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In the framework of voting theory, Young’s method consists in removing a minimum number of voters in order to obtain a Condorcet winner. We study here another method, consisting in removing a minimum number of candidates in order to obtain a Condorcet winner. We show that this method leads to the same winners as Copeland’s tournament solution, which selects a candidate who defeats a maximum number of other candidates in the pairwise comparison method advocated by Condorcet.

Keywords: Social choice, elections, voting theory, tournament solutions, pairwise comparison method, Condorcet winner, Young’s method, Copeland’s solution, removal of voters, removal of candidates.

In linear continuoustime systems with the control and exogenous inputs, this paper proposes a new Hinfinity tracking control algorithm.In this study, we extended the existing quadratic tracking control method to the Hinfinity tracking control method. The variational method gives the integral equation of the second kind as a necessary condition of the control law for the quadratic performance function regarding the Hinfinity tracking control problem. The Hinfinity tracking control algorithm differs from other Hinfinity tracking control algorithms in that it is derived using the integral equation of the second kind. In Theorem 3 the proposed Hinfinity tracking control algorithm is designed in conjunction with the Luenberger state observer. In Theorem 2 the Luenberger state observer, which is based on linear matrix inequalities (LMIs), is shown as an example in state estimation. In Theorem 1 it is shown that the integral equation of the second kind for the tracking control problem is transformed into the EulerLagrange equation, where the equations for the control and exogenous inputs are given.

A numerical simulation example demonstrates the feasibility of the Hinfinity tracking control algorithm presented in this research for linear continuoustime system. Here, it is noteworthy that the exogenous input is additional to the control input. The output of the system approaches the desired value asymptotically as time passes.

Keywords: Hinfinity tracking control, Control input, Exogenous input, Linear matrix inequalities, EulerLagrange equation.

Mathematics Subject Classification: 49N10, 49N70.

The object of this paper is to prove that wavelets are adapted for the study of other functional spaces (other then L2(????)) as Lp(????) (1 < p < ????), Cs(????) or Hs(????) where s depends of the regularity of the basis. To realize this object, we prove at first some lemmas of functional analysis then we characterize some functional spaces with wavelet series. Wavelet, Vaguelet, Regular lemma, Sobolev space.

MSC Code: 42C15; 44A15.

The key purpose of the present work is to examine a fractionalorder pantograph equation to ????type fractional derivative in RiemannLiouville sense. The existence of globally attractive solutions for fractionalorder pantograph equations is discussed.

Keywords: ????fractional derivative; Pantograph equations; Attractivity; Measure of noncompactness.

2010 AMS 26A33, 34K40, 34K14.

In this paper, we apply the Adomian decomposition method (ADM) for solving nonlinear fractional di£erential equations (FDEs) of Caputo sense. The existence and uniqueness of the solution are proved. The convergence of the series solution and the error analysis are discussed.

Keywords: Fractional di£erential equation; Adomian Method; Existence; Uniqueness; Error analysis.

Given that §(H) is the partially ordered set of cyclic subgroups of a nite group H. Suppose that A is the class of pgroups whose order is pn for integer n > 3. Dene a map; ???? : A ???? (0; 1] by ????(H) = §(H) H . This work in an eort to make investigations on the second minimum and maximum value of ????alongside their corresponding minimum and maximum points, applies the eulers totient function as to nding the number of cyclic subgroups of nite pgroups.

Key words and phrases: Finite p Groups, Cyclic subgroups, Dihedral subgroup, Abelian subgroups, Quaternion group, Semidihedral group.

AMS Mathematics Subject Classication 2020: Primary : 20D60. Secondary : 20D15

This paper deals with the problem of finding nonzero distinct integer solutions to the nonhomogeneous ternary sextic equation given by 3(x2 + y2) – 2xy = 972z6.

Keywords: Nonhomogeneous sextic, Ternary sextic, Integer solutions.

In this paper, we present two new statistical probability models connecting n + 1, nonnegative real parameters, p0, p1, p2, ..., pn such that 0 < pn ???? pn ???? ... p2 ???? p1 ???? 0. The rst model concerns with random variable of discrete type and the second model is its analogue for the random variable of continuous type.General formulas the mean, variance, m.g.f., the rth moment, the skewness and the kurtosis of the discrete model are obtained. In case of continuous model for n = 1, we also obtain the mean, variance and the m.g.f.

Key words: Probability density function, statistical model, moments, kurtosis, skewness and moment generating functions.