Mathematic Model and Analytic Solution for a Cylinder Subject to Exponential Function

Hollow cylinders are widely used in spacecraft, rockets, weapons, metallurgy, materials, and mechanical manufacturing industries, and so on, hydraulic bulging roll cylinder and hydraulic press work all belong to hollow cylinders. However, up till now, the solution of the cylinder subjected to the pressures in the three-dimensional space is still at the stage of the analytical solution to the normal pressure or the approximate solution to the variable pressure by numerical method. The analytical solution to the variable pressure of the cylinder has not yet made any breakthrough in theory and can not meet accurate theoretical analysis and calculation requirements of the cylindrical in Engineering. In view of their importance, the precision calculation and theoretical analysis are required to investigate on engineering. A stress function which meets both the biharmonic equations and boundary conditions is constructed in the three-dimensional space. Furthermore, the analytic solution of a hollow cylinder subjected to exponential function distributed variable pressure on its inner and outer surfaces is deduced. By controlling the pressure subject to exponential function distributed variable pressure in the hydraulic bulging roller without any rolling load, using a static tester to record the strain supported hydraulic bulging roll, and comparing with the theoretical calculation, the experimental test result has a higher degree of agreement with the theoretical calculation. Simultaneously, the famous Lam6 solution can be deduced when given the unlimited length of cylinder along the axis. The analytic solution paves the way for the mathematic building and solution of hollow cylinder with randomly uneven pressure.

Degenerations of rational Bézier surface with weights in the form of exponential function

Rational Bezier surface is a widely used surface fitting tool in CAD. When all the weights of a rational B@zier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bezier surfaces. In this paper, we study on the degenerations of the rational Bezier surface with weights in the exponential function and indicate the difference of our result and the work of Garcia-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bezier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.

A multiplicative Gauss-Newton minimization algorithm:Theory and application to exponential functions

Multiplicative calculus(MUC)measures the rate of change of function in terms of ratios,which makes the exponential functions significantly linear in the framework of MUC.Therefore,a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC.Taking this as motivation,this paper lays mathematical foundation of well-known classical Gauss-Newton minimization(CGNM)algorithm in the framework of MUC.This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization(MGNM)method along with its convergence properties.The proposed method is generalized for n number of variables,and all its theoretical concepts are authenticated by simulation results.Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions.From simulation results,it has been observed that proposed MGNM method converges for 12972 points,out of 19600 points considered while optimizing multiplicatively-linear exponential function,whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points,respectively.Furthermore,for a given set of initial value,the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods.A similar pattern is observed for multiplicatively-non-linear exponential function.Therefore,it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods.

AN EME BLIND SOURCE SEPARATION ALGORITHM BASED ON GENERALIZED EXPONENTIAL FUNCTION

This letter investigates an improved blind source separation algorithm based on Maximum Entropy （ME） criteria. The original ME algorithm chooses the fixed exponential or sigmoid ftmction as the nonlinear mapping function which can not match the original signal very well. A parameter estimation method is employed in this letter to approach the probability of density function of any signal with parameter-steered generalized exponential function. An improved learning rule and a natural gradient update formula of unmixing matrix are also presented. The algorithm of this letter can separate the mixture of super-Gaussian signals and also the mixture of sub-Gaussian signals. The simulation experiment demonstrates the efficiency of the algorithm.

Using negative exponential function to characterize built-up land density along slope gradient for 19 urban areas in China

In China,numerous cities are expanding into sloping land,yet the quantitative distribution patterns of urban built-up land density along the slope gradient remain unclear,limiting the understanding of sloping land urbanization.In this paper,a simple negative exponential function was presented to verify its applicability in 19 typical sloping urban areas in China.The function fits well for all case urban areas(R^(2)≥0.951,p<0.001).The parameters of this function clearly describe two fundamental attributes:initial value a and decline rate b.Between 2000 and 2020,a tends to increase,while b tends to decrease in all urban areas,confirming the hypothesis of mutual promotion between flatland densification and sloping land expansion.Multiple regression analysis indicates that the built-up land density and the ruggedness of background land can explain 70.7%of a,while the average slope ratio of built-up land to background land,the built-up land density and the built-up land area can explain 82.1%of b.This work provides a quantitative investigative tool for distribution of urban built-up land density along slope gradient,aiding in the study of the globally increasing phenomenon of sloping land urbanization from a new perspective.

Approximation Properties of Vectorial Exponential Functions

This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,and Applications”.With respect to classical Finite Elements Methods,Trefftz methods are unconventional methods because of the way the basis functions are generated.Trefftz discontinuous Galerkin(TDG)methods have recently shown potential for numerical approximation of transport equations[6,26]with vectorial exponential modes.This paper focuses on a proof of the approximation properties of these exponential solutions.We show that vectorial exponential functions can achieve high order convergence.The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.

