Expansion Theorem Research Articles

Fully Analytical Evaluation of Second Virial Coefficient with Stockmayer Potential and Its Applications

In this study, fully analytical treatment for evaluating the second virial coefficient with Stockmayer potential is presented. This approach is based on the binomial expansion theorems and basic integrals for the analytical representation of the second virial coefficient. The presented relationships for the second virial coefficient over Stockmayer potential can be useful especially in the studies of the interaction of polar molecules with higher dipole moments. The usefulness of the method is confirmed by H2O, NH3, and CHCl3 molecules. The obtained results of second virial coefficient are in good agreement with literature data.

Mechanics of inspection robot traveling over the electrical line with constant velocity

This article aims to the construction of mathematical models for a situation in which the electrical line stretched and supported by two isolators, one horizontal and the other vertical and robot inspector moves with a constant speed on it. The expansion theorem is used to construct the equation of motion of the system and has been solved by two methods: expansion method and Galerkin method. As a result of even constant speed of motion of robot inspector along the electrical line, vibrations have been observed. These vibrations can cause damage to the line or the robot structure as well. Then, the effect of the resistance force, wind force, has been studied. As shown in results, resistance force acts as a damper to the system and causes a decrease in the amplitude of vibrations. The practical results are useful for engineers to select proper parameters and speed for designing and constructing the robot inspectors.

Monotone positive solution of fourth order boundary value problem with mixed integral and multi-point boundary conditions

In this paper, we investigate the existence of monotone positive solutions for a fourth order boundary value problem with dependence on the derivative in nonlinearity under integral and multi-point boundary conditions. By applying the fixed point theorem in a cone, some criteria on the existence of positive solutions are acquired. These criteria are given by explicit conditions which are generally weaker than those derived by using the classical norm-type expansion and compression theorem. As applications, three examples are presented to illustrate the validity of our mains results.

A complete symplectic approach for a class of partial differential equations arising from the elasticity

The symplectic approach is used to establish the unified framework for solving governing equations of some thin plate problems in elasticity. By introducing appropriate functions, the given partial differential equation is first transferred into a separable Hamiltonian system. The completeness of the system of generalized eigenvectors of corresponding Hamiltonian operator matrix is proved, which serves as the theoretical foundation of symplectic approach. Utilizing the expansion theorems, the general solutions of the boundary value problem for partial differential equation under consideration are obtained. Moreover, the bending, buckling, and free vibration problems of fully clamped rectangular thin plates governed by the given partial differential equation are solved analytically by the technique of superposition. Numerical results for bending, buckling, and free vibration plates are presented to demonstrate the availability and validity of the approach by comparison with those available in the literatures.

NONSTATIONARY HEAT CONDUCTION IN MULTILAYER GLAZING SUBJECTED TO DISTRIBUTED HEAT SOURCES

A method for calculation of nonstationary thermal fields in a multilayer glazing of vehicles under the effect of impulse film heat sources is offered. The glazing is considered as a rectangular multilayer plate made up of isotropic layers with constant thickness. Film heat sources are arranged on layers' interfaces. The heat conduction equation is solved using the Laplace transformation, series expansion and the second expansion theorem. The method offered can be used for designing a safe multilayer glazing under operational and emergency thermal and force loading in vehicles.

Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: Local setting

In this paper we study unfoldings of planar vector fields in a neighbourhood of a hyperbolic resonant saddle. We give a structure theorem for the asymptotic expansion of the local Dulac time (as well as the local Dulac map) with the remainder uniformly flat with respect to the unfolding parameters. Here local means close enough to the saddle in order that the normalizing coordinates provided by a suitable normal form can be used. The principal part of the asymptotic expansion is given in a monomial scale containing a deformation of the logarithm, the so-called Roussarie-Ecalle compensator. Especial attention is paid to the remainder's properties concerning the derivation with respect to the unfolding parameters.

A Two-Dimensional Gauss–Kuzmin Theorem Associated with the Random Fibonacci-Type Sequences

In this article, we attempt to investigate a two-dimensional Gauss–Kuzmin theorem for continued fraction expansions associated with random Fibonacci-type sequences introduced by Chan (2006). More precisely speaking, our focus is to obtain a Gauss–Kuzmin theorem concerning the natural extension of corresponding interval maps $$\{\tau _l: l\in \mathbb {N},l\ge 2\}$$ . Then, we give a local betterment of this theorem. It matters that together with characteristic properties of the Perron–Frobenius operator of $$\tau _l$$ under its invariant measure on the Banach space of functions of bounded variation, we are in a position to conclude unambiguous lower and upper bounds for the error term linked to distribution function in the case when $$2\le l\le 257$$ , which show that the desired optimal convergence rate is $$\mathcal {O}(\vartheta _l^n)$$ as $$n\rightarrow \infty $$ with $$\vartheta _l=\frac{2(l-1)}{2l-1+\sqrt{1+4(l-1)}}$$ .

