###### On the perturbed restricted three-body problem

Bhatnagar [ 19 ], and Sharma et al. [ 24 ]. Various researchers made studies in the restricted three-body problem under the effects of small perturbations in centrifugal and Coriolis forces such as in Szebehely [ 29 ], Bhatnagar and Hallan [ 12 ], Devi and Singh [ 14 ], and Shu and Lu [ 25 ], to quote some of them. The effect of small perturbations ε , ε ′ in Coriolis and centrifugal forces with variable mass in the restricted three-body problem has been studied by Singh [ 26 ]. He found that in the nonlinear sense the triangular points are stable for all mass ratios

###### The Triaxiality Role in the Spin-Orbit Dynamics of a Rigid Body

, when dealing with orbital dynamics applied to artificial satellites, some lines of research on intermediaries arose during the seventies by Garfinkel, Aksnes, Cid, Sterne, etc. (see review in [ 14 ]), whose benefits are now seen in areas such as the relative motion in formation flights, an example is given in [ 25 ]. Nevertheless, less work has been done when dealing with attitude dynamics, where the proposal of intermediaries is more recent [ 2 , 17 ] and, to our knowledge, no systematic study has been done on them. In this paper, we continue our work on the

###### Accuracy Problems of Numerical Calculation of Fractional Order Derivatives and Integrals Applying the Riemann-Liouville/Caputo Formulas

. Table 1 D (1/2) f ( t ), f ( t ) = t , t ∈ (0,1), relative error in % N GL NCm Diet Odiba 8 8.11 10.68 0.0004 0.0024 15 4.25 7.81 0.0004 0.0023 21 3.02 6.6 0.0004 0.0023 61 1.03 3.88 0.0004 0.0022 300 0.21 1.75 0.0002 0.0021 600 0.11 1.24 0.0002 0.0021 1000 0.06 0.96 0.0002 0.0021 Table 2 D (1/2) f ( t ), f ( t ) = e −t , t ∈ (0,5), relative error in % N GL NCm 8 61.87 13.52 15 29

###### Benefits of a dance group intervention on institutionalized elder people: a Bayesian network approach

the network that will be considered later. 2.1 Bayesian Network Model Let us consider a finite set X = { X 1 , X 2 … X n of discrete random variables and P a joint probability distribution over the variables in X . A Bayesian network (BN) is a directed acyclic graph that encodes a joint probability distribution over the set of random variables X . Formally, a BN for X is a pair B = ( G , Θ) where G is the graph whose vertexes are the random variables X 1 , X 2 … X n , and the Θ represents the set of parameters that qualify the network

###### Centers: their integrability and relations with the divergence

center at the origin. (a) If the system has a formal first integral, then it has a formal first integral of the form H = y 2 + F ( x , y ), where F starts with terms of order higher than two. (b) If the system has a local analytic first integral defined at the origin, then it has a local analytic first integral of the form H = y 2 + F ( x , y ), where F starts with terms of order higher than two. (c) If X = yf ( x , y 2 ) and Y = g ( x , y 2 ), then the system has a local analytic first integral of the form H = y 2 + F ( x , y ), where F

###### Simulation analysis of resource-based city development based on system dynamics: A case study of Panzhihua

simulation of situation 1 shows that in 2030, the GDP will reach 211.046 billion yuan, the amount of GDP growth will be 8.775 billion yuan, the industrial output value will be 25.732 billion yuan, the air pollution index will be 94.4925, the industrial solid waste pollution index will be 102.921, and the urban sewage pollution index will be 101.578. In this situation, there is room for further improvement in Panzhihua in terms of economic development and pollution treatment. Under future policies and social constraints, this development will have certain drawbacks in the

###### On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

k is meromorphically integrable, then the pair ( k , λ ) belongs to one of the following list : k , p + k p ( p − 1 ) / 2 , ( 2 , λ ) , ( − 2 , λ ) , − 5 , 49 / 40 − 5 ( 1 + 3 p ) 2 / 18 , − 5 , 49 / 40 − ( 2 + 5 p ) 2 / 10 , − 4 , 9 / 8 − 2 ( 1 + 3 p ) 2 / 9 , − 3 , 25 / 24 − ( 1 + 3 p ) 2 / 6 , − 3 , 25 / 24 − 3 ( 1 + 4 p ) 2 / 32 , − 3 , 25 / 24 − 3 ( 1 + 5 p ) 2 / 50 , − 3 , 25 / 24 − 3 ( 2 + 5 p ) 2 / 50 , 3 , − 1 / 24 + ( 1 + 3 p ) 2 / 6 , 3 , − 1 / 24 + 3 ( 1 + 4 p ) 2 / 32 , 3 , − 1 / 24 + 3 ( 1 + 5 p ) 2 / 50 , 3 , − 1 / 24 + 3 ( 2 + 5 p ) 2 / 50

###### Multigrid method for the solution of EHL line contact with bio-based oils as lubricants

{H}_0+c_2\left[G^\Delta-\frac{\Delta}{\pi}\sum_{j=1}^{N-1}(P_j+P_{j+1})\right], \end{array} $$ where c 2 is the relaxation factor and G Δ is the nondimensional load on the coarsest grid. Following procedure is followed for the determination of pressure cavitation point [ 25 ]. Let X c be a point in ( X in , X out ). If pressure gradient d P d X | X c = 0 , $\begin{array}{} \frac{dP}{dX}|_{X_c}=0, \end{array} $ then X c is the required pressure cavitation point. Otherwise, shift the point X c on next interval to the left or right depending on d P

###### Analysis of models for viscoelastic wave propagation

1 ( x < 1/2), 0 ( x ≥ 1/2) ν 1 1 0.3 1 ρ 10 10 10 10 11.3 3D numerical simulation We now present a numerical simulation for viscoelastic waves propagating in the parallelepiped Ω = (0, 1) × (0, 10) × (0, 1) with a Dirichlet boundary on one of the small faces Γ D := (0, 1) × {0} × (0, 1). The PDE we are simulating is u ¨ ( t ) = d i v σ ( t ) Ω × [ 0 , 50 ] , γ D u ( t ) = 0.25 w ( t ) , 0 , 0 ⊤ Γ D × [ 0 , 50 ] , γ N σ ( t ) = 0 Γ N × [ 0 , 50 ] , $$\begin{array}{} \displaystyle \quad

###### Oscillatory flow of a Casson fluid in an elastic tube with variable cross section

}}.\frac{{\partial A}}{{\partial t}} \end{array}$$ (24) c 2 ( z ) = A ρ 0 ∂ p ∂ A = S 2 ρ 0 ∂ p ∂ S $$\begin{array}{} \displaystyle {c^2}(z) = \frac{A}{{{\rho _0}}}\frac{{\partial p}}{{\partial A}} = \frac{S}{{2{\rho _0}}}\frac{{\partial p}}{{\partial S}} \end{array}$$ (25) here, c ( z ) is the local value of the wave speed. From Eqs. (24) and (25) , we have ∂ A ∂ t = A c 2 ν U 0 L a 0 2 i e i t p e $$\begin{array}{} \displaystyle \frac{{\partial A}}{{\partial t}} = \frac{A}{{{c^2}}}\frac{{\nu {U_0}L}}{{{a_0}^2}}i{e^{it}}{p_e} \end{array}$$ (26) using Eqs. (22