The Orthogonal Bases of Exponential Functions Based on Moran-Sierpinski Measures

Let An∈M2(ℤ)be integral matrices such that the infinite convolution of Dirac measures with equal weightsμ{A_(n),n≥1}δA_(1)^(-1)D*δA_(1)^(-1)A_(2)^(-2)D*…is a probability measure with compact support,where D={(0,0)^(t),(1,0)^(t),(0,1)^(t)}is the Sierpinski digit.We prove that there exists a setΛ⊂ℝ2 such that the family{e2πi〈λ,x〉:λ∈Λ} is an orthonormal basis of L^(2)(μ{A_(n),n≥1})if and only if 1/3(1,-1)A_(n)∈Z^(2)for n≥2 under some metric conditions on A_(n).

Algebraic Independence of Certain Values of Exponential Function

The algebraic independence of e^θ1,…,e^θs is proved, where θ1,… ,θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements.

THE RAMANUJAN'S Q-EXTENSION FOR THE EXPONENTIAL FUNCTION AND STATISTICAL DISTRIBUTIONS

In this paper. we shall propose a q-extension for the exponential function and develop q-analogs for families of statistical distribution, such as,the normal, and Poisson distribution etc.Many properties of these families will be studied.

Optimal investment with transaction costs based on exponential utility function:a parabolic double obstacle problem

This paper concerns optimal investment problem with proportional transaction costs and finite time horizon based on exponential utility function. Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. Numerical examples are obtained by the binomial method.

Atomic Exponential Basis Function Eup(x,w) - Development and Application

This paper presents exponential Atomic Basis Functions(ABF),which are called Eup(x;w).These functions are infinitely differentiable finite functions that unlike algebraic up(x)basis functions,have an unspecified parameter-frequency w.Numerical experiments show that this class of atomic functions has good approximation properties,especially in the case of large gradients(Gibbs phenomenon).In this work,for the first time,the properties of exponential ABF are thoroughly investigated and the expression for calculating the value of the basis function at an arbitrary point of the domain is given in a form suitable for implementation in numerical analysis.Application of these basis functions is shown in the function approximation example.The procedure for determining the best frequencies,which gives the smallest approximation error in terms of the least squares method,is presented.

The Class of Atomic Exponential Basis Functions EFup_(n)(x,ω)-Development and Application

The purpose of this paper is to present the class of atomic basis functions(ABFs)which are of exponential type and are denoted by EFupn(x,ω).While ABFs of the algebraic type are already represented in the numerical modeling of various problems inmathematical physics and computationalmechanics,ABFs of the exponential type have not yet been sufficiently researched.These functions,unlike the ABFs of the algebraic type Fupn(x),contain the tension parameterω,which gives them additional approximation properties.Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFupn(x,ω).The function EFupn for n=0 is called the“mother”ABF of the exponential type,i.e.,EFup0(x,ω)≡Eup(x,ω).In other words,the functions EFupn(x,ω)are elements of the linear vector space EUPn and retain all the properties of their“mother”function Eup(x,ω).Thus,this paper,in terms of its content and purpose,can be understood as a sequel of the article by Brajcic Kurbasa et al.,which shows the basic properties and application of the basis function Eup(x,ω).This paper presents,in an analogous way,the development and application of the exponential basis functions EFupn(x,ω).Here,for the first time,expressions for calculating the values of the functions EFupn(x,ω)and their derivatives are given in a form suitable for application in numerical analyses,which is shown in the verification examples of the approximations of known functions.

Order Exponential Evaluation Model for Road Traffic Safety in City Clusters

This study presents an order exponential model for estimating road traffic safety in city clusters.The proposed model introduces the traffic flow intrinsic properties and uses the characteristics and regular patterns of traffic development to identify road traffic safety levels in city clusters.Additionally,an evaluation index system of city cluster road traffic safety was constructed based on the spatial and temporal distribution.Then Order Exponential Evaluation Model(OEEM),a comprehensive model using order exponent function for road traffic safety evaluation,was put forward,which considers the main characteristics and the generation process of traffic accidents.The model effectively controlled the unsafe behavior of the traffic system.It could define the levels of city cluster road traffic safety and dynamically detect road safety risk.The proposed model was verified with statistical data from three Chinese city clusters by comparing the common model for road traffic safety with an ideal model.The results indicate that the order exponent approach undertaken in this study can be extended and applied to other research topics and fields.

Existence and Exponential Stability of Almost Periodic Solution for Hopfield Neural Network Equations with Almost Periodic Imput

By constructing Liapunov functions and building a new inequality, we obtain two kinds of sufficient conditions for the existence and global exponential stability of almost periodic solution for a Hopfield-type neural networks subject to almost periodic external stimuli. Irt this paper, we assume that the network parameters vary almost periodically with time and we incorporate variable delays in the processing part of the network architectures.