An exact solution method for the enumeration of connected Feynman diagrams

We completely generalize previous results related to the counting of connected Feynman diagrams. We use a generating function approach, which encodes the Wick contraction combinatorics of the respective connected diagrams. Exact solutions are found for an arbitrary number of external legs, and a general algorithm is implemented for this calculus. From these solutions, we calculate many asymptotics expansion terms for a simple analytical tool (Taylor expansion theorem). Our approach offers new perspectives in the realm of Feynman diagrams enumeration (or zero-dimensional quantum field theory).

Expansions of real closed fields that introduce no new smooth functions

We prove the following theorem: let R˜ be an expansion of the real field R‾, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a “semialgebraic chunk”. Then every definable smooth function f:X⊆Rn→R with open semialgebraic domain is semialgebraic.Conditions (I) and (II) hold for various d-minimal expansions R˜=〈R‾,P〉 of the real field, such as when P=2Z, or P⊆R is an iteration sequence. A generalization of the theorem to d-minimal expansions R˜ of Ran fails. On the other hand, we prove our theorem for expansions R˜ of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as 〈R‾,Ralg,2Z〉.

Research on semi-active vibration isolation system based on electromagnetic spring

This paper proposes a semi-active variable stiffness vibration isolation system based on electromagnetic spring for the low-frequency vibration isolation of mass-varying objects. It is achieved by four straight leaf springs in parallel to an electromagnetic spring system composed of a single electromagnet and a permanent magnet. The equivalent magnetic circuit method is used to compute electromagnetic force of the electromagnetic spring system, and mathematical model of the semi-active vibration isolation system is established according to Maxwell's equations. The nonlinear mathematical model is linearized at the equilibrium point by using the Taylor series expansion theorem to establish linear state-space representation of the system, and then using the traditional PID control method, a double closed-loop feedback control system of the inner current loop and outer location loop is designed. By controlling the current in the coil, the equivalent stiffness and electromagnetic force of the system are variable to achieve semi-active control. Furthermore, the control block diagram of the semi-active vibration isolation system is built based on Simulink software, then make a simulation analysis to the vibration isolation performance of the system and compare the effects of vibration isolation with inner current loop control and without inner current loop control, respectively. Finally, the experiments prove the correctness of the theory. It concludes that this semi-active vibration isolation system is a vibration isolation system with broad application prospects, which has fast current response, high vibration isolation efficiency, and an excellent vibration isolation effect for the low-frequency disturbance of mass-varying objects.

Logcf: An Efficient Tool for Real Root Isolation

Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem. The authors give a new algorithm for computing an upper bound of positive roots in this paper. The complexity of the algorithm is O(n log(u+1)) additions and multiplications where u is the optimal upper bound satisfying Theorem 3.1 of this paper and n is the degree of the polynomial. The method together with some tricks have been implemented as a software package logcf using C language. Experiments on many benchmarks show that logcf is competitive with RootIntervals of Mathematica and the function realroot of Maple averagely and it is much faster than existing open source real root solvers in many test cases.

Cluster Expansions with Renormalized Activities and Applications to Colloids

We consider a binary system of small and large objects in the continuous space interacting via a non-negative potential. By integrating over the small objects, the effective interaction between the large ones becomes multi-body. We prove convergence of the cluster expansion for the grand canonical ensemble of the effective system of large objects. To perform the combinatorial estimate of hypergraphs (due to the multi-body origin of the interaction) we exploit the underlying structure of the original binary system. Moreover, we obtain a sufficient condition for convergence which involves the surface of the large objects rather than their volume. This amounts to a significant improvement in comparison to a direct application of the known cluster expansion theorems. Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.

Calculation of Temperature Fields in Multilayer Plates and Shells with Distributed Heat Sources

The aircraft multilayer glazing is considered as arectangular multilayer plate made up of isotropic layers withconstant thickness. The temperature on the side surface of theplate is zero. Convective heat transfer occurs on outer surfaces ofthe plate; on layers' interfaces film heat sources are arranged.The heat conduction equation for an arbitrary plate layer isreduced to the functional equation. A solution of the functionalequation we search in the form of three space functions product.We get the system of ordinary differential equations. Seriesexpansion factors are determined from a system of linearalgebraic equations. A transform of the required function isfound by the second expansion theorem, and the problemsolution has the form of double trigonometrical series.The comparative analysis of the results is carried out with theresults of other method. The method offered can be used fordesigning a safe multilayer glazing under operational andemergency thermal and force loading in different vehicles.

