# Volume 64, № 12, 2012

### Delayed feedback makes neuronal firing statistics non-Markovian

Kravchuk K. G., Vidybida A. K.

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1587-1609

The instantaneous state of a neural network consists of both the degree of excitation of each neuron and the positions of impulses in communication lines between the neurons. In neurophysiological experiments, the neuronal firing moments are registered, but not the state of communication lines. However, future spiking moments depend substantially on the past positions of impulses in the lines. This suggests that the sequence of intervals between firing moments (interspike intervals, ISI) in the network can be non-Markovian. In this paper, we address this question for a simplest possible neural "network", namely, a single neuron with delayed feedback. The neuron receives excitatory input both from the input Poisson process and from its own output through the feedback line. We obtain exact expressions for the conditional probability density $P (t_{n+1} | t_n,... ,t_1, t_0) dt_{n+1}$ and prove that $P (t_{n+1} | t_n,... ,t_1, t_0)$ does not reduce to $P (t_{n+1} | t_n,... ,t_1)$ for any $n \geq 0$. This means that the output ISI stream cannot be represented as a Markov chain of any finite order.

### On impulsive Sturm - Liouville operators with singularity and spectral parameter in boundary conditions

Amirov R. Kh., Güldü Y., Topsakal N.

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1610-1629

We study properties and the asymptotic behavior of spectral characteristics for a class of singular Sturm-Liouville differential operators with discontinuity conditions and an eigenparameter in boundary conditions. We also determine the Weyl function for this problem and prove uniqueness theorems for a solution of the inverse problem corresponding to this function and spectral data.

### A study on tensor product surfaces in low-dimensional Euclidean spaces

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1630-1640

We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have a harmonic Gauss map.

### Three-Dimensional Matrix Superpotentials

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1641-1640

We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have a harmonic Gauss map. We present а classification of matrix superpotentials that correspond to exactly solvable systems of Schrodinger equations. Superpotentials of the following form are considered: $W_k = kQ + P \frac 1k$, where $k$ is a parameter and $P, Q$ and $R$ are Hermitian matrices that depend on a variable $x$. The list of three-dimensional matrix superpotentials is explicitly presented.

### Representations of Algebras Defined by a Multiplicative Relation and Corresponding to the Extended Dynkin Graphs $\tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$

Livins'kyi I. V., Radchenko D. V.

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1654-1675

We describe, up to unitary equivalence, all $k$-tuples $(A_1, A_2,..., A_k)$ of unitary operators such that $A^{n_i}_i = I$ for $i = \overline{1, k}$ and $A_1 A_2 ... A_k = \lambda I$, where the parameters $(n_1,... ,n_k)$ correspond to one of the extended Dynkin diagrams $\tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$, and $\lambda \in \mathbb{C}$ is a fixed root of unity.

### Injectivity Classes of the Pompeiu Transformation

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1676-1684

New conditions for the injectivity of the Pompeiu transform for integral ball means are obtained. The main results substantially improve some known uniqueness theorems for functions with vanishing integrals over balls of fixed radius.

### $S^1$-bott functions on manifolds

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1685-1698

We study $S^1$$ -Bott functions on compact smooth manifolds. In particular, we investigate $S_1$-invariant Bott functions on manifolds with circle action.

### On convolutions on configuration spaces. II. spaces of locally finite configurations

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1699-1719

We consider the convolution of probability measures on spaces of locally finite configurations (subsets of a phase space) as well as their connection with the convolution of the corresponding correlation measures and functionals. In particular,the convolution of Gibbs measures is studied. We also describe a relationship between invariant measures with respect to some operator and properties of the corresponding image of this operator on correlation functions.

### Mykola Ivanovych Shkil' (on his 80th birthday)

Korolyuk V. S., Lukovsky I. O., Perestyuk N. A., Pratsiovytyi M. V., Samoilenko A. M., Yakovets V. P.

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1720-1722

### Index of volume 64 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1723 - 1728