Terms Of Exponential Functions Research Articles

ANALISIS KEMAMPUAN PEMECAHAN MASALAH MATEMATIS PADA MATERI PERPANGKATAN DAN BENTUK AKAR DITINJAU DARI MINAT BELAJAR SISWA KELAS IX DI SMPS KRISTEN BNKP TELUKDALAM TA. 2022/2023

The problem of this research is the ability of students' mathematical problem solving which is influenced by interest in learning. The aims of this study were 1) to see how well students could apply problem-solving skills, 2) to see students' learning interests and 3) to understand what obstacles students experienced in working on problem-solving ability tests. This research uses a qualitative approach and applies descriptive research. The subjects in this study were 22 students in class VIII of Christian Private Middle School BNKP Telukdalam 2022/2023 Academic Year. There are three research instruments used, namely tests of students' problem solving abilities, questionnaires of student interest in learning and interviews. Data analysis techniques according to Miles and Huberman. The results showed that the students' abilities in terms of exponentials and root forms were sufficient, where there were 17 students who were in the sufficient category and 5 in the insufficient category. The results of the analysis of students' interest in learning were 2 students with high learning interest, 14 students with moderate learning interest and 6 students with low learning interest. So it can be concluded that students' problem-solving abilities are still quite sufficient and students' learning interest is in the medium category. So the suggestions put forward by researchers are 1) it is hoped that students often practice problem solving skills and increase their interest in learning mathematics, 2) Teachers are more aware of the challenges facing students.

Euler–Poisson equations of a dancing spinning top, integrability and examples of analytical solutions

Equations of a rotating body with one point constrained to move freely on a plane (dancing top) are deduced from the Lagrangian variational problem. They formally look like the Euler–Poisson equations of a heavy body with fixed point, immersed in a fictitious gravity field. Using this analogy, we have found examples of analytical solutions for the case of a heavy symmetrical dancing top. They describe the motions with center of mass keeping its height fixed above the supporting plane. General solution to equations of a dancing top in terms of exponential of Hamiltonian field is given. An extra constraint, that take into account the reaction of supporting plane, leads to modification of the canonical Poisson structure and therefore the integrability according to Liouville is under the question.

Phase-type representations of stochastic interest rates with applications to life insurance

The purpose of the present paper is to incorporate stochastic interest rates into a matrix-approach to multi-state life insurance, where formulas for reserves, moments of future payments and equivalence premiums can be obtained as explicit formulas in terms of product integrals or matrix exponentials. To this end we consider the Markovian interest model, where the rates are piecewise deterministic (or even constant) in the different states of a Markov jump process, and which is shown to integrate naturally into the matrix framework. The discounting factor then becomes the price of a zero-coupon bond which may or may not be correlated with the biometric insurance process. Another nice feature about the Markovian interest model is that the price of the bond coincides with the survival function of a phase-type distributed random variable. This, in particular, allows for calibrating the Markovian interest rate models using a maximum likelihood approach to observed data (prices) or to theoretical models like e.g. a Vasiček model. Due to the denseness of phase-type distributions, we can approximate the price behaviour of any zero-coupon bond with interest rates bounded from below by choosing the number of possible interest rate values sufficiently large. For observed data models with few data points, lower dimensions will usually suffice, while for theoretical models the dimensionality is only a computational issue.

Green’s function for thick laminated composite plates with coupled stretching-bending and transverse shear deformation

In this paper, the Green’s function of a thick laminated composite plate is derived for the first time in the literature. The plate can be unsymmetric with respect to the mid-plane such that the coupling of in-plane stretching and out-of-plane bending deformations occurs. Also, the plate is thick enough that the transverse shear deformation cannot be ignored. Through the plane wave decomposition method, the partial differential governing equations based upon the first order shear deformation plate theory can be reorganized into the ordinary differential equations in terms of the transformed variable. With the governing equations reorganized into matrix form, the Green’s functions are expressed in terms of matrix exponential. Through eigen-decomposition, the matrix exponential including its related integrals can be evaluated explicitly. The explicit solutions of Green’s functions in transformed domain are obtained accordingly. In order to improve the accuracy and efficiency of numerical integration for the required transform integrals, a special quadrature rule which can eliminate the integral singularity is suggested. To have a fair verification and to extend the applicability of newly derived Green’s functions, the associated boundary element method is also introduced in this study.

