Legendre-Jacobi’s Elliptic Integrals Shed Light on the Luminosity Distance in Cosmology

This article concerns the integral related to the transverse comoving distance and, in turn, to the luminosity distance both in the standard non-flat and flat cosmology. The purpose is to determine a straightforward mathematical formulation for the luminosity distance as function of the transverse comoving distance for all cosmology cases with a non-zero cosmological constant by adopting a different mindset. The applied method deals with incomplete elliptical integrals of the first kind associated with the polynomial roots admitted in the comoving distance integral according to the scientific literature. The outcome shows that the luminosity distance can be obtained by the combination of an analytical solution followed by a numerical integration in order to account for the redshift. This solution is solely compared to the current Gaussian quadrature method used as basic recognized algorithm in standard cosmology.

Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b,and K(r)and E(r)are respectively the complete elliptic integrals of the first and second kinds.

ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS

Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.

ON GENERALIZED COMPLETE(p,q)-ELLIPTIC INTEGRALS

In this paper,we study the generalized complete(p,q)-elliptic integrals of the first and second kind as an application of generalized trigonometric functions with two parameters,and establish the monotonicity,generalized convexity and concavity of these functions.In particular,some Turán type inequalities are given.Finally,we also show some new series representations of these functions by applying Alzer and Richard's methods.

REFINEMENTS OF TRANSFORMATION INEQUALITIES FOR ZERO-BALANCED HYPERGEOMETRIC FUNCTIONS

In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The non-linear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge-Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance （10^-5）. The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.

SHARP BOUNDS FOR TOADER-TYPE MEANS IN TERMS OF TWO-PARAMETER MEANS

In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2) sin^(2))1/2 tdt.

ON THE BOUNDS OF THE PERIMETER OF AN ELLIPSE

In this paper,we present new bounds for the perimeter of an ellipse in terms of harmonic,geometric,arithmetic and quadratic means;these new bounds represent improvements upon some previously known results.

Analysis of the periodic solutions of a smooth and discontinuous oscillator

In this paper, the periodic solutions of the smooth and discontinuous （SD） oscillator, which is a strongly irra- tional nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the un- perturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equa- tion using Lyapunov method. The results presented herein this paper are valid for both smooth （e 〉 0） and discontin- uous （ce = 0） stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.

PERTURBATION SOLUTION TO 3-D NONLINEAR SUPERCAVITATING FLOW PROBLEMS

A 3-D nonlinear problem of supercavitating flow past an axisymmetric body at a small angle of attack is investigated by means of the perturbation method and Fourier-cosine-expansion method. The first three order perturbation equations are derived in detail and solved numerically using the boundary integral equation method and iterative techniques. Computational results of the hydrodynamic characteristics and cavity shapes of each order are presented for nonaxisymmetric supercavitating flow past cones with various apex-angles at differ- ent cavitation numbers. The numerical results are found in good agreement with experimental data.

Calculation and Optimization of ITER Upper VS Feeder Under an Electromagnetic Load

The upper vertical stability （VS） feeder is a part connected to the upper VS coil by a welding joint. The function of the feeder is to transfer current and coolant water to the VS coil. A giant electron^agnetic force will be generated during normal operation by the current flowing in the VS coils, interacting with the external background field. The Lorentz force will induce Tresca stress in the feeder. The amplitudes of the magnetic field and Lorentz force along the conductor running direction have been calculated based on Maxwell＇s equations. To extract the Tresca stress in the feeder, a finite element model was created using the software ANSYS and an electromagnetic load was applied on the model. According to the analytical design, the stresses were classified and evaluated based on ASME. In order to reduce the Tresca stress, some optimization works have been done and the Tresca stress has had a significant reduction in the optimized model. This analytical work figured out the stress distribution in the feeder and checked the feasibility of the prototype design model. The ANSYS analysis results will provide a guidance for later improvement and fabrication.

