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Extremal properties of distance-based graph invariants for k -trees

Minjie Zhang, Shuchao Li (2018)

Mathematica Bohemica

Sharp bounds on some distance-based graph invariants of n -vertex k -trees are established in a unified approach, which may be viewed as the weighted Wiener index or weighted Harary index. The main techniques used in this paper are graph transformations and mathematical induction. Our results demonstrate that among k -trees with n vertices the extremal graphs with the maximal and the second maximal reciprocal sum-degree distance are coincident with graphs having the maximal and the second maximal reciprocal...

Five limit cycles for a simple cubic system.

Noel G. Lloyd, Jane M. Pearson (1997)

Publicacions Matemàtiques

We resolve the centre-focus problem for a specific class of cubic systems and determine the number of limit cycles which can bifurcate from a fine focus. We also describe the methods which we have developed to investigate these questions in general. These involve extensive use of Computer Algebra; we have chosen to use REDUCE.

Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations

Luisa Malaguti, Valentina Taddei (2005)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The paper deals with the quasi-linear ordinary differential equation ( r ( t ) ϕ ( u ' ) ) ' + g ( t , u ) = 0 with t ∈ [ 0 , ∞ ) . We treat the case when g is not necessarily monotone in its second argument and assume usual conditions on r ( t ) and ϕ ( u ) . We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano, [Acta...

Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states

David Avanessoff, Jean-Baptiste Pomet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and two controls. Some of them are known to be flat, and this implies admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about “(x,u)-flatness”...

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