Accurate Solution for the Pantograph Delay Differential Equation via Laplace Transform

The Pantograph equation is a fundamental mathematical model in the field of delay differential equations. A special case of the Pantograph equation is well known as the Ambartsumian delay equation which has a particular application in Astrophysics. In this paper, the Laplace transform is successfully applied to solve the Pantograph delay equation. The solution is obtained in a closed series form in terms of exponential functions. This closed form reduces to the corresponding solution in the relevant literature for the Ambartsumian delay equation as a special case. In addition, the convergence of the obtained series is proved theoretically and validated graphically. Furthermore, the accuracy of the numerical results are estimated through several computations of the residual errors. It is shown that such residuals tend to zero, even in a huge domain. The obtained results reveal that the Laplace transform is a powerful approach to solve linear delay differential equations, including the Pantograph model.

Geodesic motion on the symplectic leaf of SO(3) with distorted e(3) algebra and Liouville integrability of a free rigid body

The solutions to the Euler–Poisson equations are geodesic lines of SO(3) manifold with the metric determined by inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We calculate its explicit form and confirm that it differs from the algebra e(3). The obtained Poisson brackets are used to demonstrate the Liouville integrability of a free rigid body. The general solution to the Euler–Poisson equations is written in terms of exponential of the Hamiltonian vector field.

Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation

In this paper, the pantograph delay differential equation y′(t)=ay(t)+byct subject to the condition y(0)=λ is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for λ=1, is well-posed if c<1, but not if c>1. In addition, the solution is available in the form of a standard power series when λ=1. In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for c∈(−1,1) such that ba<1 and it also converges for c>1 when a<0. For c=−1, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity 2πb2−a2 when b>a. The current results are introduced for the first time and have not been reported in the relevant literature.

An analytical solution for vibration response of CNT/GPL/fibre/polymer hybrid composite micro/nanoplates

In the present article, a closed-form solution is carried out for the vibration response of CNT/GPL/fiber/polymer hybrid composite macro/micro/nanoplates resting on elastic support. Multi-layered fiber/polymer hybrid plates with either functionally graded carbon nanotube reinforced composite (FG-CNTRC) material or functionally graded graphene platelets reinforced composite (FG-GPLRC) material are two different types of plate considered here. The distribution of reinforcing nanocomposites can be uniform or functionally graded through the plate layers. The governing vibration equations are developed in the framework of the nonlocal elasticity theory of Eringen and using the first-order shear deformation plate theory. In the analytical solution procedure, the partial differential equations of motion are simplified to a series of homogeneous ordinary differential equations using a displacement field for Levy-type of the boundary conditions. The general solution of the obtained equations is expressed in terms of exponential functions. Using that the displacements should satisfy the remaining boundary conditions, the vibrational frequencies are derived. Finally, the influences of different parameters including geometry and material properties such as nonlocal parameter, number and angle of layers, volume fraction, and type of distribution of reinforcing nanocomposites are investigated and discussed. Natural frequencies for various boundary conditions are illustrated. The numerical results illustrate that the effects of the nonlocal parameter on the frequency decrease when one of the boundaries is free, and these effects increase when one of the boundaries is clamped for both FG-CNTRC and FG-GPLRC plates.

On the computation of Hopf 2-cocycles with an example of diagonal type

AbstractWe present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$ , a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$ . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.

