of the energy of the system. Our decay result extends and improves some decay rates obtained in the literature such as the one in [<xref ref-type="bibr" rid="b27">27</xref>], [<xref ref-type="bibr" rid="b4">4</xref>], [<xref ref-type="bibr" rid="b33">33</xref>], [<xref ref-type="bibr" rid="b58">58</xref>] and [<xref ref-type="bibr" rid="b34">34</xref>]. The proof is based on the energy method together with convexity arguments. Numerical simulations are given to illustrate the theoretical decay result.</p>

Holographic nonlocal rotating observables and their renormalization

We analyze nonlocal rotating observables in holography corresponding to spinning bound states. To renormalize their energies and momenta we suggest and discuss different holographic renormalization schemes motivated by the static nonlocal observables. Namely the holographic renormalization and the rotating color singlet mass subtraction scheme. In the holographic renormalization we identify the infinite boundary terms and subtract them. In the mass subtraction scheme we evaluate the energy of a spinning trailing string corresponding to the color charged singlet which experiences dragging phenomena and we subtract it from the energy of the bound state to obtain the renormalized finite energy. Then we apply our generic framework to certain strongly coupled thermal theories with broken rotational symmetry. We find numerical solutions corresponding to spinning bound states with a fixed size while varying their angular frequency. By applying numerically the renormalization schemes, we find that there is a critical frequency where the bound state ceases to exist or dissociates. We also note that bound states require lower angular frequencies to dissociate when the theory has less symmetry.

Rank-to-rank embeddings, Laver tables, and Galton-Watson processes

Laver introduced a family of finite left self-distributive algebras which he used to approximate the algebra generated by a rank-to-rank elementary embedding. This article will demonstrate how to unambiguously evaluate randomly generated, potentially infinite terms in these finite algebras. Here random generation refers to the Galton-Watson process associated to a probability 0<p<1.

Spacetime instability and quantum gravity as low energy effective field theory

We discuss spacetime instability for effective field theories of quantum gravity. The effective action of gravity introduces infinite higher derivative curvature terms $R^{2}, { R }_{ \mu \nu }{ R }^{ \mu \nu }, R_{\mu\nu\kappa\lambda} R^{\mu\nu\kappa\lambda}\dots$. Although these higher derivative curvature terms are indispensable to construct the self-consistent renormalizable theory of quantum gravity, they lead to several pathologies. We clearly show that even if they are written as the Planck-suppressed operators they lead to serious consequences and and de Sitter or radiation-dominated Universe is highly unstable. We show that the couplings of these higher derivative curvatures must satisfy $\left|a_{1,2,3}\right|\gtrsim 10^{118}$ to be consistent with the cosmological observations. Thus, the standard effective field theories of quantum gravity fail to describe the observed Universe unless introducing a specific technique dealing with the higher derivative curvature terms.

Exponential stabilization of a flexible structure with a local interior control and under the presence of a boundary infinite memory

In this paper, we deal with the interior stabilization problem of a flexible structure governed by a hyperbolic partial differential equation coupled to two ordinary differential equations. Contrary to the previous works on the system, the boundary control is subject to the presence of an infinite memory term. In order to deal with such a nonlocal term, the minimal state approach is invoked. Specifically, a localized interior control is proposed in order to compensate the infinite memory effect. Thereafter, reasonable assumptions on the memory kernel are evoked so that the closed‐loop system is shown to be well‐posed thanks to semigroups theory of linear operators. Furthermore, the resolvent method is used to establish the exponential stability of the system.

Order-theoretic Trees: Monadic Second-order Descriptions and Regularity

An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch (a maximal linearly ordered subset) can be any countable linear order. Such generalized infinite trees yield convenient definitions of the rank-width and the modular decomposition of countable graphs. We define an algebra based on only four operations that generate up to isomorphism and via infinite terms these order-theoretic trees and forests.We prove that the associated regular objects, i.e., those defined by regular terms, are exactly the ones that are the unique models of monadic second-order sentences. We adapt some tools that we have used in a previous article for proving a similar result for join-trees, i.e., for order-theoretic trees such that any two nodes have a least upper-bound.

Fast multi-qubit global-entangling gates without individual addressing of trapped ions

We propose and study ways speeding up of the entangling operations in the trapped ions system with high fidelity. First, we find a scheme to increase the speed of a two-qubit gate without the limitation of trap frequency, which was considered as the fundamental limit. Second, we study the fast gate scheme for entangling more than two qubits simultaneously. We apply the method of applying multiple frequency components on laser beams for the gate operations. In particular, in order to avoid infinite terms from the coupling to carrier transition, we focus on the phase-insensitive gate scheme here. We carefully study the effect of large excitation of motional mode beyond the limit of Lamb–Dicke approximation by including up to second order terms of the Lamb–Dicke parameter. We study the speed limit of multi-qubit global entangling gates without individual addressing requirements. Furthermore, our gates can be made insensitive to the fluctuation of initial motional phases which are difficult to stabilise in the phase-insensitive gate scheme.

Stability of multi-dimensional nonlinear piezoelectric beam with viscoelastic infinite memory

The longtime behavior of a kind of fully magnetic effected nonlinear multi-dimensional piezoelectric beam with viscoelastic infinite memory is considered. The well-posedness of this nonlinear coupled PDEs’ system is showed by means of the semigroup theories and Banach fixed-point theorem. Based on frequency-domain analysis, it is proved that the corresponding coupled linear system can be indirectly stabilized exponentially by only one viscoelastic infinite memory term, which is located on one equation of these strongly coupled PDEs. Then, the exponential decay of the solution to the nonlinear coupled PDEs’ system is established by the energy estimation method under certain conditions.