We study an eccentric and elastically damped rotor from both a statical and dynamical point of view. The system, whose genesis is in the re-loading mechanism of an automatic watch, behaves like a generalized physical pendulum with the addition of eccentricity and damping. The static analysis is performed by settling the statical equilibria and defining their nature, whose effective computation can be done numerically. The dynamical analysis leads to a nonlinear differential initial-value problem whose integration is carried out by means of Jacobi elliptic functions. It reveals that, starting from both positional and kinetic zero initial conditions, only periodical motions, see formulae (4.8) or (4.13), are allowed and all confined inside a potential well. Closed form expressions of the oscillation period have been obtained through complete elliptic integrals of the first kind. In such a way a further treatment is added to the non-rich collection of 1-D nonlinear oscillators suitable of closed form integration.
We apply the recently developed sampling algorithm, called random orthogonal matrix (ROM) simulation by Ledermann et al. , to compute VaR of a market risk portfolio. Typically, the covariance matrix has a large influence on ROM VaR. But VaR, being a lower quantile of the portfolio return distribution, is also much impacted by the skewness and kurtosis of the risk factor returns. With ROM VaR it is possible to stress test risk factors under adverse market conditions by targeting other sample moments that are consistent with periods of financial crisis. In particular, the important effects of skewness or kurtosis in the tail of the portfolio returns can be incorporated in ROM VaR. In a simulation study, we integrate ROM VaR into other methods that take into account skewness and kurtosis, namely the Cornish-Fisher VaR approximation and a robust approximation to the Chebyshev-Markov VaR upper bound in Hürlimann .
In this paper, a novel auxiliary equation: φ′′ = a + bφ + cφ3 which has mutiple function solutions including trigonometric function, hyperbolic function and other functions, is considered. It is applied to a series of partial differential equations easily and effectively. It helps physicists to obtain complexiton solutions of nonlinear partial equations and analyze special phenomena accurately in their fields.
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin’s probability law. Results are given for sequences of partial quotients of positive irrational numbers and with m a natural number. A big partial quotient in one sequence finds a connection in the other.
Decomposer functions in algebraic structures were introduced and studied in the paper ”Decomposer and associative functional equations” in 2007. If (G, ) is a group, and f : G → G is a map, then f is called a right [resp. left] decomposer if and only if f(f*(x)f(y)) = f(y) [resp. f(f(x)f*(y)) = f(x)] for all x, y ∈ G, where f*(x)f(x) = f(x)f*(x) = x. Also, f is called a decomposer if it is left and right decomposer. There are many important connections between these functions and decomposition of groups by subsets. Now, we observe that if the structure is a group, then there are more important properties for them and also many connections among decomposer functions, multiplicative symmetric functions (introduced by J. G. Dhombres in 1973), separator functions and so on. For instance, every idempotent endomorphism in groups is (strong) decomposer. We also introduce some other related types and generalizations of these functions such as semi-strong decomposer and weak decomposer which help us in the study of decomposer type functions in groups. Then, several important properties and relations for these functions will be proved. Finally, we completely characterize the (two-sided) decomposer functions in arbitrary groups, and so we give a general solution of the decomposer equations that were not solved in the paper 2007 (only left and right cases were characterized).
A simple and direct process is derived to compute the determinant of any square matrix of high order. The approach involves successive applying the matrix order condensation algorithm. A computer program listing in MATLAB is included and examples for finding the determinant of a given 7x7 matrix are given here for illustration.
In this paper, the qualitative analysis methods of dynamical systems are used to investigate the periodic travelling wave solutions of ZK (2, 4,-2) equation. The phase portrait bifurcation of the travelling wave system corresponding to the equation is given. The explicit expressions of the periodic travelling wave solutions are obtained by using the portraits. The graph of the solutions are given with the numerical simulation.
B. Alspach, C.C. Chen and Kevin Mc Avaney  have discussed the Hamiltonian laceability of the Brick product C(2n, m, r) for even cycles. In , the authors have shown that the (m, r)-Brick Product C(2n + 1, 1, 2) is Hamiltonian-t-laceable for 1 ≤ t ≤ diamn. In  the authors have defined and discussed Hamiltonian-t-laceability properties of cyclic product C(2n, m) cyclic product of graphs. In this paper we explore Hamiltonian-t*-laceability of (W1,n,k) graph and Cyclo Product Cy(n, mk) of graph.
Aims: The aim of this paper is to develop a novel method for dealing with multiple attribute group decision making (MAGDM) problems with hesitant fuzzy information, in which the attribute values provided by the decision makers take the form of hesitant fuzzy elements (HFEs), the information about the weights of decision makers is unknown, and the information about attribute weights is incompletely known or completely unknown. Study Design: The developed method includes the following three stages. Place and Duration of Study: The hesitant fuzzy set (HFS), originally proposed by Torra and Narukawa, is an efficient tool to deal with situations in which experts hesitate between several possible values to evaluate the membership degree of an element to a given set. Methodology: The first stage establishes a quadratic programming model to determine the weights of decision makers by maximizing group consensus between the individual hesitant fuzzy decision matrices and the group hesitant fuzzy decision matrix. The second stage uses the maximizing deviation method to establish an optimization model, which derives the optimal weights of attributes under hesitant fuzzy environments. After obtaining the weights of decision makers and attributes through the above two stages, the third stage develops a hesitant fuzzy TOPSIS (HFTOPSIS) method to determine a solution with the shortest distance to the hesitant fuzzy positive ideal solution (HFPIS) and the greatest distance from the hesitant fuzzy negative ideal solution (HFNIS). Results: A practical example is provided to illustrate the proposed method. Conclusion: The comparison analysis with the other methods shows that the developed method has its great superiority in handling the MAGDM problems with hesitant fuzzy information.
The equations describing the two-phase magnetohydrodynamic steady flow and heat transfer through a horizontal channel consisting of two parallel porous walls subject to the action of uniform magnetic field, applied in the direction normal to the plane of flow/axis of rotation are written down, assuming that the magnetic Reynolds number is small. Also, it is assumed that the fluids in the two regions are incompressible, immiscible and electrically conducting, having different viscosities, thermal and electrical conductivities. Further, assumed that the transport properties of the two fluids are constant having constant boundary wall temperatures. Exact solutions for velocity and temperature distributions are obtained and are calculated numerically for different values of parameters. The results are presented graphically. We observe that the effect of increasing suction parameter is to increase the temperature in both phases. Also, as the suction parameter increases, there is a significant change in the primary velocity at the upper region but is insignificant in the lower region.