Impact of Navier’s slip and chemical reaction on the hydromagnetic hybrid nanofluid flow and mass transfer due to porous stretching sheet

Hybrid nanofluids (HNFs) comprise combinations of different nanoparticles suspended in base fluid. Applications of such nanofluids are rising in the areas of energy and biomedical engineering including smart (functional) coatings. Motivated by these developments, the present article examines theoretically the magnetohydrodynamic coating boundary layer flow of HNFs from a stretching sheet under the transverse magnetic field in porous media with chemically reactive nanoparticles. Darcy’s law is deployed. Momentum slips of both first and second order are included as is solutal slip. The transformed boundary value problem is solved analytically. Closed form solutions for velocity are derived in terms of exponential functions and for the concentration field in terms of incomplete Gamma functions by the application of the Laplace transformation technique. The influence of selected parameters e.g. suction/injection, magnetic field and slips on velocity and concentration distributions are visualized graphically. Concentration magnitudes are elevated with stronger magnetic field whereas they are suppressed with greater wall solutal slip. Magnetic field suppresses velocity and increases the thickness of the hydrodynamic boundary layer. The flow is accelerated with reduction in inverse Darcy number and stronger suction direct to reduce in skin friction. The concentration magnitudes are boosted with magnetic field whereas they are depleted with increasing solutal slip. The analysis provides a good foundation for further investigations using numerical methods.

Crystal melting, BPS quivers and plethystics

We study the refined and unrefined crystal/BPS partition functions of D6-D2-D0 brane bound states for all toric Calabi-Yau threefolds without compact 4-cycles and some non-toric examples. They can be written as products of (generalized) MacMahon functions. We check our expressions and use them as vacuum characters to study the gluings. We then consider the wall crossings and discuss possible crystal descriptions for different chambers. We also express the partition functions in terms of plethystic exponentials. For ℂ3 and tripled affine quivers, we find their connections to nilpotent Kac polynomials. Similarly, the partition functions of D4-D2-D0 brane bound states can be obtained by replacing the (generalized) MacMahon functions with the inverse of (generalized) Euler functions.

The cutoff phenomenon for the stochastic heat and wave equation subject to small Lévy noise

This article generalizes the small noise cutoff phenomenon obtained recently by Barrera, Högele and Pardo (JSP2021) to the mild solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and Lévy noises in the Wasserstein distance. The methods rely on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the multiplicative stochastic solution flows in terms of stochastic exponentials.

Simplified approach to estimation of organ absorbed doses for patients undergoing abdomen and pelvis CT examination

The volumetric computed tomography (CT) dose index (CTDIvol) is the measure of output displayed on CT consoles relating to dose within a standard phantom. This gives a false impression of doses levels within the tissues of smaller patients in Southeast Asia. A size-specific dose estimate (SSDE) can be calculated from the CTDIvol to provide an assessment of doses at specific positions within a scan using size-specific conversion factors. SSDE is derived using the water equivalent diameter (D w) of the patient, but calculation of D w requires sophisticated computer software. This study aimed to evaluate relationships between D W and effective diameter (D Eff), which can be measured more readily, in order to estimate SSDE at various positions within a routine clinical abdomen and pelvis CT examination for Thai patients. An in-house ImageJ algorithm was developed to measure D w, effective diameter (D Eff), and SSDE on CT slices located at the heart, liver, kidneys, colon, and bladder, on 181 CT examinations of abdomen and pelvis. Relationships between D Eff and D w were determined, and values of organ absorbed dose using D Eff were estimated. This approach was validated using a second cohort of 54 patients scanned on a different CT scanner. The results revealed that ratios between D Eff and D w at the heart level were 1.11–1.13 and those for the others were about 1.00. Additionally, the SSDE/CTDIvol ratio was estimated for each organ in terms of exponential functions using the relationships between D w and D Eff for individual organs. In summary, this study proposed a simple method for estimation of organ absorbed doses for Southeast Asian patients undergoing abdomen and pelvis CT examinations where sophisticated computer software is not available.

On the role of Hermite-like polynomials in the Fock representations of Gaussian states

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.

A new Wilson line-based action for gluodynamics

We perform a canonical transformation of fields that brings the Yang-Mills action in the light-cone gauge to a new classical action, which does not involve any triple-gluon vertices. The lowest order vertex is the four-point MHV vertex. Higher point vertices include the MHV and overline{mathrm{MHV}} vertices, that reduce to the corresponding amplitudes in the on-shell limit. In general, any n-leg vertex has 2 ≤ m ≤ n − 2 negative helicity legs. The canonical transformation of fields can be compactly expressed in terms of path-ordered exponentials of fields and their functional derivative. We apply the new action to compute several tree-level amplitudes, up to 8-point NNMHV amplitude, and find agreement with the standard methods. The absence of triple-gluon vertices results in fewer diagrams required to compute amplitudes, when compared to the CSW method and, obviously, considerably fewer than in the standard Yang-Mills action.