Exponential Continuous Non-Parametric Neural Identifier With Predefined Convergence Velocity

This paper addresses the design of an exponential function-based learning law for artificial neural networks(ANNs)with continuous dynamics.The ANN structure is used to obtain a non-parametric model of systems with uncertainties,which are described by a set of nonlinear ordinary differential equations.Two novel adaptive algorithms with predefined exponential convergence rate adjust the weights of the ANN.The first algorithm includes an adaptive gain depending on the identification error which accelerated the convergence of the weights and promotes a faster convergence between the states of the uncertain system and the trajectories of the neural identifier.The second approach uses a time-dependent sigmoidal gain that forces the convergence of the identification error to an invariant set characterized by an ellipsoid.The generalized volume of this ellipsoid depends on the upper bounds of uncertainties,perturbations and modeling errors.The application of the invariant ellipsoid method yields to obtain an algorithm to reduce the volume of the convergence region for the identification error.Both adaptive algorithms are derived from the application of a non-standard exponential dependent function and an associated controlled Lyapunov function.Numerical examples demonstrate the improvements enforced by the algorithms introduced in this study by comparing the convergence settings concerning classical schemes with non-exponential continuous learning methods.The proposed identifiers overcome the results of the classical identifier achieving a faster convergence to an invariant set of smaller dimensions.

Ulam-Hyers Stability of Trigonometric Functional Equation with Involution

The present work aims to determine the solution of trigonometric functional equation f with involution from group to field by using the properties of involution function, and the solution and Ulam-Hyers stability of the trigonometric functional equation are also discussed. Furthermore, this method generalizes the main theorem and gives the supplement in some reference.

The Proof of the Generalized Piemann’s Hypothesis

The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential functional series and finite exponential functional progression.

Beach Profiles Characteristics Along Giao Thuy and Hai Hau Coasts,Vietnam:A Field Study

Giao Thuy and Hai Hau coasts are located in Nam Dinh province, Vietnam, with a total coastline of 54.42 km in length. The sea-dike system has been seriously damaged and there have been many dike breaches which caused floods and losses. This situation is considered of a general representative for coastal area in the northern part of Vietnam. A variety of studies have shown that the gradient in the longshore sediment transport rate and the offshore fine sediment lost are the main mechanisms causing the beach erosion. This study presents a field investigation of the beach profiles at Giao Thuy and Hai Hau beaches. Three types of empirical functions for the equilibrium beach profile are applied and compared with the observations. Results show that all observed beach profiles can be described by a single function. However, one specific equilibrium profile equation is not sufficient to assess all beach profiles. In Section 1 of Giao Thuy and Section 3 of Hai Thinh beaches, beach profiles are consistent with the logarithmic function, while the exponential function fits well in Section 2. This difference is explained with respect to coastal morphology, sediment characteristics and hydrodynamic conditions which vary in site. An analysis of the validity of the beach profile functions is recommended for the numerical modeling and engineering designs in this area.

Precise integration method without inverse matrix calculation for structural dynamic equations

The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.

Breakdown Voltage Prediction by Utilizing the Behavior of Natural Ester for Transformer Applications

This research investigates the dielectric performance of Natural Ester(NE)using the Partial Differential Equation(PDE)tool and analyzes dielectric performance using fuzzy logic.NE nowadays is found to replace Mineral Oil(MO)due to its extensive dielectric properties.Here,the heat-tolerant Natural Esters Olive oil(NE1),Sunflower oil(NE2),and Ricebran oil(NE3)are subjected to High Voltage AC(HVAC)under different electrodes configurations.The break-down voltage and leakage current of NE1,NE2,and NE3 under Point-Point(P-P),Sphere-Sphere(S-S),Plane-Plane(PL-PL),and Rod-Rod(R-R)are measured,and survival probability is presented.The electricfield distribution is analyzed using the Partial Differential Equation(PDE)tool.NE shows better HVAC with stand capacity under all the electrodes configuration,especially in the S-S shape geometry.The exponential function is developed for the oils under different elec-trode geometry;NE shows a higher survival probability.Likewise,the most influ-ential dielectric properties such as breakdown voltage,kinematic viscosity,and water content are used to develop a Mamdani-based control system model that combines two variables to produce the surface model.The surface model indi-cates that the NE subjected for investigation is less susceptible to moisture effect and higher kinematic viscosity.Based on the surface models of PDE and fuzzy,it is concluded that NE possesses a high survival rate since its breakdown voltage does not react to changes in water content.Hence the application of NE in the transformer application is highly safe and possesses extended life.