On the generalized Radimacher-Menchoff Theorem for general spectral decomposition of the elliptic differential operators

The spectral decompositions of elliptic operators are considered. An analogue of the Radimacher-Menchoff Theorem for general spectral expansions corresponding to a self-adjoint extension of elliptic operators is obtained. The theorem obtained allows us to obtain a result on the almost everywhere convergence of the spectral decompositions from the Liouville classes. The estimation of the maximal operator corresponding to the spherical sums of multiple Fourier series from Liouville classes , 1 ⩽ p ⩽ 2, is derived.

The Marker–Steinhorn Theorem via Definable Linear Orders

We give a short proof of the Marker–Steinhorn theorem for o-minimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.

A nuclear matter calculation with the tensor-optimized Fermi sphere method with central interaction

Abstract The tensor-optimized Fermi sphere (TOFS) theory is applied first for the study of the property of nuclear matter using the Argonne V4$^\prime$$NN$ potential. In the TOFS theory, the correlated nuclear matter wave function is taken to be a power-series type of the correlation function $F$, where $F$ can induce central, spin–isospin, tensor, etc. correlations. This expression has been ensured by a linked cluster expansion theorem established in the TOFS theory. We take into account the contributions from all the many-body terms arising from the product of the nuclear matter Hamiltonian $\mathcal{H}$ and $F$. The correlation function is optimally determined in the variation of the total energy of nuclear matter. It is found that the density dependence of the energy per particle in nuclear matter is reasonably reproduced up to the nuclear matter density $\rho \simeq 0.20$ fm$^{-3}$ in the present numerical calculation, in comparison with other methods such as the Brueckner–Hartree–Fock approach.

Analytical evaluation of the Uehling potential using binomial expansion theorems

In this paper, we have introduced a new method to study of Uehling potential using binomial expansion theorems. Note that, the Uehling potential is a powerful tool to determine the effect of vacuum polarization in atomic and muon-atomic systems. The correcting of vacuum-polarization for an electron in a nuclear Coulomb field can be defined more precisely by the use of Uehling potential. From this point of view, the determination of explicit and closed-form analytical expressions for Uehling potential is very important. Therefore, presented method is illustrated by analytically calculation of the Uehling potential with the simple binomial coefficients and exponential integral functions. As can be seen from table and figure, the newly derived analytical expression well avoids the computational difficulties. An evaluation analysis of the Uehling potential is reported for arbitrary values of parameters. Because of its simple form the suggested method can be generally applied to the quantum thermodynamical problems.

Q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analogs of Lidstone polynomials which are in fact [Formula: see text]-Bernoulli polynomials as in the classical case. The definitions of these [Formula: see text]-Bernoulli polynomials and numbers are introduced. We also introduce [Formula: see text]-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points [Formula: see text] and [Formula: see text] either in terms of the polynomials [Formula: see text] or in terms of the polynomials [Formula: see text]. As an application, we introduce a new series expansion for the basic hypergeometric series [Formula: see text].

Positive periodic solutions for Lotka–Volterra systems with a general attack rate

The paper deals with a non-autonomous Lotka–Volterra type system, which in particular may include logistic growth of the prey population and hunting cooperation between predators. We focus on the existence of positive periodic solutions by using an operator approach based on the Krasnosel’skii homotopy expansion theorem. We give sufficient conditions in order that the localized periodic solution does not reduce to a steady state. Particularly, two typical expressions for the functional response of predators are discussed.

Bs(d) \u2212 overline{B} s(d) mixing and Bs \u2192 \u03bc+\u03bc\u2212 decay in the NMSSM with the Flavour Expansion Theorem

In this paper, motivated by the observation that the Standard Model predictions are now above the experimental data for the mass difference ΔMs(d), we perform a detailed study of Bs(d) − overline{B} s(d) mixing and Bs → μ+μ− decay in the ℤ3-invariant NMSSM with non-minimal flavour violation, using the recently developed procedure based on the Flavour Expansion Theorem, with which one can perform a purely algebraic mass-insertion expansion of an amplitude written in the mass eigenstate basis without performing any diagrammatic calculations in the interaction/flavour basis. Specifically, we consider the finite orders of mass insertions for neutralinos but the general orders for squarks and charginos, under two sets of assumptions for the squark flavour structures (i.e., while the flavour-conserving off-diagonal element δ33LR is kept in both of these two sectors, only the flavour-violating off-diagonal elements δ23LL and δi3RR (i = 1, 2) are kept in the LL and RR sectors, respectively). Our analytic results are then expressed directly in terms of the initial Lagrangian parameters in the interaction/flavour basis, making it easy to impose the experimental bounds on them. It is found numerically that the NMSSM effects with the above two assumptions for the squark flavour structures can accommodate the observed deviation for ΔMs(d), while complying with the experimental constraints from the branching ratios of Bs → μ+μ− and B → Xsγ decays.