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1

Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity

Nursultanov M., Rozenblum G.

Opuscula Mathematica

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2017

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Vol. 37, no. 1

109--139

EN

We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.

2

Feedback stabilization of one-dimensional parabolic systems related to formations

Sano H.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2015

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Vol. 63, nr 1

295--303

EN

This paper is concerned with the problem of stabilizing one-dimensional parabolic systems related to formations by using finitedimensional controllers of a modal type. The parabolic system is described by a Sturm-Liouville operator, and the boundary condition is different from any of Dirichlet type, Neumann type, and Robin type, since it contains the time derivative of boundary values. In this paper, it is shown that the system is formulated as an evolution equation with unbounded output operator in a Hilbert space, and further that it is stabilized by using an RMF (residual mode filter)-based controller which is of finite-dimension. A numerical simulation result is also given to demonstrate the validity of the finite-dimensional controller.

3

Antilogarithms of second order in algebras with logarithms and their applications to special functions

Przeworska-Rolewicz D.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2004

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Vol. 44, nr spec.

167-191

EN

In the paper [PR5] it was shown that the so-called special functions of Mathematical Physics can be obtained by means of antilogarithms of the second order for the usual differential operator ^j. The same method applied to a right invertible operator D in a commutative Leibniz algebra with logarithms permits to determine eigenvectors of linear equations of order two in D with coefficients in the algebra X under consideration by a reduction to the generalized Sturm-Liouville operator. It seems that, in a sense, the proposed method is an answer for the question of Gian-Carlo Rota concerning a unified approach to special functions (cf. [Rl], problem 4). Section 6 of the present paper is devoted to some summations formulae expressing special functions by means of exponentials. Note that, in general, we do not need any assumption about the Hilbert structure of the algebra X.

4

A nonstandard difference Sturm-Liouville operator

Javorskij J., Ljance V.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 3

285-290

EN

After Nelson's Radically Elementary Probability Theory [1] a natural question arises: whether a hyperfinite-dimensional space is sufficiently rich to be used for the same goal as an infinite-dimensional one. Here a hyperfinite 3-diagonal matrix is investigated, which spectral properties are simular to the Naimarks's singular nonselfadjoint Sturm-Liouville differential operator on semi-axis [2, 3].