Electromagnetic Analyses of ITER Lower ELM Right Bottom Corner

ELM （edge localized mode） coils are key components of ITER that suppress the edge localized mode phenomenon. A giant electromagnetic force is generated during normal operations by the current flowing in the ELM coils interacting with the external background field. The Lorentz force will induce Tresca stress in the ELM coils. If the load goes beyond the allowable threshold, the coils can hardly satisfy the safety requirements. The right-hand bottom corner was chosen to perform our electromagnetic analyses. Based on the Maxwell equation, the detailed magnetic field and Lorentz force were calculated. By use of the finite element software ANSYS, the Tresca stress was extracted and evaluated based on our analytical design. The present analysis aims to verify the feasibility of the current design. It can also serve as guidance for fabrication and structural optimization.

Identities for Catalan’s Constant Arising from Integrals Depending on a Parameter

In this paper we provide some relationships between Catalan’s constant and the 3F2 and4F3 hypergeometric functions,deriving them from some parametric integrals.In particular,using the complete elliptic integral of the first kind,we found an alternative proof of a result of Ramanujan for3F2,a second identity related to 4F3 and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.

Singular integrals in axisymmetric problems of elastostatics

Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper.These equations are received after applying the integral transformation and Gauss-Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium.Initially,three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate.So,the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only.The thorough analysis of both displacement and traction kernels is accomplished,and similarity in behavior of both kernels is established.The kernels are expressed in terms of complete elliptic integrals of first and second kinds.The second kind elliptic integrals are nonsingular,and standard Gaussian quadratures are applied for their numerical evaluation.Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities.The numerical treatment of these integrals takes into account the presence of this integrable singularity.The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity,Catalan’s constant,the Gaussian surface integral.The comparison between analytical and numerical data has proved high precision and availability of the proposed method.

Minimizing stress concentrations around an elliptical hole by concentrated forces acting on the uniaxially loaded plate

When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy＇s integral formula. And then, based on the superposition principle, the analyt- ical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained. Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concen- trated forces are applied in the reversed directions of the uniform loads, tangential stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces. Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the op- timized concentrated forces are applied.

Geometric-Harmonic Mean and Characterizations of Some Mean-Values

The purpose of this paper is to provide a direct proof on the fact that the geometric-harmonic mean of any two positive numbers can be calculated by a first complete elliptical integral, and then to give new characterizations of some mean-values.

The Explicit Periodic Wave Solutions and Their Limit Forms for a Generalized b-equation

In this paper, for b ∈ （-∞,∞） and b ≠ -1, -2, we investigate the explicit periodic wave solutions for the generalized b-equation ut ＋ 2kux - uxxt ＋ （1 ＋ b）u2ux =buxuxx ＋ uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable a. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for a. When a tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = -1 or b = -2, we guess that the equation does not have any one of above solutions.

Tangential Developable Surfaces as Bonnet Surfaces

We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry.

Technical note：Large amplitude free vibration of a flexible panel coupled with a leaking cavity

This paper addresses the effect of leakage on the natural frequencies of a large amplitude vibrating panel backed by a cavity, which has not been considered in many other related studies. The structural-acoustic governing equations are employed to study this nonlinear problem. An elliptical integral method, which was recently developed for the nonlinear panel cavity problem, is introduced here to solve for the structural-acoustics responses. The present results agree reasonably well with those obtained from the classical harmonic balance method. Modal convergences of the nonlinear solutions are performed to verify the proposed method. The effects of vibration amplitude and leakage size are studied and discussed. It is found that （1） the edge leakages in a panel cavity system significantly affect the natural frequency properties, and （2） the edge leakages induce a low frequency acoustic resonance.

Closed-Form Expression for the Exact Period of a Nonlinear Oscillator Typified by a Mass Attached to a Stretched Wire

The exact analytical expression of the period of a conservative nonlinear oscillator with a non-polynomial potential,is obtained.Such an oscillatory system corresponds to the transverse vibration of a particle attached to the center of a stretched elastic wire.The result is given in terms of elliptic functions and validates the approximate formulae derived from various approximation procedures as the harmonic balance method and the rational harmonic balance method usually implemented for solving such a nonlinear problem.