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Namig J. Guliyev

Publications and source records attributed to Namig J. Guliyev.

13 recordsLinked to original sources
arXiv2021

Approximation capability of two hidden layer feedforward neural networks with fixed weights

We algorithmically construct a two hidden layer feedforward neural network (TLFN) model with the weights fixed as the unit coordinate vectors of the $d$-dimensional Euclidean space and having $3d+2$ number of hidden neurons in total, which can approximate any continuous $d$-variable function with an arbitrary precision. This result, in particular, shows an advantage of the TLFN model over the single hidden layer feedforward neural network (SLFN) model, since SLFNs with fixed weights do not have the capability of approximating multivariate functions.

cs.NE↗
arXiv2020

On two-spectra inverse problems

We consider a two-spectra inverse problem for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter and provide a complete solution of this problem.

math.SP↗
arXiv2019

Inverse square singularities and eigenparameter dependent boundary conditions are two sides of the same coin

We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter with "a negative number of poles". More precisely, we treat in a unified manner one-dimensional Schrödinger operators with either an inverse square singularity or a boundary condition containing a rational Herglotz--Nevanlinna function of the eigenvalue parameter at each endpoint, and define Darboux-type transformations between such operators. These transformations allow one, in particular, to transfer almost any spectral result from boundary value problems with eigenparameter dependent boundary conditions to those with inverse square singularities, and vice versa.

math-ph↗
arXiv2019

Essentially isospectral transformations and their applications

We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

math.SP↗
arXiv2019

Spectral identities for Schrödinger operators

We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter. These identities can be thought of as a kind of mini version of the Gelfand--Levitan integral equation for boundary coefficients only.

math-ph↗
arXiv2018

On extensions of symmetric operators

We give an explicit description of all minimal self-adjoint extensions of a densely defined, closed symmetric operator in a Hilbert space with deficiency indices $(1, 1)$.

math.FA↗
arXiv2017

On the approximation by single hidden layer feedforward neural networks with fixed weights

Feedforward neural networks have wide applicability in various disciplines of science due to their universal approximation property. Some authors have shown that single hidden layer feedforward neural networks (SLFNs) with fixed weights still possess the universal approximation property provided that approximated functions are univariate. But this phenomenon does not lay any restrictions on the number of neurons in the hidden layer. The more this number, the more the probability of the considered network to give precise results. In this note, we constructively prove that SLFNs with the fixed weight $1$ and two neurons in the hidden layer can approximate any continuous function on a compact subset of the real line. The applicability of this result is demonstrated in various numerical examples. Finally, we show that SLFNs with fixed weights cannot approximate all continuous multivariate functions.

cs.NE↗
arXiv2016

A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function

The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can approximate an arbitrary continuous function provided that an unlimited number of neurons in a hidden layer is permitted. In this paper, we consider constructive approximation on any finite interval of $\mathbb{R}$ by neural networks with only one neuron in the hidden layer. We construct algorithmically a smooth, sigmoidal, almost monotone activation function $σ$ providing approximation to an arbitrary continuous function within any degree of accuracy. This algorithm is implemented in a computer program, which computes the value of $σ$ at any reasonable point of the real axis.

cs.NE↗
arXiv2008

Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions

Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

math.SP↗