Biorthogonal systems for solving Volterra integral equation systems of the second kind

With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.     

Bounds for the generalized Marcum function of the second kind

The Ramanujan Journal - This paper provides elementary proofs for several types of congruences involving multipartitions and self-convolutions of the divisor function. Our computations use...     

Cycles of the second kind for uncertain pendulum-like systems with several nonlinearities

The existence of cycles of the second kind was considered for uncertain pendulum-like systems with several nonlinearities. On the basis of the Kalman–Yakubovich–Popov (KYP) lemma, linear matrix inequality (LMI) conditions guaranteeing the existence of cycles of the second kind for such nonlinear systems under parameter uncertainties are established. By virtue of these results, an interesting conclusion is reached: that the synthesis problem ensuring the existence of cycles of the second kind for such an uncertain nonlinear system can be converted into a synthesis problem for a system without uncertainties. A concrete application to a synchronous machine demonstrates the validity of the proposed approach.     

一类三次系统的极限环与分支问题

讨论了E31系统中一类属于广义Liénard型方程x。 f1(x)x。 f2(x)x。2 g(x)=0的系统x。=yy。=-x δy nx2 mxy ly2 bxy2在m,n,l同号及b≠0情况下的极限环的存在惟一性与分支问题.     

Periodic solutions of some autonomous second order Hamiltonian systems

In this paper, we study the existence of periodic solutions of some autonomous second order Hamiltonian systems $$\left\{\begin{array}{l}\ddot{u}(t)=\nabla{H(u(t)),}\\[3pt]u(0)-u(T)=\dot{u}{(0)}-\dot{u}{(T)}=0.\end{array}\right.$$ We obtain some new existence theorems by the least action principle.     

Limit cycle uniqueness for a class of planar dynamical systems

A uniqueness theorem for limit cycles of a class of plane differential systems is proved. The main result is applicable to second order O.D.E.’s with a dissipative term depending both on the position and on the velocity.     

一个包含Smarandache函数及第二类伪Smarandache函数的方程

主要研究方程Z2（n）＋1=S（n）的可解性,利用初等方法以及Smarandache函数的性质,证明了该方程有无穷多个正整数解,并获得了所有正整数解的具体表现形式．     

Interpolational systems of functions of the second kind and their topology

Mathematical Notes -     

Mixed products of multiplicative systems without discontinuities of the second kind

Mixed products of multiplicative evolution operator systems with values in a Banach ring R that satisfy the orthogonality condition are studied with the aid of their infinitesimal additive systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 332–340, March, 1990.     

The Liapunov function for the second order difference systems

The Liapunov function, whose fixed sign property together with its first difference is explicitly defined by the conditions of asymptotic stability and instability, is derived for a stationary difference system of second order. Examples are given of the use of that function in the analysis of nonlinear and nonstationary problems.     

Numerical treatment of second kind Fredholm integral equations systems on bounded intervals

In this paper the authors propose numerical methods to approximate the solutions of systems of second kind Fredholm integral equations. They prove that such methods are stable and convergent. Error estimates in weighted L^p$L^p$ norm, 1?p?+∞$1?p?+\infty$, are given and some numerical tests are shown.     

Limit cycles and chaotic invariant sets in autonomous hybrid planar systems

This paper demonstrates that complex dynamical behaviors such as limit cycles and chaotic invariant sets can exist in some simple autonomous hybrid planar systems governed by simple transition laws.     

Hybrid function method for solving Fredholm and Volterra integral equations of the second kind

Numerical solutions of Fredholm and Volterra integral equations of the second kind via hybrid functions, are proposed in this paper. Based upon some useful properties of hybrid functions, integration of the cross product, a special product matrix and a related coefficient matrix   with optimal order, are applied to solve these integral equations. The main characteristic of this technique is to convert an integral equation into an algebraic; hence, the solution procedures are either reduced or simplified accordingly. The advantages of hybrid functions are that the values of n$n$ and m$m$ are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function, for the solutions of integral equations. We propose that the available optimal values of n$n$ and m$m$ can minimize the relative errors of the numerical solutions. The high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples. The hybrid function method is superior to other piecewise constant orthogonal functions [W.F. Blyth, R.L. May, P. Widyaningsih, Volterra integral equations solved in Fredholm form using Walsh functions, Anziam J. 45 (E) (2004) C269–C282; M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200 (2007) 12–20] for these problems.     

On a relation connecting the second solution of Tchebicheff's equations of the second kind and Bessel function

Summary The object of this note is to obtain by the method of operational Calculus a relation connecting the second solution of Tchebycheff's equation of the second kind and Bessel function. Some other allied relations have also been given.

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Riassunto Lo scopo di questa nota è di ottenere col metodo del calcolo operazionale, una relazione fra la seconda soluzione dell'equazione di Tschebicheff di secondo genere, e la funzione di Bessell. Viene data inoltre qualche altra relazione.

     

Projection methods for equations of the second kind

A class of projection methods, differing from the classical projection methods, is studied for the equation y = f + Ky, where K is a compact linear operator in a Banach space E, and f?E, In these methods K is approximated by a finite-rank operator K_n, which is constructed with the aid of certain projection operators, and which satisfies K_nz = Kz for all z belonging to a chosen subspace U_n ? E. Under certain conditions, it is shown that the convergence of the approximate solution is faster than that of any classical projection method based on the subspace U_n. In an example, U_n is taken to consist of piecewise constant functions, and the projections are so chosen that the method becomes equivalent to a single iteration of a classical method, the collocation method; in this case the error (in the supremum norm) is $\text{O(}\text{1}\text{n}{}^2\text{)}$, compared with $\text{O(}\text{1}\text{n}\text{)}$ for the collocation method.     

Differentials of the second kind on mumford curves

We define ap-adic analytic Hodge decomposition for the cohomology of Mumford curves, with values in a local system. When the local system is trivial, we give a new proof of Gerritzen’s theorem, that this decomposition forms a variation of Hodge structure, in a family of Mumford curves.     

关于一类非多项式平面微分系统的极限环及分支问题

旨在讨论一类非多项式平面微分系统.通过使用Dulac准则和Bendixson准则获得极限环不存在性的充分条件,引入广义Liénard系统理论以研究极限环的存在性及稳定性,应用Hopf分岔理论证明自原点分岔出极限环的充分条件.此外,给出一个范例以验证分析和结果的有效性.     

Chebyshev polynomials of the second kind

Supported by the University of the Basque Country.