Results in Mathematics

A new non-linear variant of a quantitative extension of the uniform boundedness principle is used to show sharpness of error bounds for univariate approximation by sums of sigmoid and ReLU functions. Single hidden layer feedforward neural networks with one input node perform such operations. Errors of best approximation can be expressed using moduli of smoothness of the function to be approximated (i.e., to be learned). In this context, the quantitative extension of the uniform boundedness principle indeed allows to construct counterexamples that show approximation rates to be best possible. Approximation errors do not belong to the little-o class of given bounds. By choosing piecewise linear activation functions, the discussed problem becomes free knot spline approximation. Results of the present paper also hold for non-polynomial (and not piecewise defined) activation functions like inverse tangent. Based on Vapnik–Chervonenkis dimension, first results are shown for the logistic function.

In this paper, we consider the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line. By using a variational method and a variant fountain theorem, we obtain some new criteria to guarantee that the impulsive problem has at least one solution and infinitely many solutions respectively. We extend and improve some recent results.

We give necessary and sufficient conditions on an m-isometry to have an invertible m-isometrical extension. As particular cases, we give a useful characterization for a general m-isometrical unilateral weighted shift and for $$\ell $$ -Jordan isometries. In particular, every $$\ell $$ -Jordan isometry operator has an invertible $$(2\ell -1)$$ -isometrical extension.

We consider a boundary value problem in a half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to the normal variable. As an application, we present a theorem of uniqueness for the problem in the class of functions of power growth.

In this paper we prove two existence theorems for a generalized hemivariational inequality involving set-valued mappings of two variables in Banach spaces. In order to prove the results, we make use of the well-known Ky Fan’s Intersection Theorem and Simon’s Minimax Theorem. To illustrate the generality and applicability of the two existence theorems, several applications are provided at the end of the paper.

We study the stability of the Drygas functional equation on a restricted domain. The main tool used in the proofs is the fixed point theorem for functional spaces.

We characterize Osserman and conformally Osserman Riemannian manifolds with the local structure of a warped product. By means of this approach we analyze the twisted product structure and obtain, as a consequence, that the only Osserman manifolds which can be written as a twisted product are those of constant curvature. Pseudo-Riemannian versions of those results are also considered, showing that four-dimensional conformally Osserman warped products are locally conformally flat. The result however fails in higher dimensions where there exist conformally Osserman products which are not locally conformally flat.

Under some simple conditions on real function f defined on an interval I, the bivariable functions given by the following formulas $$\begin{aligned} A_{f}\left( x,y\right):= & {} f\left( x\right) +y-f\left( y\right) , \\ G_{f}\left( x,y\right):= & {} \frac{f\left( x\right) }{f\left( y\right) }\,y, \\ \text{ and } \quad H_{f}\left( x,y\right):= & {} \frac{xy}{f\left( x\right) +y-f\left( y\right) }, \end{aligned}$$ for all $$x,y\in I$$ , generalize, respectively, the classical weighted arithmetic, geometric and harmonic means. The invariance equations $$\begin{aligned} A_{f}\circ \left( G_{g},H_{h}\right) =A_{f}, \quad G_{g}\circ \left( A_{f},H_{h}\right) =G_{g} \quad \text{ and } \quad H_{h}\circ \left( A_{f},G_{g}\right) =H_{h}, \end{aligned}$$ where f, g,  h are the unknown functions are, in some special cases, solved. The convergence of iterates of the relevant mean-type mappings is considered. As an application the solutions of some functional equations are determined.