Multi-parametric Rational Solutions to the KdV Equation

We construct multi-parametric rational solutions to the KdV equation. For this, we use solutions in terms of exponentials depending on several parameters and take a limit when one of these parameters goes to 0. Here we present degenerate rational solutions and give a result without the presence of a limit as a quotient of polynomials depending on 3N parameters. We give the explicit expressions of some of these rational solutions.

Entire solutions of certain nonlinear differential and delay-differential equations

Let f(z) be an entire solution of the nonlinear differential equation f(z)n+P(z,f)=b1eλ1z+b2eλ2z, where n≥2, P(z,f) is a differential polynomial in f of degree ≤n−1 with meromorphic functions of order less than 1 as coefficients, and b1, b2, λ1, λ2 are nonzero constants and t=λ2/λ1 is real. By utilizing Nevanlinna theory, we show that t=−1 or t is a positive rational number and that in either case f(z) is expressed in terms of exponential functions. This partially answers a question in the literature. We also show that these results extend to finite-order entire solutions of the above equation when P(z,f) is replaced by a delay-differential polynomial in f with meromorphic functions of order less than 1 as coefficients.

Application of the Exp-Function and Generalized Kudryashov Methods for Obtaining New Exact Solutions of Certain Nonlinear Conformable Time Partial Integro-Differential Equations

The key objective of this paper is to construct exact traveling wave solutions of the conformable time second integro-differential Kadomtsev–Petviashvili (KP) hierarchy equation using the Exp-function method and the (2 + 1)-dimensional conformable time partial integro-differential Jaulent–Miodek (JM) evolution equation utilizing the generalized Kudryashov method. These two problems involve the conformable partial derivative with respect to time. Initially, the conformable time partial integro-differential equations can be converted into nonlinear ordinary differential equations via a fractional complex transformation. The resulting equations are then analytically solved via the corresponding methods. As a result, the explicit exact solutions for these two equations can be expressed in terms of exponential functions. Setting some specific parameter values and varying values of the fractional order in the equations, their 3D, 2D, and contour solutions are graphically shown and physically characterized as, for instance, a bell-shaped solitary wave solution, a kink-type solution, and a singular multiple-soliton solution. To the best of the authors’ knowledge, the results of the equations obtained using the proposed methods are novel and reported here for the first time. The methods are simple, very powerful, and reliable for solving other nonlinear conformable time partial integro-differential equations arising in many applications.

New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM

In this article, a modification of the rapidly convergent approximation method is proposed to solve a coupled Korteweg–de Vries equations with conformable derivative that govern shallow-water waves. Based on the Leibniz and chain rule of conformable derivative, these equations reduced into ODEs with integer-order using traveling wave transformation. Adopting the modified scheme a new novel exact solution of the reduced coupled ordinary differential equations is obtained in terms of exponential functions. Finally, by putting them back into traveling wave transformation the solutions of the considered partial differential equations with conformable derivative are derived. To ensure the boundedness of the derived solutions few theorems have been proposed and proved. The derived results of the theorems are utilized to plot the solutions. Graphics exhibit that solutions have variant multi-hump soliton peculiarities and their tails decay to zero exponentially in a monotonic manner. These results not only show the efficiency of the modified scheme but also establish that the solution is enriched with new multi-hump features.

Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time

In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process by $$X=\{X_{t},t\ge 0\}$$ and set $$\tau _{l}=\inf \{t\ge 0, X_{t}\le l\}$$ and $$\tau _{l,u}=\inf \{t\ge 0, X_{t}\notin (l,u)\}$$ with $$l<u$$ . We first establish the existence and uniqueness of strong solutions of the stochastic differential equation (SDE) satisfied by X. Then we investigate the Laplace transforms associated with $$\tau _{l}$$ and $$\tau _{l,u}$$ . It turns out that the explicit expressions for those Laplace transforms can be expressed in terms of exponential functions.